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)