Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses 2m-1$2m-1$ random variables for one step in the m-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.     

Superprocesses arising from interactive stochastic flows

In this paper, we give a unified construction for superprocesses with dependent spatial motion constructed by Dawson, Li, Wang and superprocesses of stochastic flows constructed by Ma and Xiang. Furthermore, we also give some examples and rescaled limits of the new class of superprocesses.     

Some approximations of stochastic θ-integrals

In this paper, we consider problems of approximation of stochastic θ-integrals (θ)∫ ₀ ^t f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑ _k=1 ^p f_n(B _n ^θ (t_k-1))[B _n ^θ (t_k)-B _n ^θ (t_k-1], where the sequences {f_n,n∈∕#x007D; and {[B _n ^θ ,n∈∕} are convolution-type approximations of the functionf and Brownian motionB. Belorussian State University, F. Skoryna ave. 4, 220050 Minsk, Belorus. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 248–256, April–June, 1999. Translated by V. Mackevičius     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework.     

Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility

We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

Method of regularization for second-order stochastic ∗

The paper is concerned with the strong solution of secondorder stochastic evolution equations in a Hilbert space. We introduce the method of regularization to prove the existence, uniqueness of strong solution for such equations without the usual coercivity assumption. The result is applied to stochastic wave and plate equations to yield the existence of a unique strong solution for each of such problems arising from physical application.     

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.     

Multitime-scale singularly perturbed linear stochastic systems ∗

By applying diagonalization transformation, generalized variation of constants formula and theory of differential inequalities,the mean square convergence of solution process of a shingularly perturbed linear stochastic differential system of Itô-type is investigated. Moreover, slow and fast modes decomposition provides an auxiliary decoupled system whose solution processes are incorporated in approximating the solution processes of the original system     

Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables

Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by (1) $$\mathop {\min }\limits_{x \in D} F(x)withF(x): = Ef(\omega ,x).$$ Here,f=f(ω,x) is a function on ? ^r depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ? ^r . Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.     

Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire Itô’s formula

In this paper, the mean-square exponential stability is investigated for multi-linked stochastic delayed complex networks with stochastic hybrid impulses. Distinct from the existing literature, we study the MSDCNs on the basis of the multi-linked stochastic functional differential equations that consider the impact of a certain past interval on the present. Moreover, the stochastic hybrid impulses we discuss possess stochastic impulsive moments and impulsive gain, which make the impulses fit better to the real-world demands for control. Also, a novel concept of average stochastic impulsive gain is proposed to measure the intensity of the stochastic hybrid impulses. By the use of Dupire Itô’s formula, based on Lyapunov method, graph theory and stochastic analysis techniques, two sufficient criteria for the mean-square exponential stability are derived, which are closely related to average stochastic impulsive gain, stochastic disturbance strength as well as the topological structure of the network itself. Finally, an application about neural networks is discussed and corresponding numerical example is presented to demonstrate the feasibility and effectiveness of the theoretical results.     

Observability estimate for stochastic Schrödinger equations

In this Note, we present an observability estimate for stochastic Schrödinger equations with nonsmooth lower order terms. The desired inequality is derived by a global Carleman estimate which is based on a fundamental weighted identity for stochastic Schrödinger-like operator. As an interesting byproduct, starting from this identity, one can deduce all the known controllability/observability results for several stochastic and deterministic partial differential equations that are derived before via Carleman estimate in the literature.     

Well-posedness of the stochastic KdV–Burgers equation

We are interested in rigorously proving the invariance of white noise under the flow of a stochastic KdV–Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by a fractional spatial derivative), we establish (almost sure) local well-posedness of the stochastic KdV–Burgers equation with white noise as initial data. Next we observe that spatial white noise is invariant under the projection of this system to the first N>0$N>0$ modes of the trigonometric basis. Finally, we prove a global well-posedness result under an additional smoothing of the noise.     

The attractor of the stochastic generalized Ginzburg-Landau equation

The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system.Then we prove the random system possesses a global random attractor in H_0~1.