34 research outputs found
Backward stochastic variational inequalities on random interval
The aim of this paper is to study, in the infinite dimensional framework, the
existence and uniqueness for the solution of the following multivalued
generalized backward stochastic differential equation, considered on a random,
possibly infinite, time interval: \cases{\displaystyle
-\mathrm{d}Y_t+\partial_y\Psi (t,Y_t)\,\mathrm{d}Q_t\ni\Phi
(t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<\tau,\cr
\displaystyle{Y_{\tau}=\eta,}} where is a stopping time, is a
progressively measurable increasing continuous stochastic process and
is the subdifferential of the convex lower semicontinuous
function . As applications, we obtain from our main
results applied for suitable convex functions, the existence for some backward
stochastic partial differential equations with Dirichlet or Neumann boundary
conditions.Comment: Published at http://dx.doi.org/10.3150/14-BEJ601 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Stochastic Variational Inequalities on Non-Convex Domains
The objective of this work is to prove, in a first step, the existence and
the uniqueness of a solution of the following multivalued deterministic
differential equation: ,
, where is a continuous
function and is the Fr\'{e}chet subdifferential of a
semiconvex function ; the domain of can be non-convex, but
some regularities of the boundary are required.
The continuity of the map , which associate the input function with the
solution of the above equation, as well as tightness criteria allow to pass
from the above deterministic case to the following stochastic variational
inequality driven by a multi-dimensional Brownian motion: , with
.Comment: 39 page
Stochastic approach for a multivalued Dirichlet-Neumann problem
We prove the existence and uniqueness of a viscosity solution of the
parabolic variational inequality with a nonlinear multivalued Neumann-Dirichlet
boundary condition:% {equation*} \{{array}{r} \dfrac{\partial u(t,x)}{\partial
t}-\mathcal{L}_{t}u(t,x) {+}{% \partial \phi}\big(u(t,x)\big)\ni
f\big(t,x,u(t,x),(\nabla u\sigma)(t,x)\big), t>0, x\in \mathcal{D},\medskip
\multicolumn{1}{l}{\dfrac{\partial u(t,x)}{\partial n}+{\partial \psi}\big(%
u(t,x)\big)\ni g\big(t,x,u(t,x)\big), t>0, x\in
Bd(\mathcal{D}%),\multicolumn{1}{l}{u(0,x)=h(x), x\in \bar{\mathcal{D}},}%
{array}%. {equation*}% where and are
subdifferentials operators and is a second differential
operator. The result is obtained by a Feynman-Ka\c{c} representation formula
starting from the backward stochastic variational inequality:% {equation*}
\{{array}{l} dY_{t}{+}F(t,Y_{t},Z_{t}) dt{+}G(t,Y_{t}) dA_{t}\in \partial \phi
(Y_{t}) dt{+}\partial \psi (Y_{t}) dA_{t}{+}Z_{t}dW_{t}, 0\leq t\leq T,\medskip
\ Y_{T}=\xi .% {array}%. {equation*}Comment: 29 page
Viscosity solutions for systems of parabolic variational inequalities
In this paper, we first define the notion of viscosity solution for the
following system of partial differential equations involving a subdifferential
operator: where
is the subdifferential operator of the proper convex lower
semicontinuous function and
is a second differential operator given by
,
. We prove the uniqueness of the viscosity solution and then,
via a stochastic approach, prove the existence of a viscosity solution
of the above parabolic variational
inequality.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ204 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Convergence of invariant measures for singular stochastic diffusion equations
It is proved that the solutions to the singular stochastic -Laplace
equation, and the solutions to the stochastic fast diffusion
equation with nonlinearity parameter on a bounded open domain
with Dirichlet boundary conditions are continuous in mean,
uniformly in time, with respect to the parameters and respectively (in
the Hilbert spaces , respectively). The highly
singular limit case is treated with the help of stochastic evolution
variational inequalities, where \mathbbm{P}-a.s. convergence, uniformly in
time, is established.
It is shown that the associated unique invariant measures of the ergodic
semigroups converge in the weak sense (of probability measures).Comment: to appear in Stoch. Proc. Appl. (in press), 18 p
Numerical Schemes for Multivalued Backward Stochastic Differential Systems
We define some approximation schemes for different kinds of generalized
backward stochastic differential systems, considered in the Markovian
framework. We propose a mixed approximation scheme for a decoupled system of
forward reflected SDE and backward stochastic variational inequality. We use an
Euler scheme type, combined with Yosida approximation techniques.Comment: 13 page