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Abstract
In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.
Highlights
Backward stochastic differential equations (BSDEs) were first introduced by Pardoux and Peng [1]. a class of backward doubly stochastic differential equations (BDSDEs) were introduced by Pardoux and Peng [2] in 1994, with two different directions of stochastic integrals, i.e., the equations involve both a forward stochastic integral dWt and a backward stochastic integral dBt
In 2005, Shi et al [14] firstly gave the comparison theorem for one-dimensional BDSDEs and by this, they showed the existence of the minimal solution of BDSDEs under linear growth conditions
The existence and uniqueness results about BDSDEs with Markov chains under Lipschitz condition and under monotone condition are given in Section 3 and Section 4 respectively
Summary
Backward stochastic differential equations (BSDEs) were first introduced by Pardoux and Peng [1]. Obtained the existence and uniqueness result of the solutions to BDSDEs with locally monotone and locally Lipschitz coefficient. In 2010, Zhang and Zhao [9] proved the existence and uniqueness of the L2ρ (Rd ; R1 ) ⊗ L2ρ (Rd ; Rd )-valued solutions for BDSDEs with linear growth and the monotonicity condition. In 2005, Shi et al [14] firstly gave the comparison theorem for one-dimensional BDSDEs and by this, they showed the existence of the minimal solution of BDSDEs under linear growth conditions. We relax the conditions of BSDEs with Markov chains and add a backward Brownian motion to drive it, which will provide a theoretical basis for the study of more general regime-switching models. The existence and uniqueness results about BDSDEs with Markov chains under Lipschitz condition and under monotone condition are given in Section 3 and Section 4 respectively. We prove the comparison theorems for these kinds of BDSDEs
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