94 research outputs found
On Backward Doubly Stochastic Differential Evolutionary System
In this paper, we are concerned with backward doubly stochastic differential
evolutionary systems (BDSDESs for short). By using a variational approach based
on the monotone operator theory, we prove the existence and uniqueness of the
solutions for BDSDESs. We also establish an It\^o formula for the Banach
space-valued BDSDESs.Comment: 33 page
Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality
A Dynkin game is considered for stochastic differential equations with random
coefficients. We first apply Qiu and Tang's maximum principle for backward
stochastic partial differential equations to generalize Krylov estimate for the
distribution of a Markov process to that of a non-Markov process, and establish
a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be
a random field of It\^o's type which takes values in a suitable Sobolev space.
We then prove the verification theorem that the Nash equilibrium point and the
value of the Dynkin game are characterized by the strong solution of the
associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a
backward stochastic partial differential variational inequality (BSPDVI, for
short) with two obstacles. We obtain the existence and uniqueness result and a
comparison theorem for strong solution of the BSPDVI. Moreover, we study the
monotonicity on the strong solution of the BSPDVI by the comparison theorem for
BSPDVI and define the free boundaries. Finally, we identify the counterparts
for an optimal stopping time problem as a special Dynkin game.Comment: 40 page
Maximum Principle for Quasi-linear Backward Stochastic Partial Differential Equations
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 BSPDE 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
Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients
We are concerned with the linear-quadratic optimal stochastic control problem
with random coefficients. Under suitable conditions, we prove that the value
field , is
quadratic in , and has the following form:
where is an essentially bounded nonnegative symmetric matrix-valued adapted
processes. Using the dynamic programming principle (DPP), we prove that is
a continuous semi-martingale of the form with being a
continuous process of bounded variation and and that with
is a solution to the associated backward stochastic
Riccati equation (BSRE), whose generator is highly nonlinear in the unknown
pair of processes. The uniqueness is also proved via a localized completion of
squares in a self-contained manner for a general BSRE. The existence and
uniqueness of adapted solution to a general BSRE was initially proposed by the
French mathematician J. M. Bismut (1976, 1978). It had been solved by the
author (2003) via the stochastic maximum principle with a viewpoint of
stochastic flow for the associated stochastic Hamiltonian system. The present
paper is its companion, and gives the {\it second but more comprehensive}
adapted solution to a general BSRE via the DDP. Further extensions to the
jump-diffusion control system and to the general nonlinear control system are
possible.Comment: 16 page
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