Existence, minimality and approximations of solutions of BSDEs with convex drivers

P. Cheridito, M.A. Stadje

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove the existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.
Original languageEnglish
Pages (from-to)1540-1565
JournalStochastic Processes and their Applications
Volume122
Issue number4
DOIs
Publication statusPublished - 2012

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Minimality
Driver
Differential equations
Approximation
Lipschitz
Supersolution
Backward Stochastic Differential Equation
Uniformly continuous
Lower Semicontinuous
Unique Solution
Existence of Solutions

Cite this

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abstract = "We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove the existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.",
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Existence, minimality and approximations of solutions of BSDEs with convex drivers. / Cheridito, P.; Stadje, M.A.

In: Stochastic Processes and their Applications, Vol. 122, No. 4, 2012, p. 1540-1565.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Existence, minimality and approximations of solutions of BSDEs with convex drivers

AU - Cheridito, P.

AU - Stadje, M.A.

PY - 2012

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AB - We study the existence of solutions to backward stochastic differential equations with drivers f(t,W,y,z) that are convex in z. We assume f to be Lipschitz in y and W but do not make growth assumptions with respect to z. We first show the existence of a unique solution (Y,Z) with bounded Z if the terminal condition is Lipschitz in W and that it can be approximated by the solutions to properly discretized equations. If the terminal condition is bounded and uniformly continuous in W we show the existence of a minimal continuous supersolution by uniformly approximating the terminal condition with Lipschitz terminal conditions. Finally, we prove the existence of a minimal RCLL supersolution for bounded lower semicontinuous terminal conditions by approximating the terminal condition pointwise from below with Lipschitz terminal conditions.

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DO - 10.1016/j.spa.2011.12.008

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EP - 1565

JO - Stochastic Processes and their Applications

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