191 research outputs found
Variational inequalities in Hilbert spaces with measures and optimal stopping problems
We study the existence theory for parabolic variational inequalities in
weighted spaces with respect to excessive measures associated with a
transition semigroup. We characterize the value function of optimal stopping
problems for finite and infinite dimensional diffusions as a generalized
solution of such a variational inequality. The weighted setting allows us
to cover some singular cases, such as optimal stopping for stochastic equations
with degenerate diffusion coefficient. As an application of the theory, we
consider the pricing of American-style contingent claims. Among others, we
treat the cases of assets with stochastic volatility and with path-dependent
payoffs.Comment: To appear in Applied Mathematics and Optimizatio
Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise
In this work, we introduce a new method to prove the existence and uniqueness
of a variational solution to the stochastic nonlinear diffusion equation
where is a bounded and open domain
in , , and is a Wiener process of the form
, e_k \in C^2(\bar\mathcal{O})\cap
H^1_0(\mathcal{O}), and , , are independent Brownian
motions. This is a stochastic diffusion equation with a highly singular
diffusivity term and one main result established here is that, for all initial
conditions in , it is well posed in a class of continuous
solutions to the corresponding stochastic variational inequality. Thus one
obtains a stochastic version of the (minimal) total variation flow. The new
approach developed here also allows to prove the finite time extinction of
solutions in dimensions , which is another main result of this
work. Keywords: stochastic diffusion equation, Brownian motion, bounded
variation, convex functions, bounded variation flow
The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation
One proves the -theorem for mild solutions to a nondegenerate, nonlinear
Fokker-Planck equation and under appropriate hypotheses on
and the convergence in ,
, respectively, for some of the solution
to an equilibrium state of the equation for a large set of nonnegative
initial data in . These results are new in the literature on nonlinear
Fokker-Planck equations arising in the mean field theory and are also relevant
to the theory of stochastic differential equations. As a matter of fact, by the
above convergence result, it follows that the solution to the McKean-Vlasov
stochastic differential equation corresponding to (1), which is a nonlinear
distorted Brownian motion, has this equilibrium state as its unique invariant
measure. Keywords: Fokker-Planck equation, -accretive operator, probability
density, Lyapunov function, -theorem, McKean-Vlasov stochastic differential
equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject
Classification: 35B40, 35Q84, 60H10
- …