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

This paper considers the setting governed by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo></mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> is the "public" flow of information, and <i>τ</i> is a random time which might not be <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> -observable. This framework covers credit risk and life insurance. In this setting, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> is assumed to be generated by a <i>d</i>-dimensional Brownian motion <i>W</i> and <i>ξ</i> is a vulnerable claim, whose payment's policy depends <i>essentially</i> on the occurrence of <i>τ</i>. The hedging problems, in many directions, for this claim led to the question of studying the linear reflected-backward-stochastic differential equations (RBSDE hereafter), <DispFormula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtable> <mml:mtr><mml:mtd><mml:mi>d</mml:mi> <mml:msub><mml:mi>Y</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>∧</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:msub><mml:mi>Z</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mi>d</mml:mi> <mml:msub><mml:mi>W</mml:mi> <mml:mrow><mml:mi>t</mml:mi> <mml:mo>∧</mml:mo> <mml:mi>τ</mml:mi></mml:mrow> </mml:msub> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:msub><mml:mi>M</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:msub><mml:mi>K</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mo>,</mml:mo> <mml:mspace></mml:mspace> <mml:msub><mml:mi>Y</mml:mi> <mml:mi>τ</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo></mml:mtd> </mml:mtr> <mml:mtr><mml:mtd><mml:mi>Y</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>S</mml:mi> <mml:mspace></mml:mspace> <mml:mtext>on</mml:mtext> <mml:mspace></mml:mspace> <mml:mo>〚</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>〚</mml:mo> <mml:mo>,</mml:mo> <mml:mspace></mml:mspace> <mml:msubsup><mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mi>τ</mml:mi></mml:msubsup> <mml:mo>(</mml:mo> <mml:msub><mml:mi>Y</mml:mi> <mml:mrow><mml:mi>s</mml:mi> <mml:mo>-</mml:mo></mml:mrow> </mml:msub> <mml:mo>-</mml:mo> <mml:msub><mml:mi>S</mml:mi> <mml:mrow><mml:mi>s</mml:mi> <mml:mo>-</mml:mo></mml:mrow> </mml:msub> <mml:mo>)</mml:mo> <mml:mi>d</mml:mi> <mml:msub><mml:mi>K</mml:mi> <mml:mi>s</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace></mml:mspace> <mml:mi>P</mml:mi> <mml:mtext>-a.s.</mml:mtext> <mml:mo>.</mml:mo></mml:mtd> </mml:mtr> </mml:mtable> </mml:math> </DispFormula> This is the objective of this paper. For this RBSDE and without any further assumption on <i>τ</i> that might neglect any risk intrinsic to its stochasticity, we answer the following: a) What are the sufficient minimal conditions on the data <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo></mml:math> that guarantee the existence of the solution to this RBSDE? b) How can we estimate the solution in norm using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo></mml:math> ? c) Is there an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> -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.