Abstract
The paper is devoted to the Cauchy problem of backward stochastic super-parabolic equations with quadratic growth. We prove two Itô formulas in the whole space. Furthermore, we prove the existence of weak solutions for the case of one-dimensional state space, and the uniqueness of weak solutions without constraint on the state space.
Highlights
Let d and d0 be integers and {Wt := (Wt1, . . . , Wtd0 )∗, 0 ≤ t ≤ T } be a d0-dimensional standard Brownian motion defined on some probability space (, F, P )
The paper is devoted to the Cauchy problem of backward stochastic superparabolic equations with quadratic growth
We prove the existence of weak solutions for the case of onedimensional state space, and the uniqueness of weak solutions without constraint on the state space
Summary
Let d and d0 be integers and {Wt := (Wt1, . . . , Wtd0 )∗, 0 ≤ t ≤ T } be a d0-dimensional standard Brownian motion defined on some probability space ( , F , P ). Bismut (1976) derived a matrix-valued BSDE of a quadratic generator—the socalled backward stochastic Riccati equation (BSRE) in the study of linear quadratic optimal control with random coefficients, while he could not solve it in general. In that paper, he described the difficulty and failure of his fixed-point techniques in the proof of the existence and uniqueness for BSDE of a quadratic generator (i.e., the so-called quadratic BSDE). The Cauchy problem of (super-parabolic) BSPDEs with quadratic growth in the second unknown variable arises naturally in the solution of the risk-sensitive optimal control problem as the associated Hamilton−Jacobi−Bellman (HJB) equation.
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