Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

Given a d -dimensional Wiener process W , with its natural filtration F t , a F T -measurable random variable ξ in R , a bounded measure x on R , and an adapted process ( s , y , z ) M h ( s , y , z ), we consider the following BSDE: Y t = ξ + Z t T h ( s , Y s , Z s ) d s + Z R ( L T a ( Y ) m L t a ( Y )) x (d a ) m Z t T Z s d W s for 0 h t h T . Here L t a ( Y ) stands for the local time of Y at level a . For h =0, we establish the existence and the uniqueness of the processes ( Y , Z ), and if h is continuous with linear growth we establish the existence of a solution. We prove limit theorems for solutions of backward stochastic differential equations of the above form. Those limit theorems permit us to deduce that any solution of that equation is the limit, in a strong sense, of a sequence of semi-martingales, which are solutions of ordinary BSDEs of the form Y t = ξ + Z t T f ( Y s ) Z s 2 d s m Z t T Z s d W s . A comparison theorem for BSDEs involving measures is discussed. As an application we obtain, with the help of the connection between BSDE and PDE, some corresponding limit theorems for a class of singular non-linear PDEs and a new probabilistic proof of the comparison theorem for PDEs.

Oscillations in quasilinear stochastic hereditary systems and their application to specific problems are investigated. The algorithm of transition to a standard system with the help of the averaging method with small parameter in the presence of fast time is justified.

The hypoellipticity of degenerate parabolic differential operators

The systems of nonlinear differential equations represent the mathematical model of several general systems in biological, physical and social sciences [1]. One of the most versatile techniques in the theory of nonlinear differential systems is the second method of Lyapunov [1,2]. The notion of vector Lyapunov functions, together with the theory of systems of differential inequalities, provides a very general comparison principle by means of which a number of qualitative properties of solutions of deterministic functional differential systems are studied in a unified way [3]. It is natural to expect such an extension to stochastic functional differential systems. The obtained comparison principles to stochastic functional differential systems have been utilized to study the stability behavior of hereditary stochastic general systems.

The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system has an asymptotically stable equilibrium point, random effects can cause the trajectories of the system to leave any bounded domain with probability one. In this paper we consider the effect of small random perturbations of the type referred to as Gaussian white noise, on a (deterministic) dynamical system $\dot x = b(x)$. The vector $x(t)$ then becomes a stochastic process $x_\varepsilon (t)$ which satisfies the stochastic differential equation $dx_\varepsilon = b(x_\varepsilon )dt + \varepsilon \sigma (x_\varepsilon )dw$. Here $w(t)$ is the n dimensional Wiener process (Brownian motion), $b(x)$ is a vector field, $\sigma (x)$ is the diffusion matrix and $\varepsilon \ne 0$ is a small real parameter. We give the first complete formal solution of the following problem originally posed by Kolmogorov: Find the asymptotic expansion in $\varepsilon $ of (i) the probability distribution of the points on the boundary of a domain, where trajectories of the perturbed system first exit, and (ii) of the expected exit times. Our method is to relate the solutions of the above problems to the solutions of various singularly perturbed elliptic boundary value problems with turning points, whose solutions are then constructed asymptotically.

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

Here w(t) is the «-dimensional Wiener process (Brownian motion), b(x) is a vector field, o(x) is the diffusion matrix and e ¥= 0 is a small real parameter. The cumulative effect of even very small random perturbations may be considerable after sufficiently long times, so that even if the deterministic dynamical system has an asymptotically stable equilibrium point, the trajectories of the system will leave any compact domain with probability one. The following problem was posed by Kolmogorov: determine the probability distribution of the points on the boundary where trajectories exit, at the first time of their exit from a compact domain, as well as the expected exit times. The random effect may be thought of as a slow diffusion of particles in the deterministic flow field given by b(x), and the results may differ according as particles are diffusing (a) with a flow, (b) across a flow, or (c) against a flow. Results on (a) were first obtained by Levinson [4] , and on (b) by Khasminskii [3] , both of whom used analytical techniques. Problem (c) seems to be the most difficult, and to date only partial results are available (cf. Ventsel and Freidlin [5] and Friedman [1] who used probabalistic methods). Using analytical techniques, we present a full solution of this problem for flows which are essentially gradients of a potential (as well as certain more general flows). Let f]l be a compact domain in R with a smooth boundary 3£2. Let a(j(x) = ie(o(x)a*(x))//-, be strictly positive definite in £2, b(x) = (bv b2,..., bn)9 and let u€(x) be the solution of the Dirichlet problem

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly.

Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

This paper studies a class of variational inequalities with degenerate parabolic operators and symmetric structure, which is an extension of the parabolic equation in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

We establish existence theorems for weak solutions of boundary value problems associated with degenerate parabolic operators.

Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects

Despite the intensive work done on developing numerical schemes for stochastic differential equations, stability analysis of these schemes when applied to stochastic differential systems is still under premature stage. Motivated by the work of Saito and Mitsui in [11], we investigate mean square stability for a class of weak second order Runge‐Kutta schemes applied to 2‐dimensional stochastic differential systems. Stability criteria will be established, and numerical examples in support of the theoretical analysis will be provided. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, a suite of theoretic tools is provided for discontinuous control design and finite-time stability analysis of a class of stochastic differential systems. The notion of Filippov&#x0027;s solutions for stochastic differential systems is proposed, and the corresponding solution existence problem is explored. The classical It&#x00F4; differentiation formula is generalized for quasi-<inline-formula><tex-math notation="LaTeX">$C_{0}^{2}(\mathbb {R}^{n},\mathbb {R})$</tex-math></inline-formula>-class functions along Filippov&#x0027;s solutions of stochastic differential systems, and two involved set-valued stochastic integrals are introduced with a study on their properties. Some finite-time stability results of stochastic differential systems are revealed with Filippov&#x0027;s solutions, and one of them is applied to neural synchronization, together with case simulations.