Abstract
In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDEs with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.
Highlights
The maximum principle is a powerful tool to study the regularity of solutions, and constitutes a beautiful chapter of the classical theory of deterministic second-order elliptic and parabolic partial differential equations
Using the technique of Moser’s iteration, Aronson and Serrin proved the maximum principle and local bound of weak solutions for deterministic quasi-linear parabolic equations, which are stated in the backward form as the following two theorems
In contrast to Denis, Matoussi, and Stoica’s Lp estimate (p ∈ (2, ∞)) for the time and space maximal norm of weak solutions of quasi-linear stochastic partial differential equations (SPDEs), we prove an L∞ estimate for that of quasi-linear backward stochastic partial differential equations (BSPDEs) (1.1)
Summary
In this paper we investigate the following quasi-linear BSPDE:. − vr(t, x) dWtr, (t, x) ∈ Q := [0, T ] × O; u(T, x) = G(x), x ∈ O. To inherit in our stochastic maximum principle the general structure of admitting the unbounded coefficients b and c in the deterministic maximum principle, we prove by approximation in Section 4 the existence and uniqueness result (Theorem 4.1) for the weak solution to the quasi-linear BSPDE (1.1) with the null Dirichlet condition on the lateral boundary, under a new rather general framework This result is invoked to prove Proposition 4.3 as the Ito’s formula for the composition of solutions of BSDEs into a class of time-space smooth functions, which is the starting point of the De Giorgi scheme in the proof of subsequent stochastic maximum principles. We use the De Giorgi iteration scheme to obtain the global maximum principles for BSPDEs (1.1) and in the second subsection, we prove the local maximum principle for our backward stochastic parabolic De Giorgi class
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