This paper treats the filtering and parameter identification for the stochastic diffusion systems with unknown boundary conditions. The physical situation of the unknown boundary conditions can be found in many …
This paper treats the filtering and parameter identification for the stochastic diffusion systems with unknown boundary conditions. The physical situation of the unknown boundary conditions can be found in many industrial problems,i.g., the salt concentration model of the river Rhine is a typical example . After formulating the diffusion systems by regarding the noisy observation data near the systems boundary region as the system’s boundary inputs, we derive the Kalman filter and the related likelihood function. The consistency property of the maximum likelihood estimate for the systems parameters is also investigated. Some numerical examples are demonstrated.
We consider the dynamics of forward rate process which is modeled by the parabolic type infinite-dimensional factor model. The parameters included in this parabolic model are estimated by using the …
We consider the dynamics of forward rate process which is modeled by the parabolic type infinite-dimensional factor model. The parameters included in this parabolic model are estimated by using the yield curve as the observation data. In this paper, we propose the filtering and identification method for the parabolic type factor model by using the particle filter algorithm.
The maximum likelihood estimation (M.L.E.) for spatially-varying parameters in stochastic parabolic systems is studied. The main result is to show the consistency property of the M.L.E. for unknown parameters by …
The maximum likelihood estimation (M.L.E.) for spatially-varying parameters in stochastic parabolic systems is studied. The main result is to show the consistency property of the M.L.E. for unknown parameters by using the method of sieves, i.e., firs
The extended least square parameter estimate for stochastic heat diffusion equations is considered. The unknown parameter is a heat diffusion coefficient which is a function of a spatial variable. Almost …
The extended least square parameter estimate for stochastic heat diffusion equations is considered. The unknown parameter is a heat diffusion coefficient which is a function of a spatial variable. Almost sure convergence for the estimated parameter is proved. A numerical example is demonstrated for supporting the theoretical resluts developed here.
An adaptive boundary control problem for a stochastic heat diffusion equation is studied. The considered system contains an unknown potential coefficient which is a function of the spatial variables. The …
An adaptive boundary control problem for a stochastic heat diffusion equation is studied. The considered system contains an unknown potential coefficient which is a function of the spatial variables. The estimation algorithm for the unknown potential coefficient is proposed by using the stochastic approximation technique. After showing the strong consistency of the estimated parameter, the cost for the adaptive control scheme presented here is shown to converge to the optimal ergodic cost. Finally some numerical examples are shown.
Motivating a simple industrial example of constructing a pipe line, we present a stochastic hyperbolic system with a moving boundary region. This hyperbolic system is formulated as an Ito stochastic …
Motivating a simple industrial example of constructing a pipe line, we present a stochastic hyperbolic system with a moving boundary region. This hyperbolic system is formulated as an Ito stochastic integral equation in Hilbert spaces. The mean square stability problem is considered under the circumstance that the moving boundary is under control.
The purpose of this paper is to study the identification problem of an infinite-dimensional parameter, more precisely a spatially varying parameter, in stochastic diffusion equations. In a previous study [S. …
The purpose of this paper is to study the identification problem of an infinite-dimensional parameter, more precisely a spatially varying parameter, in stochastic diffusion equations. In a previous study [S. I. Aihara and Y. Sunahara, SIAM J. Control Optim., 26 (1988), pp. 1062–1075], some explicit conditions for the consistency property of the maximum likelihood estimate (MLE) is explored. Here, an algorithm for generating the MLE is developed with the aid of the regularization technique proposed by [C. Kravaris and J. H. Seinfeld, SIAM J. Control Optim., 23 (1985), pp. 217–241]. After the consistency property of the MLE by a regularization is proved, necessary conditions for the regularized MLE (RMLE) are derived. Proposed is an iterative algorithm for computing one of the solutions of the necessary conditions derived. The convergence property of the sequence generated by the proposed algorithm is also shown. Finally, numerical examples are presented.
The infinite dimensional parameter estimation for stochastic heat diffusion equation is considered using the method of sieves. The consistency property is also studied for a long run data.
The infinite dimensional parameter estimation for stochastic heat diffusion equation is considered using the method of sieves. The consistency property is also studied for a long run data.
A nonlinear filtering problem where the signal is the solution of a nonlinear stochastic variational inequality is investigated. The finitely additive white noise framework is used to model the observation …
A nonlinear filtering problem where the signal is the solution of a nonlinear stochastic variational inequality is investigated. The finitely additive white noise framework is used to model the observation process. The Zakai equation for the unnormalized conditional density belongs to a class of second order partial differential equations with homogeneous Dirichlet boundary conditions.
The purpose of this note is to derive the least-square state estimator for a class of stochastic pseudoparabolic systems under noisy observations. Within the framework of function spaces, properties of …
The purpose of this note is to derive the least-square state estimator for a class of stochastic pseudoparabolic systems under noisy observations. Within the framework of function spaces, properties of the solution with regard to the state equation are studied. By showing the results of digital simulation experiments, the dynamics of the least-square estimator is given under noisy distributed observations.
Let x t be a diffusion process observed via a noisy sensor, whose output is yt We consider the problem of evaluating the maximum a posteriori trajectory {xs0≤ s ≤ …
Let x t be a diffusion process observed via a noisy sensor, whose output is yt We consider the problem of evaluating the maximum a posteriori trajectory {xs0≤ s ≤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter
Consider the problem of maximizing a functional that depends on a control function $k(x,t)$ and on the solution $u(x,t)$ of a parabolic variational inequality with k appearing in the data. …
Consider the problem of maximizing a functional that depends on a control function $k(x,t)$ and on the solution $u(x,t)$ of a parabolic variational inequality with k appearing in the data. Necessary conditions are obtained for the maximizers $k_0 (x,t)$, and the structure of $k_0 $ is then analyzed. An application to the Stefan problem is given.
Consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data. The …
Consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data. The variational problem for k is nondiflerentiable and nonconvex. We obtain necessary conditions on a maximizes $k_0 $ and then use them to determine the structure of $k_0 $ in some cases.
Next article Full AccessPseudoparabolic Partial Differential EquationsR. E. Showalter and T. W. TingR. E. Showalter and T. W. Tinghttps://doi.org/10.1137/0501001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Shmuel Agmon, Lectures on elliptic …
Next article Full AccessPseudoparabolic Partial Differential EquationsR. E. Showalter and T. W. TingR. E. Showalter and T. W. Tinghttps://doi.org/10.1137/0501001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965v+291 MR0178246 0142.37401 Google Scholar[2] S. Agmon, , A. Douglis and , L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623–727 MR0125307 0093.10401 CrossrefISIGoogle Scholar[3] M. S. Agronovich and , M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russian Math. 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Series B, Vol. 21, No. 3 | 1 Jan 2016 Cross Ref Cauchy problems of pseudo-parabolic equations with inhomogeneous termsZeitschrift für angewandte Mathematik und Physik, Vol. 66, No. 6 | 21 July 2015 Cross Ref Global existence and nonexistence of solutions for the nonlinear pseudo-parabolic equation with a memory termMathematical Methods in the Applied Sciences, Vol. 38, No. 17 | 11 December 2014 Cross Ref Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in Rn with a sublinear sourceApplied Mathematics Letters, Vol. 48 | 1 Oct 2015 Cross Ref Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential sourceCommunications on Pure and Applied Analysis, Vol. 14, No. 6 | 1 Sep 2015 Cross Ref Blow-up phenomena for a pseudo-parabolic equationMathematical Methods in the Applied Sciences, Vol. 38, No. 12 | 12 August 2014 Cross Ref Small perturbation of a semilinear pseudo-parabolic equationDiscrete and Continuous Dynamical Systems, Vol. 36, No. 2 | 1 Aug 2015 Cross Ref Global and blow-up solutions of superlinear pseudoparabolic equations with unbounded coefficientNonlinear Analysis: Theory, Methods & Applications, Vol. 122 | 1 Jul 2015 Cross Ref On some boundary value problems for systems of pseudoparabolic equationsSiberian Mathematical Journal, Vol. 56, No. 4 | 16 August 2015 Cross Ref Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearityJournal of Differential Equations, Vol. 258, No. 12 | 1 Jun 2015 Cross Ref Time Periodic Solutions for a Pseudo-parabolic Type Equation with Weakly Nonlinear Periodic SourcesBulletin of the Malaysian Mathematical Sciences Society, Vol. 38, No. 2 | 7 November 2014 Cross Ref Initial-boundary-value problems for anisotropic elliptic-parabolicpseudoparabolic equations with variable exponents of nonlinearityJournal of Mathematical Sciences, Vol. 206, No. 1 | 27 February 2015 Cross Ref Numerically solving an equation for fractional powers of elliptic operatorsJournal of Computational Physics, Vol. 282 | 1 Feb 2015 Cross Ref A quasi-boundary value regularization method for identifying an unknown source in the Poisson equationJournal of Inequalities and Applications, Vol. 2014, No. 1 | 20 March 2014 Cross Ref The local well-posedness of solutions for a nonlinear pseudo-parabolic equationBoundary Value Problems, Vol. 2014, No. 1 | 24 September 2014 Cross Ref On solvability of the first initial-boundary value problem for quasilinear pseudoparabolic equationsApplied Mathematics and Computation, Vol. 246 | 1 Nov 2014 Cross Ref A quasi-boundary value regularization method for determining the heat sourceMathematical Methods in the Applied Sciences, Vol. 37, No. 18 | 20 December 2013 Cross Ref A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equationComputational Mathematics and Mathematical Physics, Vol. 54, No. 9 | 12 September 2014 Cross Ref Non-uniqueness results for entropy two-phase solutions of forward–backward parabolic problems with unstable phaseJournal of Mathematical Analysis and Applications, Vol. 413, No. 2 | 1 May 2014 Cross Ref Inverse problem for a pseudoparabolic equation with integral overdetermination conditionsDifferential Equations, Vol. 50, No. 4 | 11 May 2014 Cross Ref Weak solutions for a high-order pseudo-parabolic equation with variable exponentsApplicable Analysis, Vol. 93, No. 2 | 17 April 2013 Cross Ref Exponential stability of the traveling fronts for a viscous Fisher-KPP equationDiscrete and Continuous Dynamical Systems - 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In this paper we deal with the existence of a nonanticipative stochastic process M, solution of the stochastic parabolic variational inequality for all v and m in some fixed spaces …
In this paper we deal with the existence of a nonanticipative stochastic process M, solution of the stochastic parabolic variational inequality for all v and m in some fixed spaces of stochastic processes (m and M are martingales). If ϕ is a convex indicator function, we obtain a “maximal solution”.
We establish basic results on existence and uniqueness for the solution of stochastic PDE's. We express the solution of a backward linear stochastic PDE in terms of the conditional law …
We establish basic results on existence and uniqueness for the solution of stochastic PDE's. We express the solution of a backward linear stochastic PDE in terms of the conditional law of a partially observed Markov diffusion process. It then follows that the adjoint forward stochastic PDE governs the evolution of the “unnormalized conditional density”
In this paper we are concerned with stability problems for infinite dimensional systems. First we review the theory for linear systems where the dynamics are governed by strongly continuous semigroups …
In this paper we are concerned with stability problems for infinite dimensional systems. First we review the theory for linear systems where the dynamics are governed by strongly continuous semigroups and then use these results to obtain globial existence and stability results for nonlinear systems. We also consider the problem of designing feedback controls to enhance the stability properties of a system.
Identification of spatially varying parameters in distributed parameter systems from noisy data is an ill-posed problem. The concept of regularization, widely used in solving linear Fredholm integral equations, is developed …
Identification of spatially varying parameters in distributed parameter systems from noisy data is an ill-posed problem. The concept of regularization, widely used in solving linear Fredholm integral equations, is developed for the identification of parameters in distributed parameter systems. A general regularization identification theory is first presented and then applied to the identification of parabolic systems. The performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation.
The paper contains a phenomenological description of the whole US forward rate curve (FRC), based on data in the period 1990–1996. It is found that the average deviation of the …
The paper contains a phenomenological description of the whole US forward rate curve (FRC), based on data in the period 1990–1996. It is found that the average deviation of the FRC from the spot rate grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a 'Value-at-Risk' type of pricing. The instantaneous FRC, however, departs from a simple square-root law. The deformation is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. It is shown that this is consistent with the volatility 'hump' around one year found by several authors (which is confirmed). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. This is rationalized by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short-scale distortions of the FRC. This shape-dependent term could lead to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. It is suggested that the presence of transaction costs (or other market 'imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.1
The purpose of this paper is to study the identification problem of an infinite-dimensional parameter, more precisely a spatially varying parameter, in stochastic diffusion equations. In a previous study [S. …
The purpose of this paper is to study the identification problem of an infinite-dimensional parameter, more precisely a spatially varying parameter, in stochastic diffusion equations. In a previous study [S. I. Aihara and Y. Sunahara, SIAM J. Control Optim., 26 (1988), pp. 1062–1075], some explicit conditions for the consistency property of the maximum likelihood estimate (MLE) is explored. Here, an algorithm for generating the MLE is developed with the aid of the regularization technique proposed by [C. Kravaris and J. H. Seinfeld, SIAM J. Control Optim., 23 (1985), pp. 217–241]. After the consistency property of the MLE by a regularization is proved, necessary conditions for the regularized MLE (RMLE) are derived. Proposed is an iterative algorithm for computing one of the solutions of the necessary conditions derived. The convergence property of the sequence generated by the proposed algorithm is also shown. Finally, numerical examples are presented.
Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. …
Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.
Levy's modulus of continuity is proved for infinite dimensional Wiener processes. Using the loglog law for a Banach space valued Wiener process in [7], we prove the loglog law for …
Levy's modulus of continuity is proved for infinite dimensional Wiener processes. Using the loglog law for a Banach space valued Wiener process in [7], we prove the loglog law for Hilbert space valued stochastic integrals, if the integrand is Holder continuous. From a corollary of Kolmogorov's law we derive the Hölder continuity of Hilbert space valued stochastic integrals if the fourth moment of the integrand is uniformly bounded. As an application we show that the mild solution of a stochastic evolution equation has a continuous version if the semigroup governing this equation is analytic.
Parameter identification is studied for infinite dimensional linear systems. An almost sure characterization of sample path-wise limit sets of maximum likelihood estimates is given.
Parameter identification is studied for infinite dimensional linear systems. An almost sure characterization of sample path-wise limit sets of maximum likelihood estimates is given.
Parameter estimation in a class of Ito Processes with a parametrized "drift" term is considered. An almost sure characterization of a sample path-wise limit sets of maximum likelihood estimates is …
Parameter estimation in a class of Ito Processes with a parametrized "drift" term is considered. An almost sure characterization of a sample path-wise limit sets of maximum likelihood estimates is given. Related problems of estimation under approximate parametrizations and estimation of a random parameter with unknown distribution are also considered.
In this paper, we apply Grenander’s method of sieves to the problem of estimation of the infinite dimensional parameter in a nonstationary linear diffusion model. We use an increasing sequence …
In this paper, we apply Grenander’s method of sieves to the problem of estimation of the infinite dimensional parameter in a nonstationary linear diffusion model. We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximize the likelihood function. We show that if the dimension of the sieves tends to infinity with the sample size with a rate not too fast then the sequence of restricted maximum likelihood estimators for the parameter is consistent and asymptotically normal.