The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.     

Differential equations driven by rough paths with jumps

We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.     

Solutions of nonlinear impulsive Itô functional-differential equations: Stability by the linear approximation

We study the p-stability (2 ≤ p < ∞) of solutions of nonlinear impulsive Itô functional-differential equations. To this end, we use the stability theory developed for deterministic functional-differential equations. The moment stability of solutions of nonlinear impulsive Itô functional-differential equations is studied with the use of the problem on the admissibility of a pair of spaces for linear impulsive Itô functional-differential equations. We prove assertions similar to traditional theorems on stability by the first approximation.     

Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Itô formula is derived. When a standard Brownian motion is the original semimartingale, classical Itô stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.     

Kinetic energy of the Langevin particle

We compute the kinetic energy of the Langevin particle using different approaches. We build stochastic differential equations that describe this physical quantity based on both the Itô and Stratonovich stochastic integrals. It is shown that the Itô equation possesses a unique solution, whereas the Stratonovich one possesses infinitely many, all but one absent of physical meaning. We discuss how this fact matches with the existent discussion on the Itô vs Stratonovich dilemma and the apparent preference toward the Stratonovich interpretation in the physical literature.     

On the inverse stochastic reconstruction problem

We consider the reconstruction problem in the class of stochastic differential equations of Itô type on the basis of given motion properties that depend only on part of the variables. We determine the set of controls providing necessary and sufficient conditions for the existence of a given integral manifold.     

Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.     

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Itô stochastic integral and solutions of stochastic different equations are obtained.     

The Peano phenomenon for Itô equations

The weak convergence of the measures generated by the solutions of stochastic Itô equations with low diffusion is studied, as the diffusion tends to zero. It is proved that the limiting measure in the presence of the Peano phenomenon for a relevant ordinary differential equation is concentrated on its extreme solutions with definite weights. The formulas for their calculation are given.     

Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus.     

Stability of stochastic partial differential equation

Stochastic partial differential equations such as occur in vibration problems for mechanical structures subjected to random loading are modelled as infinite dimensional stochastic Itô differential equations using a semigroup approach. Sufficient conditions for exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths of the displacement and velocity, are given. Under these same conditions it is shown that the zero solution is pathwise asymptotically stable relative to finite dimensional initial conditions. Illustrative examples are included.     

Stochastic inverse problem with indirect control

In the class of stochastic differential systems of equations of Itô type with indirect control by the first or second derivative, we consider the inverse dynamic problem of constructing a system controller such that a given manifold is an integral manifold of the system. In both cases, the quasi-inversion method is used to construct the whole set of equations of controllers providing the solution of this problem.     

Modified Equations for Stochastic Differential Equations

We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itô SDEs with additive noise, and extensions to other types of equation and approximation are discussed.     

On the Moments of the Modulus of Continuity of Itô Processes

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.     

Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.     

Stochastic Taylor Expansions for Functionals of Diffusion Processes

In this article, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean-square and the mean truncation errors are given.     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

On pathwise super-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. In particular, we provide conditions for solutions to converge to zero at a given rate, which is faster than any exponential rate of decay. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.