38 research outputs found
Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media
We establish an averaging principle for a family of
solutions
of a system of
SDE-BSDEwith a null recurrent fast component . Incontrast
to the classical periodic case, we can not rely on aninvariant probability and
the slow forward component cannot be approximated by a
diffusion process.On the other hand, we assume that the coefficients admit a
limit in a\`{C}esaro sense. In such a case, the limit coefficients may
havediscontinuity. We show that we can approximate the
triplet bya
system of SDE-BSDE where is aMarkov diffusion
which is the unique (in law) weak solution of theaveraged forward component and
is the unique solution to the averaged backward component. This is done
with a backward component whosegenerator depends on the variable .
Asapplication, we establish an homogenization result for semilinearPDEs when
the coefficients can be neither periodic nor ergodic. Weshow that the averaged
BDSE is related to the averaged PDE via aprobabilistic representation of the
(unique) Sobolev --solution of the
limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich
are based on stability property and weak convergence of BSDEsin the S-topology
Optimality necessary conditions in singular stochastic control problems with nonsmooth data
AbstractThe present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau–Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients
Lp-solutions to BSDEs with super-linear growth coefficient. Application to degenerate semilinear PDEs
We consider multidimensional backward stochastic differential equations (BSDEs). We prove the existence and uniqueness of solutions when the coefficient grow super-linearly, and moreover, can be neither locally Lipschitz in the variable y nor in the variable z. This is done with super-linear growth coefficient and a p-integrable terminal condition (p > 1). As application, we establish the existence and uniqueness of solutions to degenerate semilinear PDEs with superlinear growth generator and an Lp-terminal data, p > 1. Our result cover, for instance, the case of PDEs with logarithmic nonlinearities
On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls
This paper is concerned with necessary and sufficient conditions for
near-optimal singular stochastic controls for systems driven by a nonlinear
stochastic differential equations (SDEs in short). The proof of our result is
based on Ekeland's variational principle and some delicate estimates of the
state and adjoint processes. This result is a generalization of Zhou's
stochastic maximum principle for near-optimality to singular control problem.Comment: 19 pages, submitted to journa
Adaptive importance sampling with forward-backward stochastic differential equations
We describe an adaptive importance sampling algorithm for rare events that is
based on a dual stochastic control formulation of a path sampling problem.
Specifically, we focus on path functionals that have the form of cumulate
generating functions, which appear relevant in the context of, e.g.~molecular
dynamics, and we discuss the construction of an optimal (i.e. minimum variance)
change of measure by solving a stochastic control problem. We show that the
associated semi-linear dynamic programming equations admit an equivalent
formulation as a system of uncoupled forward-backward stochastic differential
equations that can be solved efficiently by a least squares Monte Carlo
algorithm. We illustrate the approach with a suitable numerical example and
discuss the extension of the algorithm to high-dimensional systems
BSDE associated with LĂ©vy processes and application to PDIE
We deal with backward stochastic differential equations (BSDE for short) driven by
Teugel's martingales and an independent Brownian motion. We study the existence,
uniqueness and comparison of solutions for these equations under a Lipschitz as well as
a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove
that if the Lipschitz constant LN behaves as log(N) in the ball B(0,N), then the corresponding BSDE has a unique solution which depends continuously on the on the coefficient and the terminal data. This is done with an unbounded terminal data. As application, we give a probabilistic interpretation for a large class of partial differential
integral equations (PDIE for short)