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

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.

scholarly journals REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A LÉVY PROCESS

2009 ◽  
Vol 50 (4) ◽  
pp. 486-500 ◽  
Author(s):  
YONG REN ◽  
XILIANG FAN
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Penalization Method ◽  
Lower Semi ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales ◽  
Continuous Convex Function

AbstractIn this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

scholarly journals Backward stochastic differential equations with unbounded generators

2019 ◽  
Vol 19 (01) ◽  
pp. 1950008 ◽  
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.

Reflected generalized BSDEs with discontinuous barriers driven by a Lévy process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales

Abstract This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.

scholarly journals Anticipated backward SDEs with jumps and quadratic-exponential growth drivers

2019 ◽  
Vol 19 (03) ◽  
pp. 1950020
Author(s):  
Masaaki Fujii ◽  
Akihiko Takahashi
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Exponential Growth ◽  
Linear Growth ◽  
Uniform Norm ◽  
Backward Stochastic Differential Equations ◽  
Forward Process ◽  
Backward Sdes

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.

Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

2012 ◽  
Vol 524-527 ◽  
pp. 3801-3804
Author(s):  
Shi Yu Li ◽  
Wu Jun Gao ◽  
Jin Hui Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Equations ◽  
Local Martingale ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
The One ◽  
Uniformly Lipschitz

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.

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