Support theorem for stochastic variational inequalities

Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities

We study a two-player zero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation of the game turns out to be a double-obstacle quasi-variational inequality, therefore the two obstacles are implicitly given. We prove that the upper and lower value functions coincide, indeed we show, by means of the dynamic programming principle for the stochastic differential game, that they are the unique viscosity solution to the HJBI equation, therefore proving that the game admits a valu…

A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of a unique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles

This paper studies a system of $m$ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is th…

On the variational interpretation of local logarithmic Sobolev inequalities

The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $Γ$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{ö}dinger problem yields a variations interpretation…

Probabilistic Representation of Viscosity Solutions to Quasi-Variational Inequalities with Non-Local Drivers

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

Stochastic differential variational inequalities with applications

In this paper, we introduce and study a stochastic differential variational inequality (SDVI) which consists of a stochastic differential equation and a stochastic variational inequality. We obtain the existence and uniqueness of the solutions for SDVI by using the iteration method and Gronwall's inequality. Moreover, we show the convergence of Euler scheme for solving SDVI under some mild conditions. Finally, we apply the obtained results to solve the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.

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 othe…

Ergodicity of robust switching control and nonlinear system of quasi variational inequalities

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the dif…

Backward stochastic variational inequalities with locally bounded generators

The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: \begin{equation} \left\{\begin{array} {l}-dY_{t}+\partial φ(Y_{t})dt \ni F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\;0\leq t<T \\ Y_{T}=η, \end{array} \right.\end{equation} where $F$ satisfies a local boundedness condition.

Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations

The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous noise. In particular, this provides a characterization of solutions to such SPDE in terms of (generalized) strong solutions. Second, for the one-dimensional stochastic mean curvature flow with normal noise we adapt the notion of stochastic variational inequalities to provide a characterization of solutions previously obtained in a limiting sense only. This so…

McKean-Vlasov随机变分不等式的适定性与混沌传播

Well-posedness and propagation of chaos for McKean-Vlasov stochastic variational inequalities

本文研究一类漂移与扩散系数均依赖时间、状态及其分布的McKean-Vlasov随机变分不等式，在系数满足相应正则性条件下建立强适定性，并首次得到该类不等式的混沌传播结果。

An analytic approach to the ergodic theory of stochastic variational inequalities

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of …