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Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes (p. 81-82)
Jan A. Van Casteren
This article is written in honor of G. Lumer whom I consider as my semi-group teacher
Abstract. In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic Differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial Differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates.
1. Introduction
Backward stochastic Differential equations, in short BSDEs, have been well studied during the last ten years or so. They were introduced by Pardoux and Peng [20], who proved existence and uniqueness of adapted solutions, under suitable squareintegrability assumptions on the coeffcients and on the terminal condition. They provide probabilistic formulas for solution of systems of semi-linear partial Differential equations, both of parabolic and elliptic type. The interest for this kind of stochastic equations has increased steadily, this is due to the strong connections of these equations with mathematical finance and the fact that they provide a generalization of the well-known Feynman-Kac formula to semi-linear partial differential equations. In the present paper we will concentrate on the relationship between time-dependent strong Markov processes and abstract backward stochastic Differential equations. The equations are phrased in terms of a martingale type problem, rather than a strong stochastic Differential equation. They could be called weak backward stochastic Differential equations. Emphasis is put on existence and uniqueness of solutions. The paper in [27] deals with the same subject, but it concentrates on comparison theorems and viscosity solutions.
The notion of squared gradient operator is implicitly used by Bally at al in [4]. The latter paper was one of the motivations to write the present paper with an emphasis on the squared gradient operator. In addition, our results are presented in such a way that the state space of the underlying Markov process, which in most of the other papers on BSDEs is supposed to be Rn, can be any diffusion with an abstract state space, which throughout our text is denoted by E. In fact in the existing literature the underlying Markov process is a (strong) solution of a (forward) stochastic Differential equation: see, e.g., [4], [8] and [7] and [19]. For more on this see Remark 2.9 below. In particular our results are applicable in case the Markov process under consideration is Brownian motion on a Riemannian manifold. Our condition on the generator (or coefficient) of the BSDE f in terms of the squared gradient is very natural. In the Lipschitz context it is more or less optimal. Moreover, our proof of existence is not based on standard regularization methods by using convolution products with smooth functions, but on a homotopy argument due to Crouzeix [11], which seems more direct than the classical approach. We also obtain rather precise quantitative estimates. Only very rudimentary sketches of proofs are given, details will appear elsewhere.
