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Stochastic processes and financial applications

Works Count: 101766

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This cluster of papers focuses on the theory and applications of option pricing models, including topics such as stochastic calculus, jump diffusion, volatility modeling, mean field games, term structure models, risk premia, Monte Carlo simulation, and market microstructure noise in the context of financial economics.

Keywords

Option Pricing; Stochastic Calculus; Jump Diffusion; Volatility Modeling; Mean Field Games; Term Structure Models; Risk Premia; Monte Carlo Simulation; Market Microstructure Noise; Financial Economics

Option valuation using the fast Fourier transform

1999-01-01

Peter Carr , Dilip B. Madan

In this paper the authors show how the fast Fourier transform may be used to value options when the characteristic function of the return is known analytically. In this paper the authors show how the fast Fourier transform may be used to value options when the characteristic function of the return is known analytically.

Stochastic differential equations and applications

2008-01-01

Xuerong Mao

This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and … This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists.

Financial Modelling with Jump Processes

2003-12-30

Peter Tankov

WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has … WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematic

Multidimensional Diffusion Processes

1997-01-01

Daniel W. Stroock , S. R. Srinivasa Varadhan

Stochastic controls : Hamiltonian systems and HJB equations

1999-01-01

Jiongmin Yong , Xun Yu Zhou

Introduction to Stochastic Control Theory

1970-01-01

Karl Johan Åström

Continuous Martingales and Brownian Motion

1999-01-01

Daniel Revuz , Marc Yor

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The Malliavin Calculus and Related Topics

2006-01-01

David Nualart

Numerical solution of stochastic differential equations

2016-04-22

Peter E. Kloeden , Eckhard Platen

1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of … 1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.

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Brownian Motion and Stochastic Calculus

2007-04-03

Ioannis Karatzas , Steven E. Shreve

Methods and Applications of Interval Analysis

1979-01-01

Ramon E. Moore

Finite representations Finite evaluation Finite convergence Computable sufficient conditions for existence and convergence Safe starting regions for iterative methods Applications to mathematical programming Applications to operator equations An application in … Finite representations Finite evaluation Finite convergence Computable sufficient conditions for existence and convergence Safe starting regions for iterative methods Applications to mathematical programming Applications to operator equations An application in finance Internal rates-of-return.

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Monte Carlo Methods in Financial Engineering

2003-01-01

Paul Glasserman

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The valuation of options for alternative stochastic processes

1976-01-01

John C. Cox , Stephen A. Ross

An introduction to stochastic partial differential equations

2006-11-23

John B. Walsh

Theory of Financial Decision Making.

1988-03-01

Eduardo S. Schwartz , Jonathan E. Ingersoll

Based on courses developed by the author over several years, this book provides access to a broad area of research that is not available in separate articles or books of … Based on courses developed by the author over several years, this book provides access to a broad area of research that is not available in separate articles or books of readings. Topics covered include the meaning and measurement of risk, general single-period portfolio problems, mean-variance analysis and the Capital Asset Pricing Model, the Arbitrage Pricing Theory, complete markets, multiperiod portfolio problems and the Intertemporal Capital Asset Pricing Model, the Black-Scholes option pricing model and contingent claims analysis, 'risk-neutral' pricing with Martingales, Modigliani-Miller and the capital structure of the firm, interest rates and the term structure, and others.

Stochastic Integration and Differential Equations

2005-01-01

Philip Protter

Pricing Interest-Rate-Derivative Securities

1990-10-01

John Hull , Alan White

This article shows that the one-state-variable interest-rate models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985b) can be extended so that they are consistent with both the current term … This article shows that the one-state-variable interest-rate models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985b) can be extended so that they are consistent with both the current term structure of interest rates and either the current volatilities of all spot interest rates or the current volatilities of all forward interest rates. The extended Vasicek model is shown to be very tractable analytically. The article compares option prices obtained using the extended Vasicek model with those obtained using a number of other models.

Weak Convergence and Empirical Processes

1996-01-01

Aad van der Vaart , Jon A. Wellner

An analytic derivation of the cost of deposit insurance and loan guarantees An application of modern option pricing theory

1977-06-01

Robert C. Merton

Mean field games

2007-03-01

Jean‐Michel Lasry , Pierre-Louis Lions

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Martingales and stochastic integrals in the theory of continuous trading

1981-08-01

J. Michael Harrison , Stanley R. Pliska

Methods of Mathematical Finance

1998-01-01

Ioannis Karatzas , Steven E. Shreve

Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model

1987-03-01

Robert F. Engle , David M. Lilien , Russell P. Robins

The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engle's ARCH model is extended to allow the conditional variance … The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engle's ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M. Estimation and infer ence procedures are proposed, and the model is applied to three interest rate data sets. In most cases the ARCH process and the time varying risk premium are highly significant. A collection of LM diagnostic tests reveals the robustness of the model to various specification changes such as alternative volatility or ARCH measures, regime changes, and interest rate formulations. The model explains and interprets the recent econometric failures of the expectations hypothesis of the term structure. Copyright 1987 by The Econometric Society.

Continuous-Time Finance.

1991-09-01

Stephen A. Ross , Robert C. Merton

Section I: Introductin to Finance and the Mathematics of Continuous-Time Models 1 Modern Finance 2 Introduction to Portfolio Selection and Capital Market Theory: Static Analysis 3 On the Mathematics and … Section I: Introductin to Finance and the Mathematics of Continuous-Time Models 1 Modern Finance 2 Introduction to Portfolio Selection and Capital Market Theory: Static Analysis 3 On the Mathematics and Economic Assumptions of Continuous-Time Financial Models Section II: Optimum Consumption and Portfolio Selection in Continuous-Time Models 4. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case 5. Optimum Consumption and Portfolio Rules in a Continuous-Time Model 6. Further Developments in theory of Optimal Consumption and Portfolio Selection Section III: Warrant and Option Pricing Theory 7. A Complete Model of Warrant Pricing the Maximizes Utility 8. Theory of Rational Option Pricing 9. Option Pricing when Underlying Stock Returns are Discontinuous 10. Further Developments in Option Pricing Theory Section IV: Contingent-Claims Analysis in the Theory of Corporate Finance and Financial Intermediation 11. A Dynamic General Equilibrium Model of the Asset Market and its Application to the Pricing of the Capital Structure of the Firm 12. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates 13. On the Pricing of Contingent Claims and the Modigliani-Miller Theorem 14. Contingent Claims Analysis in the Theory of Corporate Finance and Financial Intermediation Section V: An Intertemporal-Equilibrium Theory of Finance 15. An Intertemporal Capital Asset Pricing Model 16. A General Equilibrium Theory of Finance in Continuous-Time Section VI: Applications of the Continuous-Time Model to Selected Issues in Public Finance 17. An Asymptotic Theory of Growth Under Uncertainty 18. On Consumption-Indexed Public Pension Plans 19. An Analytic Derivation of the Cost of Loan Guarantees and Deposit Insurance 20. On the Cost of Deposit Insurance when there are Surveillance Costs

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

1993-04-01

Steven L. Heston

Journal Article A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options Get access Steven L. Heston Steven L. Heston Yale University Address correspondence to … Journal Article A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options Get access Steven L. Heston Steven L. Heston Yale University Address correspondence to Steven L. Heston, Yale School of Organization and Management, 135 Prospect Street, New Haven, CT06511. Search for other works by this author on: Oxford Academic Google Scholar The Review of Financial Studies, Volume 6, Issue 2, April 1993, Pages 327–343, https://doi.org/10.1093/rfs/6.2.327 Published: 02 April 2015

A YIELD‐FACTOR MODEL OF INTEREST RATES

1996-10-01

Darrell Duffie , Rui Kan

This paper presents a consistent and arbitrage‐free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process … This paper presents a consistent and arbitrage‐free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process with “stochastic volatility.” the yield of any zero‐coupon bond is taken to be a maturity‐dependent affine combination of the selected “basis” set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as well as numerical techniques for calculating the prices of term‐structure derivative prices. the case of jump diffusions is also considered.

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An equilibrium characterization of the term structure

1977-11-01

Oldrich A Vasicek

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Adapted solution of a backward stochastic differential equation

1990-01-01

Étienne Pardoux , Shigē Péng

Limit Theorems for Stochastic Processes.

1988-12-01

Ditlev Monrad , Jean Jacod , Albert N. Shiryaev

I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems and Changes of Measures.- IV. Hellinger Processes, … I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems and Changes of Measures.- IV. Hellinger Processes, Absolute Continuity and Singularity of Measures.- V. Contiguity, Entire Separation, Convergence in Variation.- VI. Skorokhod Topology and Convergence of Processes.- VII. Convergence of Processes with Independent Increments.- VIII. Convergence to a Process with Independent Increments.- IX. Convergence to a Semimartingale.- X. Limit Theorems, Density Processes and Contiguity.- Bibliographical Comments.- References.- Index of Symbols.- Index of Terminology.- Index of Topics.- Index of Conditions for Limit Theorems.

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Dynamic Asset Pricing Theory.

1993-12-01

Jonathan E. Ingersoll , Darrell Duffie

Dynamic Asset Pricing Theory is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multiperiod settings under uncertainty. The asset pricing results … Dynamic Asset Pricing Theory is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multiperiod settings under uncertainty. The asset pricing results are based on the three increasingly restrictive assumptions: absence of arbitrage, single-agent optimaltiy, and equilibrium. These results are unified with two key concepts, state prices and martingales. Technicalities are given relatively little emphasis so as to draw connections between these concepts and to make plain the similarities between discrete and continuous-time models. For simplicity, all continuous-time models are based on Brownian motion. Applications include term structure models, derivative valuation and hedging methods, and dynamic programming algorithms for portfolio choice and optimal exercise of American options. Numerical methods covered include Monte Carlo simulation and finite-difference solvers for partial differential equations. Each chapter provides extensive problem exercises and notes to the literature. This second edition is substantially longer, while still retaining the consciseness for which the first edition was praised. All chapters from the first edition have been revised. Two new chapters have been added on term structure modeling and on derivative securities. References have been updated throughout. With this new edition, Dynamic Asset Pricing Theory remains the definitive textbook in the field.

ON THE PRICING OF CORPORATE DEBT: THE RISK STRUCTURE OF INTEREST RATES*

1974-05-01

Robert C. Merton

The value of a particular issue of corporate debt depends essentially on three items: (1) the required rate of return on riskless (in terms of default) debt (e.g., government bonds … The value of a particular issue of corporate debt depends essentially on three items: (1) the required rate of return on riskless (in terms of default) debt (e.g., government bonds or very high grade corporate bonds); (2) the various provisions and restrictions contained in the indenture (e.g., maturity date, coupon rate, call terms, seniority in the event of default, sinking fund, etc.); (3) the probability that the firm will be unable to satisfy some or all of the indenture requirements (i.e., the probability of default). While a number of theories and empirical studies has been published on the term structure of interest rates (item 1), there has been no systematic development of a theory for pricing bonds when there is a significant probability of default. The purpose of this paper is to present such a theory which might be called a theory of the risk structure of interest rates. The use of the term “risk” is restricted to the possible gains or losses to bondholders as a result of (unanticipated) changes in the probability of default and does not include the gains or losses inherent to all bonds caused by (unanticipated) changes in interest rates in general. Throughout most of the analysis, a given term structure is assumed and hence, the price differentials among bonds will be solely caused by differences in the probability of default. In a seminal paper, Black and Scholes 1 present a complete general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of “observable” variables. Therefore, the model is subject to direct empirical tests which they 2 performed with some success. Merton 5 clarified and extended the Black-Scholes model. While options are highly specialized and relatively unimportant financial instruments, both Black and Scholes 1 and Merton 5, 6 recognized that the same basic approach could be applied in developing a pricing theory for corporate liabilities in general. In Section II of the paper, the basic equation for the pricing of financial instruments is developed along Black-Scholes lines. In Section III, the model is applied to the simplest form of corporate debt, the discount bond where no coupon payments are made, and a formula for computing the risk structure of interest rates is presented. In Section IV, comparative statics are used to develop graphs of the risk structure, and the question of whether the term premium is an adequate measure of the risk of a bond is answered. In Section V, the validity in the presence of bankruptcy of the famous Modigliani-Miller theorem 7 is proven, and the required return on debt as a function of the debt-to-equity ratio is deduced. In Section VI, the analysis is extended to include coupon and callable bonds. The dynamics for the value of the firm, V, through time can be described by a diffusion-type stochastic process with stochastic differential equation α is the instantaneous expected rate of return on the firm per unit time, C is the total dollar payouts by the firm per unit time to either its shareholders or liabilities-holders (e.g., dividends or interest payments) if positive, and it is the net dollars received by the firm from new financing if negative; σ 2 is the instantaneous variance of the return on the firm per unit time; dz is a standard Gauss-Wiener process. Many of these assumptions are not necessary for the model to obtain but are chosen for expositional convenience. In particular, the “perfect market” assumptions (A.1-A.4) can be substantially weakened. A.6 is actually proved as part of the analysis and A.7 is chosen so as to clearly distinguish risk structure from term structure effects on pricing. A.5 and A.8 are the critical assumptions. Basically, A.5 requires that the market for these securities is open for trading most of time. A.8 requires that price movements are continuous and that the (unanticipated) returns on the securities be serially independent which is consistent with the “efficient markets hypothesis” of Fama 3 and Samuelson 9.11 Of course, this assumption does not rule out serial dependence in the earnings of the firm. See Samuelson 10 for a discussion. In closing this section, it is important to note which variables and parameters appear in (7) (and hence, affect the value of the security) and which do not. In addition to the value of the firm and time, F depends on the interest rate, the volatility of the firm's value (or its business risk) as measured by the variance, the payout policy of the firm, and the promised payout policy to the holders of the security. However, F does not depend on the expected rate of return on the firm nor on the riskȁpreferences of investors nor on the characteristics of other assets available to investors beyond the three mentioned. Thus, two investors with quite different utility functions and different expectations for the company's future but who agree on the volatility of the firm's value will for a given interest rate and current firm value, agree on the value of the particular security, F. Also all the parameters and variables except the variance are directly observable and the variance can be reasonably estimated from time series data. As a specific application of the formulation of the previous section, we examine the simplest case of corporate debt pricing. Suppose the corporation has two classes of claims: (1) a single, homogenous class of debt and (2) the residual claim, equity. Suppose further that the indenture of the bond issue contains the following provisions and restrictions: (1) the firm promises to pay a total of B dollars to the bondholders on the specified calendar date T; (2) in the event this payment is not met, the bondholders immediately take over the company (and the shareholders receive nothing); (3) the firm cannot issue any new senior (or of equivalent rank) claims on the firm nor can it pay cash dividends or do share repurchase prior to the maturity of the debt. For a given maturity, the risk premium is a function of only two variables: (1) the variance (or volatility) of the firm's operations, σ 2 and (2) the ratio of the present value (at the riskless rate) of the promised payment to the current value of the firm, d. Because d is the debt-to-firm value ratio where debt is valued at the riskless rate, it is a biased upward estimate of the actual (market-value) debt-to-firm value ratio. Since Merton 5 has solved the option pricing problem when the term structure is not “flat” and is stochastic, (by again using the isomorphic correspondence between options and levered equity) we could deduce the risk structure with a stochastic term structure. The formulae (13) and (14) would be the same in this case except that we would replace “ exp [ − r τ ]” by the price of a riskless discount bond which pays one dollar at time τ in the future and “ σ 2 τ ” by a generalized variance term defined in 5. In the derivation of the fundamental equation for pricing of corporate liabilities, (7), it was assumed that the Modigliani-Miller theorem held so that the value of the firm could be treated as exogeneous to the analysis. If, for example, due to bankruptcy costs or corporate taxes, the M-M theorem does not obtain and the value of the firm does depend on the debt-equity ratio, then the formal analysis of the paper is still valid. However, the linear property of (7) would be lost, and instead, a non-linear, simultaneous solution, F = F [ V ( F ) , τ ] , would be required. Fortunately, in the absence of these imperfections, the formal hedging analysis used in Section II to deduce (7), simultaneously, stands as a proof of the M-M theorem even in the presence of bankruptcy. To see this, imagine that there are two firms identical with respect to their investment decisions, but one firm issues debt and the other does not. The investor can “create” a security with a payoff structure identical to the risky bond by following a portfolio strategy of mixing the equity of the unlevered firm with holdings of riskless debt. The correct portfolio strategy is to hold ( F v V ) dollars of the equity and ( F – F v V ) dollars of riskless bonds where V is the value of the unlevered firm, and F and F v are determined by the solution of (7). Since the value of the “manufactured” risky debt is always F, the debt issued by the other firm can never sell for more than F. In a similar fashion, one could create levered equity by a portfolio strategy of holding ( f v V ) dollars of the unlevered equity and ( f – f v V ) dollars of borrowing on margin which would have a payoff structure identical to the equity issued by the levering firm. Hence, the value of the levered firm's equity can never sell for more than f. But, by construction, f + F = V, the value of the unlevered firm. Therefore, the value of the levered firm can be no larger than the unlevered firm, and it cannot be less. Note, unlike in the analysis by Stiglitz 11, we did not require a specialized theory of capital market equilibrium (e.g., the Arrow-Debreu model or the capital asset pricing model) to prove the theorem when bankruptcy is possible. Contrary to what many might believe, the relative riskiness of the debt can decline as either the business risk of the firm or the time until maturity increases. Inspection of (33) shows that this is the case if d > 1 (i.e., the present value of the promised payment is less than the current value of the firm). To see why this result is not unreasonable, consider the following: for small T (i.e., σ 2 or τ: small), the chances that the debt will become equity through default are large, and this will be reflected in the risk characteristics of the debt through a large g. By increasing T (through an increase in σ 2 or τ), the chances are better that the firm value will increase enough to meet the promised payment. It is also true that the chances that the firm value will be lower are increased. However, remember that g is a measure of how much the risky debt behaves like equity versus debt. Since for g large, the debt is already more aptly described by equity than riskless debt. (E.g., for d > 1 , g > 1 2 and the “replicating” portfolio will contain more than half equity.) Thus, the increased probability of meeting the promised payment dominates, and g declines. For d < 1 , g will be less than a half, and the argument goes just the opposite way. In the “watershed” case when d = 1 , g equals a half; the “replicating” portfolio is exactly half equity and half riskless debt, and the two effects cancel leaving g unchanged. In closing this section, we examine a classical problem in corporate finance: given a fixed investment decision, how does the required return on debt and equity change, as alternative debt-equity mixes are chosen? Because the investment decision is assumed fixed, and the Modigliani-Miller theorem obtains, V, σ 2 , and α (the required expected return on the firm) are fixed. For simplicity, suppose that the maturity of the debt, τ, is fixed, and the promised payment at maturity per bond is $1. Then, the debt-equity mix is determined by choosing the number of bonds to be issued. Since in our previous analysis, F is the value of the whole debt issue and B is the total promised payment for the whole issue, B will be the number of bonds (promising $1 at maturity) in the current analysis, and F /B will be the price of one bond. In the usual analysis of (default-free) bonds in term structure studies, the derivation of a pricing relationship for pure discount bonds for every maturity would be sufficient because the value of a default-free coupon bond can be written as the sum of discount bonds' values weighted by the size of the coupon payment at each maturity. Unfortunately, no such simple formula exists for risky coupon bonds. The reason for this is that if the firm defaults on a coupon payment, then all subsequent coupon payments (and payments of principal) are also defaulted on. Thus, the default on one of the “mini” bonds associated with a given maturity is not independent of the event of default on the “mini” bond associated with a later maturity. However, the apparatus developed in the previous sections is sufficient to solve the coupon problem. Moreover, even for those cases where closed-form solutions cannot be found, powerful numerical integration techniques have been developed for solving equations like (7) or (41). Hence, computation and empirical testing of these pricing theories is entirely feasible. Note that in deducing (40), it was assumed that coupon payments were made uniformly and continuously. In fact, coupon payments are usually only made semi-annually or annually in discrete lumps. However, it is a simple matter to take this into account by replacing “ C ¯ ” in (40) by “ Σ i C ¯ i δ ( τ − τ i ) ” where δ( ) is the dirac delta function and τ i is the length of time until maturity when the i th coupon payment of C ¯ i dollars is made. We have developed a method for pricing corporate liabilities which is grounded in solid economic analysis, requires inputs which are on the whole observable; can be used to price almost any type of financial instrument. The method was applied to risky discount bonds to deduce a risk structure of interest rates. The Modigliani-Miller theorem was shown to obtain in the presence of bankruptcy provided that there are no differential tax benefits to corporations or transactions costs. The analysis was extended to include callable, coupon bonds.

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Valuing American Options by Simulation: A Simple Least-Squares Approach

2001-01-01

Francis A. Longstaff , Eduardo S. Schwartz

This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to … This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.

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Empirical Performance of Alternative Option Pricing Models

1997-12-01

Gurdip Bakshi , Charles Cao , Zhiwu Chen

ABSTRACT Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and … ABSTRACT Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time‐series data, (2) out‐of‐sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance.

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Stochastic Differential Equations

2003-01-01

Bernt Øksendal

An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

2001-01-01

Desmond J. Higham

A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to … A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed. The article is built around $10$ MATLAB programs, and the topics covered include stochastic integration, the Euler--Maruyama method, Milstein's method, strong and weak convergence, linear stability, and the stochastic chain rule.

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Option pricing: A simplified approach

1979-09-01

John C. Cox , Stephen A. Ross , Mark Rubinstein

The Pricing of Options on Assets with Stochastic Volatilities

1987-06-01

John Hull , Alan White

ABSTRACT One option‐pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The … ABSTRACT One option‐pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black‐Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity.

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Backward Stochastic Differential Equations in Finance

1997-01-01

Nicole El Karoui , Shigē Péng , M.C. Quenez

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory … We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

Stochastic Differential Equations: An Introduction with Applications.

1987-09-01

Saul Jacka , Bernt Øksendal

Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary … Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation

1992-01-01

David Heath , Robert A. Jarrow , A. J. Morton

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an … This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

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A Theory of the Term Structure of Interest Rates

1985-03-01

John C. Cox , Jonathan E. Ingersoll , Stephen A. Ross

This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the … This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually

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Transform Analysis and Asset Pricing for Affine Jump-diffusions

2000-11-01

Darrell Duffie , Jun Pan , Kenneth J. Singleton

In the setting of 'affine' jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow … In the setting of 'affine' jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option 'smirks' of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options

1996-01-01

David S. Bates

An efficient method is developed for pricing American options on stochastic volatility/jump-diffusion processes under systematic jump and volatility risk. The parameters implicit in deutsche mark (DM) options of the model … An efficient method is developed for pricing American options on stochastic volatility/jump-diffusion processes under systematic jump and volatility risk. The parameters implicit in deutsche mark (DM) options of the model and various submodels are estimated over the period 1984 to 1991 via nonlinear generalized least squares, and are tested for consistency with $/DM futures prices and the implicit volatility sample path. The stochastic volatility submodel cannot explain the “volatility smile” evidence of implicit excess kurtosis, except under parameters implausible given the time series properties of implicit volatilities. Jump fears can explain the smile, and are consistent with one 8 percent DM appreciation “outlier” observed over the period 1984 to 1991.

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Continuous Martingales and Brownian Motion

1991-01-01

Daniel Revuz , Marc Yor

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Numerical Solution of Stochastic Differential Equations

1992-01-01

Peter E. Kloeden , Eckhard Platen

Brownian Motion and Stochastic Calculus

1998-01-01

Ioannis Karatzas , Steven E. Shreve

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Limit Theorems for Stochastic Processes

2003-01-01

Jean Jacod , Albert N. Shiryaev

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Limit L of Real Functions when x→a (or x→a− or x→a+)

2025-06-25

Nicolas A. Pereyra | CRC Press eBooks

Well-posedness and strong convergence analysis of stochastic space-time fractional wave problems driven by fractional Brownian sheet

2025-06-25

Yi Yang , Jin Huang , Hu Li | Computers & Mathematics with Applications

Analysis of threshold Ornstein–Uhlenbeck process with piecewise linear drift and piecewise constant diffusion

2025-06-24

Dingwen Zhang , Zhuowei Sun | Chaos Solitons & Fractals

Yield Curve Modeling: Applicability of the Traditional Factor Models for Sri Lanka Government Bonds

2025-06-24

K. P. N. S. Dayarathne , Uthayasanker Thayasivam | SN Computer Science

Estimation bias in the Ornstein-Uhlenbeck process with flow data

2025-06-24

Milena Hoyos | Econometric Reviews

Applying the Generalized Laplace Residual Power Series Method to the Time-Fractional Multi-Asset Black-Scholes European Option Pricing Model

2025-06-24

Nitithorn Sukwong , Wannika Sawangtong , Thanin Sitthiwirattham +1 more | Contemporary Mathematics

It is a well-known fact that the Black-Scholes model is used in order to analyse the behavior of the financial market with regard to the pricing of options. An explicit … It is a well-known fact that the Black-Scholes model is used in order to analyse the behavior of the financial market with regard to the pricing of options. An explicit analytical solution to the Black-Scholes equation is known as the Black-Scholes formula. The Black-Scholes equation is modified by mathematicians in the form of fractional Black-Scholes equations. Unfortunately, there are certain cases in which the fractional-order Black-Scholes equation does nothave a closed-form formula. This article demonstrates the method for deriving analytical solutions to the fractional multi-asset Black-Scholes equation with the left-side Caputo-type Katugampola fractional derivative. The -Laplace residual power series approach, which blends the residual power series method with the-Laplace transform, is the methodology used to find analytical solutions to this equation. The suggested method is remarkably precise and efficient for the fractional multi-asset Black-Scholes equation, according to numerical analyses. This confirms that the-Laplace residual power series method is among the most effective techniques for finding analytical solutions to fractional-order differential equations.

Binomial Tree Method for American Option Pricing: Discrete Cosine Transform Approach

2025-06-24

Yoshifumi Muroi , Shintaro Suda | Computational Economics

Abstract This study introduces a novel methodology for pricing options with early exercise features, specifically American and Bermudan options, using the discrete cosine transform (DCT) within the binomial tree model … Abstract This study introduces a novel methodology for pricing options with early exercise features, specifically American and Bermudan options, using the discrete cosine transform (DCT) within the binomial tree model framework. The research begins by addressing the limitations of traditional binomial tree methods when applied to complex models, such as Lévy processes, which have been perceived as inefficient for option pricing. We outline a systematic approach that incorporates the DCT to enhance computational efficiency and accuracy in option pricing. The procedure involves first establishing the binomial tree model, followed by the integration of the DCT to estimate option prices rapidly. We demonstrate the effectiveness of this method by applying it to various models, including the classic Black-Scholes model, as well as jump-diffusion and exponential Lévy process models, such as the exponential CGMY and exponential normal inverse Gaussian models. The key highlights of our research include the significant improvement in pricing speed and precision, as well as the versatility of the DCT in adapting to both standard and complex financial models. This study not only expands the applicability of the binomial tree model but also paves the way for future research in option pricing methodologies.

Parameters estimation of the Rayleigh diffusion process: Inference aspects and application to real data

2025-06-23

Yassine Chakroune , Ahmed Nafidi , Abdenbi El Azri | Monte Carlo Methods and Applications

Abstract In this paper, a stochastic model related to the Rayleigh density function curve is proposed. First, we determined the explicit form of the process by solving the stochastic differential … Abstract In this paper, a stochastic model related to the Rayleigh density function curve is proposed. First, we determined the explicit form of the process by solving the stochastic differential equation by applying the Itô method. Then we determined the probabilistic characteristics such as the density function, the mean and the conditional mean functions. Unlike other processes in the same context, this one allowed us to find the explicit form of the estimators of these parameters by solving the maximum likelihood equations system. In addition, an estimation study on simulated data is carried out in order to validate the efficiency of the estimators proposed by the maximum likelihood methodology. Finally, an application to real data is presented.

Valuation of Euro-Convertible Bonds in a Markov-Modulated, Cox–Ingersoll–Ross Economy

2025-06-23

Yu-Min Lian , Jun-Home Chen , Szu‐Lang Liao | Mathematics

This study investigates the valuation of Euro-convertible bonds (ECBs) using a novel Markov-modulated cojump-diffusion (MMCJD) model, which effectively captures the dynamics of stochastic volatility and simultaneous jumps (cojumps) in both … This study investigates the valuation of Euro-convertible bonds (ECBs) using a novel Markov-modulated cojump-diffusion (MMCJD) model, which effectively captures the dynamics of stochastic volatility and simultaneous jumps (cojumps) in both the underlying stock prices and foreign exchange (FX) rates. Furthermore, we introduce a Markov-modulated Cox–Ingersoll–Ross (MMCIR) framework to accurately model domestic and foreign instantaneous interest rates within a regime-switching environment. To manage computational complexity, the least-squares Monte Carlo (LSMC) approach is employed for estimating ECB values. Numerical analyses demonstrate that explicitly incorporating stochastic volatilities and cojumps significantly enhances the realism of ECB pricing, underscoring the novelty and contribution of our integrated modeling approach.

Pricing Convertible Bonds Based on GAN and Transformer

2025-06-23

Li Liang , Zhiqian Hu , Hao Guo +3 more | Computational Economics

On randomization of affine diffusion processes with application to pricing of options on VIX and S&P 500

2025-06-23

Lech A. Grzelak | Applied Mathematics and Computation

None

2025-06-23

Michel Benaı̈m , Claude Lobry , Tewfik Sari +1 more | Annales de la faculté des sciences de Toulouse Mathématiques

In a recent paper [6], P. Carmona gives an asymptotic formula for the top Lyapunov exponent of a linear <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-periodic cooperative differential equation, in the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math>. This … In a recent paper [6], P. Carmona gives an asymptotic formula for the top Lyapunov exponent of a linear <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-periodic cooperative differential equation, in the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math>. This short note discusses and extends this result. The assumption that the system is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-periodic is replaced by the more general assumption that it is driven by a continuous time uniquely ergodic Feller Markov process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub><mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow> </mml:msub></mml:math>. When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ω</mml:mi> <mml:mi>t</mml:mi> </mml:msub></mml:math> is replaced by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>ω</mml:mi> <mml:mi>t</mml:mi> <mml:mi>T</mml:mi> </mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>ω</mml:mi> <mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>T</mml:mi></mml:mrow> </mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math> asymptotic formulas for the top Lyapunov exponent in the fast (i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math>) and slow (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>) regimes are given.

Path-Dependent Controlled Mean-Field Coupled Forward-Backward SDEs: The Associated Stochastic Maximum Principle

2025-06-23

Rainer Buckdahn , Juan Li , Junsong Li +1 more | SIAM Journal on Control and Optimization

A stochastic reconstruction theorem on rectangular increments with an application to a mixed hyperbolic SPDE

2025-06-22

Carlo Bellingeri , Hannes Kern | Stochastic Partial Differential Equations Analysis and Computations

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Stochastic equations with singular drift driven by fractional Brownian motion

2025-06-21

Oleg Butkovsky , Khoa Lê , Leonid Mytnik | Probability and Mathematical Physics

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Mean-splitting partial differential equations: A solution approach for the Cox–Ingersoll–Ross stochastic process with applications to data forecast

2025-06-21

Patrick Chidzalo , Phillip O. Ngare , Joseph Mung’atu +2 more | Research in Mathematics

A backward stochastic differential equation driven by fractional Brownian motion with the Osgood condition

2025-06-20

Yaya Sagna , Sadibou Aïdara , Djibril Ndiaye | Stochastics

On Weak Convergence of Stochastic Wave Equation with Colored Noise on $$\mathbb {R}$$

2025-06-20

Wenxuan Tao | Journal of Theoretical Probability

Abstract In this paper, we study the following stochastic wave equation on the real line: $$\partial _t^2 u_{\alpha }=\partial _x^2 u_{\alpha }+b\left( u_\alpha \right) +\sigma \left( u_\alpha \right) \eta _{\alpha … Abstract In this paper, we study the following stochastic wave equation on the real line: $$\partial _t^2 u_{\alpha }=\partial _x^2 u_{\alpha }+b\left( u_\alpha \right) +\sigma \left( u_\alpha \right) \eta _{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mfenced> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> </mml:math> . The noise $$\eta _\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> is white in time and colored in space with a covariance structure $$\mathbb {E}[\eta _\alpha (t,x)\eta _\alpha (s,y)]=\delta (t-s)f_\alpha (x-y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>-</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where $$f_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> is continuous with respect to $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $$u_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> , in terms of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of $$f_{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> to which our theorem applies.

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Robust optimal control for hidden Markovian-switching jump diffusions: the finite-horizon payoff

2025-06-20

Beatris Adriana Escobedo-Trujillo , Javier Garrido , Francisco Alejandro Alaffita-Hernández | International Journal of Control

StochasticDominance.jl: A Julia Package for Higher Order Stochastic Dominance

2025-06-20

Rajmadan Lakshmanan , Alois Pichler | The Journal of Open Source Software

Accelerating Level-Value Adjustment for the Polyak Stepsize

2025-06-19

Anbang Liu , Mikhail A. Bragin , Xi Chen +1 more | Journal of Optimization Theory and Applications

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Non-Gaussianity of invariant measures to SPDEs in Da Prato–Debussche regime

2025-06-19

Ajay Chandra , Ilya Chevyrev | Stochastic Partial Differential Equations Analysis and Computations

Abstract We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato–Debussche regime. The approach is essentially … Abstract We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato–Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the $$\Phi ^4_\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Φ</mml:mi> <mml:mi>δ</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> measures in dimensions $$\delta <\frac{14}{5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mn>14</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.

Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes.

2025-06-19

Letizia Angeli , Dan Crisan , Michela Ottobre | ESAIM. Mathematical modelling and numerical analysis

We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion … We prove a general criterion providing sufficient conditions under which a time-discretiziation of a given Stochastic Differential Equation (SDE) is a uniform in time approximation of the SDE. The criterion is also, to a certain extent, discussed in the paper, necessary. Using such a criterion we then analyse the convergence properties of numerical methods for solutions of SDEs; we consider Explicit and Implicit Euler, split-step and (truncated) tamed Euler methods. In particular, we show that, under mild conditions on the coefficients of the SDE (locally Lipschitz and strictly monotone), these methods produce approximations of the law of the solution of the SDE that converge uniformly in time. The bounds we provide are non-asymptotic. The theoretical results are verified by numerical examples.

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Correction terms for parametric white noise processes. Gaussian, Poissonian and α-stable

2025-06-19

Rossella Laudani , Mario Di Paola , G. Falsone | Research Square (Research Square)

<title>Abstract</title> In this work, the important rules of the stochastic differential calculus, when the external and parametric (or multiplicative) excitations are modeled as Gaussian white noise processes, are generalized to … <title>Abstract</title> In this work, the important rules of the stochastic differential calculus, when the external and parametric (or multiplicative) excitations are modeled as Gaussian white noise processes, are generalized to the case of non-Gaussian white noise excitations. This extension began some years ago, when the classical Gaussian stochastic differential calculus was generalized to thePoisson delta-correlated actions. In the present work, the extension has been completed considering the more general class of white noise excitations, which are α-stable Lévy white noises. It is shown that, in the case of parametric excitations, this extension also regards the correction terms necessary for using the Ito-type integration. These results have been possible thanks to the evidence that all the white processes here considered belong to a class of processes (the motion ones), whose formal derivative is just an element of the white noise class.

A Fast and Accurate Numerical Approach for Pricing American-Style Power Options

2025-06-19

Tsvetelin Zaevski , Hristo Sariev , Mladen Savov | Mathematics

In this paper, we present a fast and accurate numerical approach applied to specific American-style derivatives, namely American power call and put options, whose main feature is that the underlying … In this paper, we present a fast and accurate numerical approach applied to specific American-style derivatives, namely American power call and put options, whose main feature is that the underlying asset is raised to a power. The study is set in the Black–Scholes framework, and we consider continuously paying dividends assets and arbitrary positive values for the power. It is important to note that although a log-normal process raised to a power is again log-normal, the resulting change in variables may lead to a negative dividend rate, and this case remains largely understudied in the literature. We derive closed-form formulas for the perpetual options’ optimal boundaries and for the fair prices. For finite maturities, we approximate the optimal boundary using some first-hitting properties of the Brownian motion. As a consequence, we obtain the option price quickly and with relatively high accuracy—the error is at the third decimal position. We further provide a comprehensive analysis of the impact of the parameters on the options’ value, and discuss ordinary European and American capped options. Various numerical examples are provided.

Stability and Error Estimates of Operator Splitting Methods on a Variable Space-Time Grid for American Options with Jumps

2025-06-18

Pradeep Kumar Sahu , Kuldip Singh Patel , Pawan Kumar Mishra | Computational Economics

Non-Uniqueness of Best-Of Option Prices Under Basket Calibration

2025-06-18

Mohammed Ahnouch , Lotfi Elaachak , Abderrahim Ghadi | Risks

This paper demonstrates that perfectly calibrating a multi-asset model to observed market prices of all basket call options is insufficient to uniquely determine the price of a best-of call option. … This paper demonstrates that perfectly calibrating a multi-asset model to observed market prices of all basket call options is insufficient to uniquely determine the price of a best-of call option. Previous research on multi-asset option pricing has primarily focused on complete market settings or assumed specific parametric models, leaving fundamental questions about model risk and pricing uniqueness in incomplete markets inadequately addressed. This limitation has critical practical implications: derivatives practitioners who hedge best-of options using basket-equivalent instruments face fundamental distributional uncertainty that compounds the well-recognized non-linearity challenges. We establish this non-uniqueness using convex analysis (extreme ray characterization demonstrating geometric incompatibility between payoff structures), measure theory (explicit construction of distinct equivalent probability measures), and geometric analysis (payoff structure comparison). Specifically, we prove that the set of equivalent probability measures consistent with observed basket prices contains distinct measures yielding different best-of option prices, with explicit no-arbitrage bounds [aK,bK] quantifying this uncertainty. Our theoretical contribution provides the first rigorous mathematical foundation for several empirically observed market phenomena: wide bid-ask spreads on extremal options, practitioners’ preference for over-hedging strategies, and substantial model reserves for exotic derivatives. We demonstrate through concrete examples that substantial model risk persists even with perfect basket calibration and equivalent measure constraints. For risk-neutral pricing applications, equivalent martingale measure constraints can be imposed using optimal transport theory, though this requires additional mathematical complexity via Schrödinger bridge techniques while preserving our fundamental non-uniqueness results. The findings establish that additional market instruments beyond basket options are mathematically necessary for robust exotic derivative pricing.

Long time behavior of optimal liquidation problems with semimartingale strategies and external flows

2025-06-18

Xin Cheng , Guanxing Fu , Xiaonyu Xia | Mathematics and Financial Economics

Robust asset-liability management games in a stochastic market with stochastic cash flows under HARA utility

2025-06-18

Ning Wang , Yumo Zhang | Insurance Mathematics and Economics

Comparison of numerical solutions of option pricing under two mixed Black–Scholes models

2025-06-17

Hossein Sahebi Fard , Elham Dastranj | International Journal of Financial Engineering

Enhancing the Black–Scholes model with other financial models is a widely used approach to improve its accuracy and adaptability to real market conditions. This enhancement is typically achieved by replacing … Enhancing the Black–Scholes model with other financial models is a widely used approach to improve its accuracy and adaptability to real market conditions. This enhancement is typically achieved by replacing the fixed parameters of the traditional Black–Scholes model with stochastic variables, allowing for greater flexibility in capturing market dynamics. However, this modification leads to nonlinear partial differential equations (PDEs), which require advanced mathematical techniques for analysis and solution. This study extends the Black–Scholes framework by incorporating two stochastic interest rate models, resulting in nonlinear PDEs that better reflect real-world financial complexities. We systematically analyze and compare the numerical solutions of these nonlinear PDEs using two distinct computational approaches, evaluating their effectiveness and convergence properties. Furthermore, to assess the practical applicability of our models, we conduct a numerical case study using real market data. For each stochastic model, we implement both solution approaches to determine how closely the computed option prices align with actual market prices. This comparative analysis provides insights into the strengths and limitations of each method and highlights the impact of stochastic interest rate modeling on option pricing accuracy.

The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions

2025-06-17

Ping Zhao , Haiyan Yuan | Mathematics

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to … In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results.

A note on an explicit numerical scheme for SDEs with superlinear coefficients

2025-06-17

Mingtian Tang , Zaiming Liu , Yunyan Wang | Journal of Applied Mathematics and Computing

An Engineering Approach to Asymptotics and Approximations: With Applications in Optical Wave Propagation and Imaging in Turbulence

2025-06-16

Melissa Beason , Larry C. Andrews | SPIE eBooks

Valuation of GLWB annuities with optional conversion to combo products providing LTC benefits

2025-06-16

Shaoying Chen , Zhenyu Cui , Zhimin Zhang +1 more | Scandinavian Actuarial Journal

Long-time behaviors of some stochastic differential equations driven by Lévy noise

2025-06-16

I. V. Orlovskyi , Frank Proske , Olena A. Tymoshenko | Journal of Differential Equations

A closed‐form pricing solution for options on assets with pricing errors

2025-06-16

Louis R. Piccotti | The Journal of Financial Research

Abstract This article examines the effects that pricing errors in the underlying asset have on options prices and their Greeks. Pricing errors can be viewed as random proportional transaction costs. … Abstract This article examines the effects that pricing errors in the underlying asset have on options prices and their Greeks. Pricing errors can be viewed as random proportional transaction costs. When pricing errors are information‐unrelated, options prices are unambiguously higher than the Black‐Scholes case and increasing in the pricing error variance. Hedging volatility is higher and the optimal exercise price for American put options is decreased. The option implied risk‐neutral density and option Greeks are materially affected, which leads to suboptimal risk management and hedging when pricing errors are not accounted for. Simulation and data evidence validate the theoretical results.

On time uniform Wong-Zakai approximation theorems

2025-06-16

Pierre Del Moral , Shulan Hu , Ajay Jasra +2 more | Mathematics of Computation

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary … We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations; e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion approximations.

Volatility modelling in a Markov-switching environment: two Ornstein–Uhlenbeck-related approaches

2025-06-16

Anita Behme | Finance and Stochastics

Multiagent Relative Investment Games in a Jump Diffusion Market with Deep Reinforcement Learning Algorithm

2025-06-16

Liwei Lu , Ruimeng Hu , Xu Yang +1 more | SIAM Journal on Financial Mathematics

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The sign and the co-monotonicity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg" display="inline" id="d1e38"><mml:mi>Z</mml:mi></mml:math> for a class of decoupled FBSDEs: Theory and applications

2025-06-16

Shuhui Liu , Defeng Sun | Systems & Control Letters

Analytically weak solutions to stochastic heat equations with spatially rough noise

2025-06-14

Máté Gerencsér | Stochastic Partial Differential Equations Analysis and Computations

Evaluation of Perpetual American Put Options with General Payoff

2025-06-13

Luca Anzilli , Lucianna Cananà | Risks

In this paper, we study perpetual American put options with a generalized standard put payoff and establish sufficient conditions for the existence and uniqueness of the solution to the associated … In this paper, we study perpetual American put options with a generalized standard put payoff and establish sufficient conditions for the existence and uniqueness of the solution to the associated pricing problem. As a key tool, we express the Black–Scholes operator in terms of elasticity. This formulation enables us to demonstrate that the considered pricing problem admits a unique solution when the payoff function exhibits strictly decreasing elasticity with respect to the underlying asset. Furthermore, this approach allows us to derive closed-form solutions for option pricing.

A Direct Approach in the Pricing Analysis and Risk Role Matching of a Guaranteed Annuity Option Under Correlated Risks

2025-06-13

Jude Martin Grozen , Rogemar Mamon | American Journal of Mathematical and Management Sciences

Mean square temporal error estimates for the two-dimensional stochastic Navier–Stokes equations with transport noise

2025-06-13

Dominic Breit , Thamsanqa Castern Moyo , Andreas Prohl +1 more | IMA Journal of Numerical Analysis

Abstract We study the two-dimensional Navier–Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time discretization showing a convergence rate … Abstract We study the two-dimensional Navier–Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time discretization showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier–Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.

Stochastic Linear-Quadratic Mean-Field Games of Controls for Delayed Systems with Jump Diffusion

2025-06-13

Na Li , Yilin Wei , Qingfeng Zhu | Journal of Optimization Theory and Applications

Mean-Field Linear-Quadratic Nonzero Sum Stochastic Differential Games with Overlapping Information

2025-06-12

Haiyang Li , Yu Si , Yufeng Shi | Journal of Optimization Theory and Applications

Weak convergence for continuous stochastic processes in the dual of a nuclear space with applications to the convergence of SPDEs

2025-06-12

C. A. Fonseca-Mora | Random Operators and Stochastic Equations

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \Phi^{\prime} denote the strong dual of a nuclear space Φ and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> … Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \Phi^{\prime} denote the strong dual of a nuclear space Φ and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C_{\infty}(\Phi^{\prime}) be the collection of all continuous mappings <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> x\colon[0,\infty)\rightarrow\Phi^{\prime} equipped with the topology of local uniform convergence. In this paper, we prove sufficient conditions for tightness of probability measures on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C_{\infty}(\Phi^{\prime}) and for weak convergence in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C_{\infty}(\Phi^{\prime}) for a sequence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \Phi^{\prime} -valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C_{\infty}(\Phi^{\prime}) for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

A unified model of SABR and mean-reverting stochastic volatility for derivative pricing

2025-06-12

Sun‐Yong Choi , Jeong‐Hoon Kim | Applied Mathematics and Computation

Capital Allocation Optimization for Operational Assets: A Risk-Adjusted Return Approach

2025-06-12

Navya Gupta | INTERANTIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT

ABSTRACT This research proposes an advanced framework for optimizing capital allocation across an organization's diverse portfolio of operational assets, moving beyond traditional approaches that often isolate return and risk considerations. … ABSTRACT This research proposes an advanced framework for optimizing capital allocation across an organization's diverse portfolio of operational assets, moving beyond traditional approaches that often isolate return and risk considerations. The core objective is to maximize the aggregate risk-adjusted return of the entire asset base, thereby enhancing long-term financial performance and operational resilience. The framework begins by acknowledging the inherent complexity of operational assets, which differ significantly in their revenue generation potential, operating costs, and exposure to various risk factors. A comprehensive risk assessment methodology is developed to quantify these exposures across multiple dimensions. Key operational risk categories considered include, but are not limited to:

Explicit solution for backward stochastic Volterra integral equations with linear time delayed generators

2025-06-12

Aman Auguste , Yong Ren , Harouna Coulibaly | Electronic Journal of Mathematical Analysis and Applications

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