Variational inequalities in Hilbert spaces with measures and optimal stopping problems

We study the existence theory for parabolic variational inequalities in weighted $L^2$ 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 $L^2$ 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 $dX(t)={\rm div} [\frac{\nabla X(t)}{|\nabla X(t)|}]dt+X(t)dW(t) in (0,\infty)\times\mathcal{O},$ where $\mathcal{O}$ is a bounded and open domain in $\mathbb{R}^N$, $N\ge 1$, and $W(t)$ is a Wiener process of the form $W(t)=\sum^\infty_{k=1}\mu_k e_k\beta_k(t)$, e_k \in C^2(\bar\mathcal{O})\cap H^1_0(\mathcal{O}), and $\beta_k$, $k\in\mathbb{N}$, 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 $L^2(\mathcal{O})$, 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 $1\le N\le 3$, 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 $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. 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, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10