Anticipated backward SDEs with jumps and quadratic-exponential growth drivers

In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple [Formula: see text] where [Formula: see text] is a semimartingale, and [Formula: see text] are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of [Formula: see text]’s future paths, as well as quadratic and exponential growth on the spot values of [Formula: see text], respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on [Formula: see text] with respect to the forward process are also obtained.

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Backward stochastic differential equations with unbounded generators

Stochastics and Dynamics ◽

10.1142/s0219493719500084 ◽

2019 ◽

Vol 19 (01) ◽

pp. 1950008 ◽

Cited By ~ 1

Author(s):

Bujar Gashi ◽

Jiajie Li

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Unique Solution ◽

Comparison Theorem ◽

Linear Growth ◽

Sufficient Conditions ◽

Backward Stochastic Differential Equations ◽

Type Condition ◽

Solution Pair ◽

Picard Iterations

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

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Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions

Advances in Applied Probability ◽

10.1239/aap/1113402403 ◽

2005 ◽

Vol 37 (1) ◽

pp. 134-159 ◽

Cited By ~ 9

Author(s):

J.-P. Lepeltier ◽

A. Matoussi ◽

M. Xu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Unique Solution ◽

Existence And Uniqueness ◽

Growth Conditions ◽

Backward Stochastic Differential Equations ◽

Contingent Claim ◽

Terminal Time ◽

Uniqueness Of The Solution ◽

American Contingent Claim

We prove the existence and uniqueness of the solution to certain reflected backward stochastic differential equations (RBSDEs) with one continuous barrier and deterministic terminal time, under monotonicity, and general increasing growth conditions on the associated coefficient. As an application, we obtain, in some constraint cases, the price of an American contingent claim as the unique solution of such an RBSDE.

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Infinite time interval BSDEs and the convergence of g-martingales

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700002172 ◽

2000 ◽

Vol 69 (2) ◽

pp. 187-211 ◽

Cited By ~ 26

Author(s):

Zengjing Chen ◽

Bo Wang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Unique Solution ◽

Backward Stochastic Differential Equations ◽

Time Interval ◽

Sufficient Condition ◽

Infinite Time ◽

Infinite Time Interval

AbstractIn this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.

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Some Properties of BSDEs Driven by a Simple Lévy Process with Continuous Coeffcient

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.50-51.288 ◽

2011 ◽

Vol 50-51 ◽

pp. 288-292

Author(s):

Shi Qiu Zheng ◽

Dian Chuan Jin ◽

Shuai Zhang ◽

Yan Mei Yang ◽

Jin Peng Wang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Comparison Theorem ◽

Lévy Process ◽

Linear Growth ◽

Backward Stochastic Differential Equations ◽

Levy Process

In this paper, we mainly study the properties of solutions of backward stochastic differential equations (BSDEs) driven by a simple Lévy process, whose coefficient coeffcient is continuous with linear growth. A comparison theorem for solutions of the equations are obtained, we also show the equation has either one or uncountably many solutions.

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Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions

Advances in Applied Probability ◽

10.1017/s0001867800000070 ◽

2005 ◽

Vol 37 (01) ◽

pp. 134-159 ◽

Cited By ~ 7

Author(s):

J.-P. Lepeltier ◽

A. Matoussi ◽

M. Xu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Unique Solution ◽

Existence And Uniqueness ◽

Growth Conditions ◽

Backward Stochastic Differential Equations ◽

Contingent Claim ◽

Terminal Time ◽

Uniqueness Of The Solution ◽

American Contingent Claim

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BACKWARD SDEs AND SOBOLEV SOLUTIONS FOR SEMILINEAR PARABOLIC PDEs WITH SINGULAR COEFFICIENTS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025711004481 ◽

2011 ◽

Vol 14 (03) ◽

pp. 517-536 ◽

Cited By ~ 1

Author(s):

TUSHENG ZHANG ◽

QIKANG RAN

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Backward Stochastic Differential Equations ◽

Parabolic Pdes ◽

Singular Coefficients ◽

Semilinear Parabolic ◽

Backward Sdes

In this paper, we construct the solutions of semilinear parabolic PDEs with singular coefficients and establish the link to solutions of backward stochastic differential equations.

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