Derivation, 𝐿^{Ψ}-bounded martingales and covering conditions

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma normal upper Sigma comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Omega ,\,\Sigma ,\,P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complete probability space. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Sigma Subscript t Baseline right-parenthesis Subscript t element-of upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>J</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{({\Sigma _t})_{t \in J}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a directed family of sub-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Phi comma normal upper Psi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Phi ,\,\Psi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal upper Psi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\Psi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales. We do not assume that either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi"> <mml:semantics> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> <mml:annotation encoding="application/x-tex">\Psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Delta _2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> condition. We show that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies condition Exp, that is when there exists an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis u right-parenthesis less-than-or-equal-to exp a u"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>exp</mml:mi> <mml:mspace width="thinmathspace" /> <mml:mi>a</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi (u) \leq \operatorname {exp} \,au</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">u \ge 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the essential convergence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal upper Psi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\Psi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales is equivalent to the classical covering condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Subscript normal upper Phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{V_\Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This covers in particular the classical case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi left-parenthesis t right-parenthesis equals t left-parenthesis log t right-parenthesis Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mspace width="thinmathspace" /> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Psi (t) = t{(\operatorname {log} \,t)^ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The growth condition Exp on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cannot be relaxed. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a countable cofinite set, we show that the essential convergence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal upper Psi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\Psi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales is equivalent to a covering condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript normal upper Phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{D_\Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (that is weaker than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Subscript normal upper Phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{V_\Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fails condition Exp, condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript normal upper Phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{D_\Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is optimal. Roughly speaking, in the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1 }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales, condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript normal upper Phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{D_\Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not contain a countable cofinal set and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fails condition Exp. In this case, it seems impossible to characterize the essential convergence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal upper Psi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">Ψ<!-- Ψ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\Psi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1 }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales, but not all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1 }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded martingales, converge essentially. Similar results are also established in the derivation setting.

Locations

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