A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains

A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Stochastic functional differential equations modelling materials with selective recall

We study a problem in stochastic functional differential equations which, in addition to a standard one-one-parameter noise term involves a random perturbation of the memory. This problem can also be regarded as a first order hyperbolic system of stochastic partial differential equations with given initial data and nonlocal boundary data. Existence and uniqueness of a solution is established and the generator of the associated Markov process is analyzed. Thereafter, for two model problems arising from first- and second-order integro-differential equations suggested by physical applications we establish asymptotic stability in probability of the associated stochastic processes.     

Application of stochastic point processes in mechanics

Stochastic point processes are the mathematical tools relevant to all problems where the phenomena have the nature of a random train of events. Applications may be found in structural dynamics where some stochastic excitations may be adequately idealized as random trains of impulses or general pulses. An example of application in mechanics of materials is the stochastic model of the grain growth processes in polycrystalline nanomaterials. Based on the stochastic differential equations formulation, analysis methods such as the moment equations method or the method of equation for the response probability density are dealt with. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions

We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples.     

On estimating the distribution of optimal traveling salesman tour lengths using heuristics

The traveling salesman problem is an important combinatorial optimization problem due to its significance in academic research and its real world applications. The problem has been extensively studied and much is known about its polyhedral structure and algorithms for exact and heuristic solutions. While most work is concentrated on solving the deterministic version of the problem, there also has been some research on the stochastic TSP. Research on the stochastic TSP has concentrated on asymptotic properties and estimation of the TSP-constant. Not much is, however, known about the probability distribution of the optimal tour length. In this paper, we present some empirical results based on Monte Carlo simulations for the symmetric Euclidean and Rectilinear TSPs. We derive regression equations for predicting the first four moments of the distribution of estimated TSP tour lengths using heuristics. We then show that a Beta distribution gives excellent fits for small to moderate sized TSP problems. We derive regression equations for predicting the parameters of the Beta distribution. Finally we predict the TSP constant using two alternative approaches.     

Stationary probability distributions of some lotka-volterra types of stochastic predation systems

This paper provides exact solutions to the stationary probability distributions in some stochastic predation systems. These are derived by solving the Fokker-Planck equations for:

(i) a generalized stochastic Lotka-Volterra predator-prey system, and

(ii) a generalised stochastic Lotka-Volterra food chain.

In all these systems the growth dynamics of all levels of species are subject to stochastic shocks. Since stationary probability distributions provide the most comprehensive characterization of a stochastic system in a steady state, system stability can be analysed accordingly     

A necessary condition for optimal control of?initial coupled forward-backward stochastic differential equations with?partial information

This paper is concerned with a class of optimal control problems of forward-backward stochastic differential equations. One feature of these problems is that they are in the case of partial information and state equations are coupled at initial time. In terms of a classical convex variational technique, we establish a partial information maximum principle for the foregoing optimization problems. We also work out an example of partial information linear-quadratic optimal control to illustrate the application of the theoretical results; meanwhile, we find a forward-backward stochastic differential filtering equation, which is essentially different from classical forward stochastic filtering equations.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

EXPONENTIAL STABILITY FOR NONLINEAR HYBRID STOCHASTIC PANTOGRAPH EQUATIONS AND NUMERICAL APPROXIMATION

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.     

Distributions for the risk process with a stochastic return on investments

In this paper, we consider a risk model with stochastic return on investments. We mainly discuss the ruin probability, the surplus distribution at the time of ruin and the supremum distribution of the surplus before ruin. We prove some properties for these distributions and derive the integro-differential equations satisfied by them. We present the relation between the ruin probability and the supremum distribution before ruin.     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

Matrix-analytic methods in queuing theory

This review describes the development during the past decade of a number of matrix-analytic methods for the study of a variety of stochastic models, primarily queues but also certain models for dams and inventories. This work originated in the search for algorithmic methods and has led to results that are well-suited for computer implementation. It has also required a reexamination of the theoretical approaches to these stochastic models, which may now be analyzed by purely probabilistic methods rather than by techniques from complex analysis.The author also argues the case for an algorithmic optic on the problems of applied probability and sketches the main steps in the construction of computer codes for the evaluation of stationary distributions of interest. The approach is built on certain nonlinear matrix equations that arise naturally in the study of structured Markov chains which are generalizations of the chains embedded in the classical GI/M/1 and M/G/1 queues.     

Characterization of kernel functions associated with operator algebraic Riccati equations for linear delay systems

In infinite time quadratic control and stochastic filtering problems for linear delay systems, operator algebraic Riccati equations play a very important role. However, since these are abstract operator equations, it is very useful, in analyzing their structure, to be able to characterize the kernel functions associated with the solutions of the operator Riccati equations. The kernel functions are given by the unique solution of a set of coupled differential equations. By comparing these kernel equations with similar ones available in the literature, it is shown that this characterization result is somewhat stronger than previously known results. Possible extentions to systems with control, observation, as well as state delays are also pointed out.     

Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.     

On quantum stochastic differential equations with unbounded coefficients

Summary We prove an existence, uniqueness and unitarity theorem for quantum stochastic differential equations with unbounded coefficients which satisfy an analyticity condition on a common dense invariant domain. This result, applied to the quantum harmonic oscillator, gives a rigorous meaning to a large class of stochastic differential equations that have been considered formally in quantum probability.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107–128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.