Stochastic Lipschitz Coefficient Research Articles

Doubly reflected BSDEs driven by RCLL martingales under stochastic Lipschitz coefficient

In this paper, we study a backward stochastic differential equation driven by a Right Continuous with Left Limits (RCLL) martingale with two completely separated RCLL barriers. When the coefficient is stochastically Lipschitz, we demonstrate the existence and uniqueness of a square-integrable adapted solution using the penalization method. Additionally, we provide a fair price for a game contingent claim between two traders with additional exogenous knowledge of the same stock price in a public financial market driven by an Azéma’s martingale. We also determine a saddle point for the game when the obstacles are left upper semi-continuous along stopping times.

Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients

Abstract This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 {\frac{1}{2}} . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

On the stochastic control-stopping problem

We study the stochastic control-stopping problem when the data are path-dependent and of polynomial growth. The approach is based on backward stochastic differential equations (BSDEs for short). The problem turns into the study of a specific reflected BSDE with a stochastic Lipschitz coefficient for which we show existence and uniqueness of the solution. We then establish its relationship with the value function of the control-stopping problem. The optimal strategy is exhibited. Finally in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

Sequential Systems of Reflected Backward Stochastic Differential Equations with Application to Impulse Control

We consider a system of finite horizon, sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem. Moreover, we show that the solution to our system of RBSDEs is connected to weak solutions of a stochastic differential game where one player implements an impulse control while the opponent plays a continuous control that enters the drift term. As all our arguments are probabilistic and hence hold in a non-markovian framework, we are able to consider the setting where the underlying uncertainty in the game stems from an impulsively and continuously controlled path-dependent stochastic differential equation driven by Brownian motion.

A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.

Predictable solution for reflected BSDEs when the obstacle is not right-continuous

AbstractIn the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.

Anticipated backward doubly SDEs with generators in stochastic Lipschitz condition

In this paper, we prove the well-posedness of the solutions to anticipated backward doubly SDEs (ABDSDEs for short) with stochastic Lipschitz coefficients, where the generators (drivers) of equations not only rely on the prospective values of the pair of solutions, but also satisfy a certain type of Lipschitz condition. This type of equation has a practical background in financial mathematics.

Reflected BSDEs with jumps and two rcll barriers under stochastic Lipschitz coefficient

In this article, we give a solution to doubly reflected backward stochastic differential equations when the barriers are right continuous with left limits, and the noise is driven by a Brownian motion and an independent Poisson random measure. We show the existence and uniqueness of solution by means of the penalization method when the coefficient is stochastic Lipschitz.

A framework of BSDEs with stochastic Lipschitz coefficients

In this paper, we suggest an effective technique based on random time-change for dealing with a large class of backward stochastic differential equations (BSDEs for short) with stochastic Lipschitz coefficients. By means of random time-change, we show the relation between the BSDEs with stochastic Lipschitz coefficients and the ones with bounded Lipschitz coefficients and stopping terminal time, so they are possible to be exchanged with each other from one type to another. In other words, the stochastic Lipschitz condition is not essential in the context of BSDEs with random terminal time. Using this technique, we obtain a couple of new results of BSDEs with stochastic Lipschitz (or monotone) coefficients.

BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients

Abstract This paper deals with a class of backward stochastic differential equation driven by two mutually independent fractional Brownian motions. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient

Abstract This paper proves the existence and uniqueness of a solution to reflected backward stochastic differential equations with a lower obstacle, which is assumed to be right upper-semicontinuous. The result is established where the coefficient is stochastic Lipschitz by using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales and other tools from optimal stopping theory.

Non-Continuous Double Barrier Reflected BSDES With Jumps under a Stochastic Lipschitz Coefficient

Non-Continuous Double Barrier Reflected BSDES With Jumps under a Stochastic Lipschitz Coefficient

Fractional anticipated BSDEs with stochastic Lipschitz coefficients

Abstract In this paper, we deal with an anticipated backward stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H ∈ ( 1 / 2 , 1 ) {H\in(1/2,1)} . We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and prove a comparison theorem in a specific case.

Double barrier reflected BSDEs with stochastic Lipschitz coefficient

This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.

Existence and uniqueness of a stationary and ergodic solution to stochastic recurrence equations via Matkowski’s FPT

AbstractWe establish the existence of a unique stationary and ergodic solution for systems of stochastic recurrence equations defined by stochastic self-maps on Polish metric spaces based on the fixed point theorem of Matkowski. The results can be useful in cases where the stochastic Lipschitz coefficients implied by the currently used method either do not exist, or lead to the imposition of unnecessarily strong conditions for the derivation of the solution.

Existence and Uniqueness of Multidimensional BSDEs and of Systems of Degenerate PDEs with Superlinear Growth Generator

In the first part of this paper, we deal with the unique solvability of multidimensional backward stochastic differential equations (BSDEs) with a $p$-integrable terminal condition $(p>1)$ and a superlinear growth generator. We introduce a new local condition on the generator (assumption (H.4)) and then show that it ensures the existence and uniqueness, as well as the $L^p$-stability, of solutions. The assumptions that we impose on the generator are local in the three variables $y, z, \omega $, and therefore we also cover the BSDEs with stochastic Lipschitz coefficient. Our conditions on the generator go beyond all existing ones in the literature. For instance, the generator is not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. Furthermore, it could be neither locally monotone in the $y$-variable nor locally Lipschitz in the $z$-variable. Although we are focused on multidimensional BSDEs, our results on uniqueness and stability are new even for one-dimensional BS...

A class of time inconsistent risk measures and backward stochastic Volterra integral equations

In this paper we derive a new class of time inconsistent dynamic coherent risk measures, allowing the interest rates involved to be random, which extends and modifies the results on risk measures in Appl. Anal. 86 (2007), 1429–1442. Since in this procedure one important mathematical tool is a kind of backward stochastic Volterra integral equations (BSVIEs in short) with stochastic Lipschitz coefficients, this paper firstly is devoted to develop the existence and uniqueness theory for the corresponding BSVIEs, which not only generalizes the results in Stoch. Proc. Appl. 116 (2006), 779–795; Appl. Anal. 86 (2007), 1429–1442; Probab. Theory Related Fields 142 (2008), 21–77; but also simplifies considerably the related arguments there.