Viscosity Solution

The rheological behaviour of periodate oxidized potato starches has been investigated in both dilute and concentrated aqueous dispersions. On heating, low viscosity solutions were obtained at low concentrations and viscoelastic gels at high concentrations. During the gelation process, two types of gel structure were distinguished. First a particle gel is formed, and on continued heating and shearing, deformation and disruption

In this article we study a decoupled forward backward stochastic differential equation (FBSDE) and the associated system of partial integro-differential obstacle problems, in a flexible Markovian set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component X interacting with a continuous-time Markov chain – like component N. The jump process N defines the so-called regime of the coefficients of X, when...

Given a bounded C2 domain G ‰ Rm and functions g 2 C(@G;R) and h 2 C( " G;R n f0g), let u denote the unique viscosity solution to the equation ¡2¢1u = h in G with boundary data g. We provide a representation for u as the value of a two-player zero-sum stochastic difierential game. AMS 2000 subject classiflcations:

Given a bounded $\mathcaligr{C}^2$ domain $G\subset{\mathbb{R}}^m$, functions $g\in\mathcaligr{C}(\partial G,{\mathbb{R}})$ and $h\in\mathcaligr {C}(\bar{G},{\mathbb{R}}\setminus\{0\})$, let $u$ denote the unique viscosity solution to the equation $-2\Delta_{\infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game.

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.

This paper is a survey on some recent aspects and developments in stochastic control. We discuss the two main historical approaches, Bellman's optimality principle and Pontryagin's maximum principle, and their modern exposition with viscosity solutions and backward stochastic differential equations. Some original proofs are presented in a unifying context including degenerate singular control problems. We emphasize key results on characterization of optimal control for diffusion processes, with a view towards applications. Some examples in finance are detailed with their explicit solutions. We also discuss numerical issues and open questions.

The fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Levy processes. Essentially there are three approaches in use that is Levy models, Monte Carlo and Partial integro-differential equations. Out of these three approaches in this paper we will focus in partial integro-differential equations. There is a large supply of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense. The contribution of this paper is to analyze weak solutions the Kolmogorov back ward equations which are related to prices of European options in (time-inhomogeneous) Levy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman-K...

In this paper, we deal with a class of backward doubly stochastic differential equations (BDSDEs, in short) involving subdifferential operator of a convex function and driven by Teugels martingales associated with a L\'evy process. We show the existence and uniqueness result by means of Yosida approximation. As an application, we give the existence of stochastic viscosity solution for a class of multivalued stochastic partial differential-integral equations (MSPIDEs, in short).

The paper concerns the infinite-dimensional Hamilton–Jacobi-Bellman equation related to an optimal control problem regulated by a linear transport equation with boundary control.
A suitable viscosity solution approach is needed in view of the presence of the unbounded control-related term in the state equation in the Hilbert setting. An existence-and-uniqueness result is obtained.

We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of L p -norms and asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows.