Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

Stochastic Differential Games with Multiple Modes and Applications to Portfolio Optimization

Abstract

We study a zero-sum stochastic differential game with multiple modes. The state of the system is governed by “controlled switching” diffusion processes. Under certain conditions, we show that the value functions of this game are unique viscosity solutions of the appropriate Hamilton–Jacobi–Isaac' system of equations. We apply our results to the analysis of a portfolio optimization problem where the investor is playing against the market and wishes to maximize his terminal utility. We show that the maximum terminal utility functions are unique viscosity solutions of the corresponding Hamilton–Jacobi–Isaac' system of equations.     

Stochastic differential games with reflection and related obstacle problems for Isaacs equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L ^p -distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)     

Stochastic scalar conservation laws

We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L¹ contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.     

Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? ⁿ . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.     

Utility maximization under g*-expectation

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp?-utility and the power utility, when g(t, z) = β_t|z|² + γ_tz.     

Elliptic gradient constraint problem

We study the existence and regularity of gradient constraint problem. It arises in elastoplasticity and finance. First, we consider linear double obstacle problem which comes from viscosity solution to Hamilton–Jacobi equation and find the solution has C^1,α regularity by estimating Campanato-type integral oscillation. Then, by perturbation method and fixed point theorem in C^1,α space, we prove the existence of C^1,α solution.     

Three-dimensional stochastic vortex flows

We discuss the stochastic structure of the Navier–Stokes flow in ?³ and prove that it can be approximated by means of a finite-dimensional stochastic process. Such a process reduces to an algorithm already discussed in Reference 4 for the Euler case, when the viscosity coefficient vanishes.     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x| ^p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x| ^p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.     

Non-random Invariant Sets for Some Systems of Parabolic Stochastic Partial Differential Equations

Abstract

In this article we investigate a class of non-autonomous, semilinear, parabolic systems of stochastic partial differential equations defined on a smooth, bounded domain 𝒪 ? ? ⁿ and driven by an infinite-dimensional noise defined from an L ²(𝒪)-valued Wiener process; in the general case the noise can be colored relative to the space variable and white relative to the time variable. We first prove the existence and the uniqueness of a solution under very general hypotheses, and then establish the existence of invariant sets along with the validity of comparison principles under more restrictive conditions; the main ingredients in the proofs of these results consist of a new proposition concerning Wong–Zakaï approximations and of the adaptation of the theory of invariant sets developed for deterministic systems. We also illustrate our results by means of several examples such as certain stochastic systems of Lotka–Volterra and Landau–Ginzburg equations that fall naturally within the scope of our theory.     

Invariant manifolds for weak solutions to stochastic equations

Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C ² submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: any weak solution, which is viable in a finite dimensional C ² submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves. Received: 15 April 1999 / Revised version: 4 February 2000 / Published online: 18 September 2000     

Stepanov‐like doubly weighted pseudo almost automorphic processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion

This paper investigates the properties of the p‐mean Stepanov‐like doubly weighted pseudo almost automorphic (S^pDWPAA) processes and its application to Sobolev‐type stochastic differential equations driven by G‐Brownian motion. We firstly prove the equivalent relation between the S^pDWPAA and Stepanov‐like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for S^pDWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the S^p DWPAA solution for a class of nonlinear Sobolev‐type stochastic differential equations driven by G‐Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.     

Convergence of a numerical scheme for SPDEs with correlated noise and global Lipschitz coefficients

The aim of this paper is to investigate the pathwise numerical solution of semilinear parabolic stochastic partial differential equations (SPDEs) with colored noise instead of the usual space–time white noise. We estimate the numerical solution in the L^∞ topology by a method that takes advantages of the smoothing effect of the dominant linear operator. We consider the case the covariance operator of the forcing does not necessarily commute with the linear operator of the SPDE because of the fact that the Brownian motions are not necessarily independent. We show convergence of this method, and numerical examples give insight into the reliability of the theoretical study. Copyright © 2015 John Wiley & Sons, Ltd.     

American option valuation in a stochastic volatility model with transaction costs

In the present paper we analyse the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton–Jacobi–Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the utility function, we shall provide a numerical example illustrating how American options prices can be computed in the present modelling framework.     

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C ^{1, α} on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C ^{2, α} on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.     

Positive and unbounded solution of stochastic delayed evolution equations

This article is concerned with the blowup phenomenon of stochastic delayed evolution equations. We first establish the sufficient condition to ensure the existence of a unique nonnegative solution of stochastic parabolic equations. Then the problem of blow-up solutions in mean L^q-norm, q ? 1, in a finite time is considered. The main aim in this article is to investigate the effect of time delay and stochastic term. A new result shows that the stochastic delayed term can induce singularities.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

Multi‐valued backward stochastic differential equations driven by G‐Brownian motion and its applications

In this paper, we prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by G‐Brownian motion with subdifferential operator by means of the Moreau–Yosida approximation method. Moreover, we give a probabilistic interpretation for the viscosity solutions of a kind of nonlinear variational inequalities. Copyright © 2017 John Wiley & Sons, Ltd.