This paper considers the setting governed by <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">F</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\mathbb{F},\tau )$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{F}$</EquationSource> </InlineEquation> is the “public” flow of information, and τ is a random time which might not be <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{F}$</EquationSource> </InlineEquation>-observable. This framework covers credit risk and life insurance. In this setting, <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{F}$</EquationSource> </InlineEquation> is assumed to be generated by a d-dimensional Brownian motion W and ξ is a vulnerable claim, whose payment’s policy depends essentially on the occurrence of τ. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mtable columnalign="right left" columnspacing="0.2em"> <mtr> <mtd> <mi>d</mi> <msub> <mi>Y</mi> <mi>t</mi> </msub> <mo>=</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>∧</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>Z</mi> <mi>t</mi> </msub> <mi>d</mi> <msub> <mi>W</mi> <mrow> <mi>t</mi> <mo>∧</mo> <mi>τ</mi> </mrow> </msub> <mo>−</mo> <mi>d</mi> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>−</mo> <mi>d</mi> <msub> <mi>K</mi> <mi>t</mi> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>Y</mi> <mi>τ</mi> </msub> <mo>=</mo> <mi>ξ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>Y</mi> <mo>≥</mo> <mi>S</mi> <mspace width="1em" /> <mtext>on</mtext> <mspace width="1em" /> <mo stretchy="false">〚</mo> <mn>0</mn> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">〚</mo> <mo>,</mo> <mspace width="1em" /> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>τ</mi> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow> <mi>s</mi> <mo>−</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>S</mi> <mrow> <mi>s</mi> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mi>d</mi> <msub> <mi>K</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi>P</mi> <mtext>-a.s.</mtext> <mo>.</mo> </mtd> </mtr> </mtable> </math></EquationSource> <EquationSource Format="TEX">$ \begin{aligned} &dY_{t}=-f(t)d(t\wedge \tau )+Z_{t}dW_{t\wedge{\tau}}-dM_{t}-dK_{t},\quad Y_{\tau}=\xi ,\\ & Y\geq S\quad \text{on}\quad [\!\![0,\tau [\!\![,\quad \displaystyle \int _{0}^{\tau}(Y_{s-}-S_{s-})dK_{s}=0\quad P\text{-a.s.}.\end{aligned} $</EquationSource> </Equation> This is the objective of this paper. For this RBSDE and without any further assumption on τ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(f, \xi , S, \tau )$</EquationSource> </InlineEquation> that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>ξ</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(f, \xi , S)$</EquationSource> </InlineEquation>? c) Is there an <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{F}$</EquationSource> </InlineEquation>-RBSDE that is intimately related to the current one and how their solutions are related to each other? This latter question has practical and theoretical leitmotivs.

Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon

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2026年1月30日

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Tahir Choulli,
Safa’ Alsheyab

摘要

This paper considers the setting governed by $(F, τ)$ $(\mathbb{F},\tau )$ , where $F$ $\mathbb{F}$ is the “public” flow of information, and τ is a random time which might not be $F$ $\mathbb{F}$ -observable. This framework covers credit risk and life insurance. In this setting, $F$ $\mathbb{F}$ is assumed to be generated by a d-dimensional Brownian motion W and ξ is a vulnerable claim, whose payment’s policy depends essentially on the occurrence of τ. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), $\begin{aligned} d Y_{t} = - f (t) d (t \land τ) + Z_{t} d W_{t \land τ} - d M_{t} - d K_{t}, Y_{τ} = ξ, \\ Y \geq S on 〚 0, τ 〚, \int_{0}^{τ} (Y_{s -} - S_{s -}) d K_{s} = 0 P -a.s. . \end{aligned}$ $ \begin{aligned} &dY_{t}=-f(t)d(t\wedge \tau )+Z_{t}dW_{t\wedge{\tau}}-dM_{t}-dK_{t},\quad Y_{\tau}=\xi ,\\ & Y\geq S\quad \text{on}\quad [\!\![0,\tau [\!\![,\quad \displaystyle \int _{0}^{\tau}(Y_{s-}-S_{s-})dK_{s}=0\quad P\text{-a.s.}.\end{aligned} $ This is the objective of this paper. For this RBSDE and without any further assumption on τ that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data $(f, ξ, S, τ)$ $(f, \xi , S, \tau )$ that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using $(f, ξ, S)$ $(f, \xi , S)$ ? c) Is there an $F$ $\mathbb{F}$ -RBSDE that is intimately related to the current one and how their solutions are related to each other? This latter question has practical and theoretical leitmotivs.