Fully coupled forward-backward stochastic differential equations on Markov chains

Advances in Difference Equations, May 2016

We define fully coupled forward-backward stochastic differential equations on spaces related to continuous-time finite-state Markov chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov chains are obtained.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.advancesindifferenceequations.com/content/pdf/s13662-016-0859-6.pdf

Fully coupled forward-backward stochastic differential equations on Markov chains

Ji et al. Advances in Difference Equations Fully coupled forward-backward stochastic differential equations on Markov chains Shaolin Ji 1 Haodong Liu 1 Xinling Xiao 0 0 School of Mathematical Sciences, Shandong Normal University , Jinan, 250014 , P.R. China 1 Qilu Institute of Finance, Shandong University , Jinan, 250100 , P.R. China We define fully coupled forward-backward stochastic differential equations on spaces related to continuous-time finite-state Markov chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov chains are obtained. forward-backward stochastic differential equations; monotone assumption; Markov chain 1 Introduction Since the first introduction by Pardoux and Peng [] in , the theory of nonlinear backward stochastic differential equations (BSDEs) driven by a Brownian motion has been intensively researched by many researchers and has achieved abundant theoretical results. Now this theory is a powerful tool in stochastic analysis. It also has many important applications, namely in stochastic control, stochastic differential games, finance, and the theory of partial differential equations (PDEs). In the classic BSDE theory, we consider a Brownian motion as the driver, but a Brownian motion is a kind of very idealized stochastic model, which limits greatly the applications of the classic BSDEs. There are many results about BSDEs associated with jump process. Tang and Li [] first discussed BSDEs driven by a Brownian motion and Poisson process; Nualart and Schoutens [] considered BSDEs driven by a Brownian motion and Lévy process. Furthermore, there are also results where the Brownian motion in the diffusion term of the BSDE is replaced by another process. For example, Cohen and Elliott [] studied BSDEs driven by a continuous-time finite-state Markov chain. After that, many results, such as a comparison theorem about this kind of BSDEs, nonlinear expected results [, ], and so on, appeared. Along with the rapid development of the BSDE theory, the theory of fully coupled forward-backward stochastic differential equations (FBSDEs), closely related to BSDEs, has been developed rapidly. Fully coupled FBSDEs with Brownian motion can be encountered in the optimization problem when applying stochastic maximum principle (see []) and in mathematical finance when considering a large investor in security market (see []). Such FBSDEs are also used in the potential theory (see []). As we know now, to get the existence and uniqueness results of fully coupled FBSDEs solutions, there are mainly three methods: the method of contraction mappings [, ], the four-step scheme [], and the method of continuation [, ]. For more details on fully coupled FBSDEs, we refer to Yong [], Ma et al. [], or Briand and Hu [] and the references therein. In this paper, we study fully coupled FBSDEs driven by a martingale generated by a continuous-time finite-state Markov chain. Inspired by Peng and Wu [], we introduce an m × n full-rank matrix G to overcome the problem caused by the different dimensions of SDE and BSDE. Using the method of continuation, the Itô product rule of semimartingales, and the fixed point principle, based on the theory of BSDEs driven by a continuoustime finite-state Markov chain, we obtain existence and uniqueness results of the FBSDEs on a Markov chain. It is worth pointing out that due to the property of the martingale generated by a finite-state Markov chain, the form of the monotone assumptions we employed here is different from that in Peng and Wu []. Recently, many works have been done on BSDEs with Markov chains. In fact, there are two major formulations of the state space of the Markov chain in the literature. One is the set of unit vectors in some Euclidean space []. The other one is the set of positive integers []. Each of the formulations was used by many researchers when they studied BSDEs driven by Markov chains. When a martingale representation was needed, the second formulation was necessary. The first one facilitates the mathematics, that is, the symbolism is nicer; the second one makes the mathematics more rigorous. Given a continuoustime Markov chain, assuming its state space be the set of positive integers, there exists an integer-valued random measure that counts the jumps of a Markov chain. Crepey and Matoussi [] and Crepey [] considered BSDEs driven by both a Brownian motion and the compensated martingale of a random measure. Tao et al. [] discussed BSDEs with a singular perturbed Markov chain; Tao et al. [, ] considered the regime-switching system modulated by a Markov chain based on Crepey and Matoussi [] and Crepey []. When the state space of a continuous-time Markov chain is described by the set of unit vectors in Rd, Zhang et al. [] considered a system of Markov regime-switching model with Poisson jumps. Different from the above works, based on Cohen (...truncated)


This is a preview of a remote PDF: http://www.advancesindifferenceequations.com/content/pdf/s13662-016-0859-6.pdf

Shaolin Ji, Haodong Liu, Xinling Xiao. Fully coupled forward-backward stochastic differential equations on Markov chains, Advances in Difference Equations, 2016, pp. 133, 2016, DOI: 10.1186/s13662-016-0859-6