Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

Wick calculus for nonlinear Gaussian functionals

This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variables.     

Averaging in parabolic systems,subjected to weakly dependent random perturbations. The L1-approach

One considers an averaging method in equations of parabolic type, situated under the action of centered, weakly dependent random perturbations so that their integrals, normalized in an appropriate manner, satisfy S. N. Bernshtein's exponential estimate. For normalized fluctuations of the solution of the initial equation relative to the solution of the averaged equation, which turns out to be deterministic, one has established S. N. Bernshtein's exponential estimates. On the basis of the obtained inequalities, for an arbitrary prescribed confidence level, one can indicate a confidence band, whose bounds are determined by the solving of the averaged equation, which contains the solution of the initial problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 167–172, February, 1991.     

Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: approximate invariant measures

Systems of Wick stochastic differential equations are studied. Using an estimate on the Wick product we apply Picard iteration to prove a general existence and uniqueness theorem for systems of Wick stochastic differential equations. We also show the solution is stable with respect to perturbations of the noise. This result is used to show that the solution of a linear system of Wick stochastic differential equations driven by smoothed Brownian motion tends to the solution of the corresponding It equation as the smoothed process tends to Brownian motion     

An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ? ^q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ? ^q (𝒫) for some q ≥ 1.     

Singular homogenization with stationary in time and periodic in space coefficients

We study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential, in a general stationary and mixing random media, which is periodic in space. It is shown that a solution of the corresponding Cauchy problem converges in law to a solution of a limit stochastic PDE.     

On an Optimal Filtration Problem for One-Dimensional Diffusion Processes

We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.     

On unique solvability of one nonlinear nonlocal with respect to the solution gradient nonstationary problem

We consider a parabolic equation whose space operator is a product of a nonlinear bounded function which depends on a nonlocal characteristic with respect to a solution gradient and a strongly monotone potential operator. We prove the existence and uniqueness of the solution in the class of the vector-valued functions with values in the Sobolev space.     

On unbiased stochastic Navier–Stokes equations

A random perturbation of a deterministic Navier?CStokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term ${u{\nabla}u}$ . This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier?CStokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron?CMartin version of the Wiener chaos expansion. It is shown that the generalized solution is a Markov process and scales effectively by Catalan numbers.     

Random evolution inclusions of the subdifferential type

In this paper, we consider random evolution inclusion of the subdifferential type with a convex valued perturbation and we establish the existence of a random strong solution. Two examples, the first a nonlinear random parabolic partial differential inclusion and the second a random differential variational inequality, are also worked out in detail     

Initial trace for a doubly nonlinear parabolic equation

This paper studies the Cauchy problem for a doubly nonlinear parabolic equation. The main result shows that if there is a nonnegative solution of the Cauchy problem, then the initial trace of the solution is uniquely given as a nonnegative Borel measure satisfying an exponential growth condition. This extends the known result for the heat equation to the nonlinear case.     

On a free-boundary problem arising in the study of flame propagation

The existence of a weak solution of a two-dimensional non-stationary free-boundary problem related to flame propagation is established. The main feature of the problem is that the nonlinear coupling of the two parabolic equations occur on the free boundary and has exponential growth.     

ASYMPTOTIC BEHAVIOR FOR GENERALIZED GINZBURG-LANDAU POPULATION EQUATION WITH STOCHASTIC PERTURBATION

In this paper, we are devoted to the asymptotic behavior for a nonlinear parabolic type equation of higher order with additive white noise. We focus on the Ginzburg-Landau population equation perturbed with additive noise. Firstly, we show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. And then, it is proved that under some growth conditions on the nonlinear term, this stochastic equation has a compact random attractor, which has a finite Hausdorff dimension.     

Slightly compressible and immiscible two-phase flow in porous media

We describe the flow of two compressible phases in a porous medium. We consider the case of slightly compressible phases for which the density of each phase follows an exponential law with a small compressibility factor. A nonlinear parabolic system including quadratic velocity terms is derived to describe compressible and immiscible two-phase flow in porous media. In one-dimensional space, we establish the existence and uniqueness of a local strong solution for the regularized system. We show also that the saturation is physically admissible. We describe the asymptotic behavior of the solutions when the compressibility factor goes to zero.     

Translated Brownian Motions and Associated Wick Products

Abstract

The concept of Wick product is strongly related to the underlying Brownian motion we have fixed on the probability space. Via the Girsanov's theorem we construct a family of new Brownian motions, obtained as translations of the original one, and to each of them we associate a Wick product. This produces a family of Wick products, named γ-Wick products, parameterized by the performed translations. We aim to describe this family of products. We also define a new family of stochastic integrals, which are related in a natural way to the γ-Wick products.     

Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources

This paper investigates the global existence of the nonnegative solution and the finite time blow-up of solutions of nonlinear parabolic equation with a more complicated source term, which is a product of localized source, local source, and weight function; we also study the blow-up rate of solution to this problem.     

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of a chance-constrained new product risk model with multiple customer classes

We consider the non-convex problem of minimizing a linear deterministic cost objective subject to a probabilistic requirement on a nonlinear multivariate stochastic expression attaining, or exceeding a given threshold. The stochastic expression represents the output of a noisy system featuring the product of mutually-independent, uniform random parameters each raised to a linear function of one of the decision vector’s constituent variables. We prove a connection to (i) the probability measure on the superposition of a finite collection of uncorrelated exponential random variables, and (ii) an entropy-like affine function. Then, we determine special cases for which the optimal solution exists in closed-form, or is accessible via sequential linear programming. These special cases inspire the design of a gradient-based heuristic procedure that guarantees a feasible solution for instances failing to meet any of the special case conditions. The application motivating our study is a consumer goods firm seeking to cost-effectively manage a certain aspect of its new product risk. We test our heuristic on a real problem and compare its overall performance to that of an asymptotically optimal Monte-Carlo-based method called sample average approximation. Numerical experimentation on synthetic problem instances sheds light on the interplay between the optimal cost and various parameters including the probabilistic requirement and the required threshold.