Stochastic Differential Equations: Theory and Applications

The following sections are included:

• Preface

• Contents

The theory of strong solutions of Itô equations in Banach spaces is expounded. The results of this theory are applied to the investigation of strongly parabolic Itô partial differential equations.

Complex systems may be subject to various uncertainties. A great effort has been concentrated on predicting the dynamics under uncertainty in initial conditions. In the present work, we consider the well-known Burgers equation with random boundary forcing or with random body forcing. Our goal is to attempt to understand the stochastic Burgers dynamics by predicting or estimating the solution processes in various diagnostic metrics, such as mean length scale, correlation function and mean energy. First, for the linearized model, we observe that the important statistical quantities like mean energy or correlation functions are the same for the two types of random forcing, even though the solutions behave very differently. Second, for the full nonlinear model, we estimate the mean energy for various types of random body forcing, highlighting the different impact on the overall dynamics of space-time white noises, trace class white-in-time and colored-in-space noises, point noises, additive noises or multiplicative noises.

We consider the space-time Wigner transform of the solution of the random Schrödinger equation in the white noise limit and for high frequencies. We analyze in particular the strong lateral diversity limit in which the space-time Wigner transform becomes weakly deterministic. We also show how to use these asymptotic results in broadband array imaging in random media.

We prove the global existence and uniqueness of solutions both in the energy space and in the space of square integrable functions for a Korteweg-de Vries equation with noise. The noise is multiplicative, white in time, and is the multiplication by the solution of a homogeneous noise in the space variable.

We prove existence and uniqueness of a mild global solution to the Cauchy problem for the stochastic fractional Burgers equation on an interval with multiplicative space time white noise and periodic boundary conditions when the power of the Laplace operator is between 3/4 and 1.

We consider the problem of an executive who receives call options as compensation. He can influence the mean return of the stock with his effort, and the level of volatility through his choice of projects. The executive wants to select the optimal effort and choice of projects to maximize the expected utility from the call option minus the disutility associated with the effort. We model this as a stochastic control problem, and present an analytical solution. In this framework, we also introduce the problem of the company that simultaneously wants to minimize overtime volatility and maximize final expected value of the price of the stock. We characterize (and compute numerically for the logarithmic case) the optimal strike price the company should choose. When the executive can affect the mean return of the stock, we find that options should be granted out-of-the-money. In a context of perfect information, the executive would not be able to affect the mean return of the stock independently of the volatility, and it would be optimal to grant options in-the-money. The mathematical techniques are those of stochastic control, convex analysis, and martingale theory.

The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process whose coeffcients are locally Lipschitz functions with super linear growth. It is assumed that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of the diffusion matrix is not required.

In this paper, the convergence analysis of a class of weak approximations of solutions of stochastic differential equations is presented. This class includes recent approximations such as Kusuoka's moment similar families method and the Lyons-Victoir cubature of Wiener Space approach. We show that the rate of convergence depends intrinsically on the smoothness of the chosen test function. For smooth functions (the required degree of smoothness depends on the order of the approximation), an equidistant partition of the time interval on which the approximation is sought is optimal. For functions that are less smooth (for example Lipschitz functions), the rate of convergence decays and the optimal partition is no longer equidistant. Our analysis rests upon Kusuoka-Stroock's results on the smoothness of the distribution of the solution of a stochastic differential equation. Finally the results are applied to the numerical solution of the filtering problem.

We prove that solutions of stochastic differential equations driven by fractional Brownian motion for H > 1/2 define flows of homeomorphisms on ℝ^d.

A 3D stochastic Navier-Stokes equation with a suitable non degenerate additive noise is considered. The regularity in the initial conditions of every Markov transition kernel associated to the equation is studied by a simple direct approach. A by-product of the technique is the equivalence of all transition probabilities associated to every Markov transition kernel.

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of implicit Euler approximations is estimated under strong monotonicity and Lipschitz conditions. The results are applied to a class of quasilinear stochastic PDEs of parabolic type.

The maximum principle for SPDEs is established in multidimensional C¹ domains. An application is given to proving the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs.

We present a review of results concerning one example of parameter estimation problem, which is the same time regular and singular depending on the chosen asymptotics. The observed process satisfes the linear stochastic differential equation with the delayed trend coeffcient and we are interested in the estimation of this delay. The trend coeffcient depends on the unknown parameter (delay) as smooth as the Wiener process depends on time, i.e., we have a continuous but not differentiable w.r.t. parameter statistical model. We study the asymptotic behavior of the estimators and tests in two asymptotics: the first one corresponds to the small noise perturbation, when the diffusion coeffcient tends to zero and the second is large samples limit, when the time of observation tends to infinity. We show that the first asymptotics corresponds well to regular statistical experiments situation and the second one is typical for non regular (discontinuous type) statistical models.

We study the Cauchy-Dirichlet problem in a smooth bounded domain for linear parabolic integro-differential equations. Suffcient conditions are derived under which the problem has a unique solution in weighted Sobolev classes. The result can be used in the regularity analysis of certain functionals arising in the theory of Markov processes.

Consider a stochastic differential equation in a Hilbert space H, with the associated differential operator K₀ determined by the coeffcients in the equation, and the corresponding Markov semigroup P_t. The paper investigates the problem of constructing a core for the (weak) generator K of P_t in the space of real, uniformly continuous, and bounded functions on H, and studies the relation between K and K₀. The paper proposes an approach based on the concept of a strict solution of the corresponding Kolmogorov equation: a suitable assumption of existence of strict solutions imply good results on the core, and the assumption is verifed in an SPDE example.

The following sections are included:

• Author Index

• Subject Index