Stochastic equations of non-negative processes with jumps

We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. Under suitable conditions, the comparison properties of solutions are proved. Those results are applied to construct continuous state branching processes with immigration as strong solutions of stochastic equations.     

Stochastic Volterra equations driven by cylindrical Wiener process

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families. Both authors are supported partially by project “Proyecto Anillo: Laboratorio de Analisis Estocastico; ANESTOC”.     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

Stochastic quasilinear evolution equations in UMD Banach spaces

In this work we prove the existence and uniqueness up to a stopping time for the stochastic counterpart of Tosio Kato's quasilinear evolutions in UMD Banach spaces. These class of evolutions are known to cover a large class of physically important nonlinear partial differential equations. Existence of a unique maximal solution as well as an estimate on the probability of positivity of stopping time is obtained. An example of stochastic Euler and Navier–Stokes equation is also given as an application of abstract theory to concrete models.     

On signed measure valued solutions of stochastic evolution equations

We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier–Stokes equations in vorticity form introduced by Kotelenez.     

Logarithmic derivatives of invariant measure for stochastic differential equations in hilbert spaces

We consider a process X solution of a semilinear stochastic evolution equation in a Hilbert space. Assuming that X has an invariant measure ν, we investigate its regularity properties. Logarithmic derivatives of ν in certain directions, are shown to exist under appropriate conditions on the nonlinear term in the equation. A set of directions of differentiability for ν is explicitly described in terms of the coefficients of the equation. In some cases, logarithmic derivatives are represented as conditional expectations of random variables related to an appropriate stationary process. An application to a system of stochastic partial differential equations in one space variable is given     

Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

Large-time behavior of solutions to stochastic evolution equations driven by two-sided regular cylindrical Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of a limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation perturbed by the two-sided Rosenblatt process.     

Controllability of neutral stochastic evolution equations driven by fractional brownian motion

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ∈(1/2,1) in a Hilbert space.We employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator theory.An example is provided to illustrate the effectiveness of the proposed result.     

Stochastic delay differential equations with jump reflection: invariant measure

In this paper, we consider a class of multi-dimensional stochastic delay differential equations with jump reflection. Based on existence and uniqueness of the strong solution to equation, we prove that the Markov semigroup generated by the segment process corresponding to the solution admits a unique invariant measure on the Skorohod space when the coefficients of equation satisfy a class of monotone conditions. Finally, we establish a relationship between the regulator and the local time of the solution and discuss a local time property at large time under the stationary setting.     

Right-continuous solutions of systems of stochastic integral equations

Unique solutions are shown to exist for systems of stochastic integral equations which allow right-continuous semimartingales (also known as quasimartingales) as differentials.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360     

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Bernstein processes and Pauli-type equations

We use ideas from a previous paper by the author to construct a Markov Bernstein process, whose probability density is the product of the solutions of the (imaginary time) Schrödinger-equation and its adjoint equation, associated to a class of Pauli-type Hamiltonians. A path integral representation of these solutions is obtained as well as the associated regularised Newton equations.     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Forward-backward stochastic differential equations with Brownian motion and poisson process

1.IntroductionLet(n,Y,{S}tZo,P)beastochasticbasissuchthatAscontainsallp-nullelementsofFand5 =nR .=h,t2o.Wesupposethatthefiltration{R}tZoisgeneratede>0bythefollowingtwOmutuallyindependentProcesses:(i)Ad-dbonsionalstandardBroedanmotion{Bt}tZo;(h)APoissonrandommeasureNonR xZ,whereZCFIisanonemptyopensetequippedwithitsBorelheldB(Z),withcompensatorN(dz,dt)=A(dz)dt,suchthatN(Ax[0,t])=(N--N)(Ax10,t])tZoisamartingaleforallAEB(Z)satisfyingA(A)相似文献   

Stochastic morphological evolution equations

The inadequacy of locally defined set-valued differential equations to describe the evolution of shapes and morphological forms in biology, which are usually neither convex or nondecreasing, was recognised by J.-P. Aubin, who introduced morphological evolution equations, which are essentially nonlocally defined set-valued differential equations with the inclusion vector field also depending on the entire reachable set. This concept is extended here to the stochastic setting of set-valued Itô evolution equations in Hilbert spaces. Due to the nonanticipative nature of Itô calculus, the evolving reachable sets are nonanticipative nonempty closed random sets. The existence of solutions and their dependence on initial data are established. The latter requires the introduction of a time-oriented semi-metric in time-space variables. As a consequence the stochastic morphological evolution equations generate a deterministic nonautonomous dynamical system formulated as a two-parameter semigroup with the complication that the random subsets take values in different spaces at different time instances due to the nonanticipativity requirement. It is also shown how nucleation processes can be handled in this conceptual framework.