Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

Continuous weak approximation for stochastic differential equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

Runge-Kutta methods for jump-diffusion differential equations

In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

Discretization and simulation of stochastic differential equations

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.     

An adaptive scheme for the approximation of dissipative systems

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, a regular explicit Euler scheme–with constant or decreasing step–may explode and implicit Euler schemes are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.     

Quantum and non-causal stochastic calculus

Summary The quantum stochastic calculus initiated by Hudson and Parthasarathy, and the non-causal stochastic calculus originating with the papers of Hitsuda and Skorohod, are two potent extensions of the Itô calculus, currently enjoying intensive development. The former provides a quantum probabilistic extension of Schrödinger's equation, enabling the construction of a Markov process for a quantum dynamical semigroup. The latter allows the treatment of stochastic differential equations which involve terms which anticipate the future. In this paper the close relationship between these theories is displayed, and a noncausal quantum stochastic calculus, already in demand from physics, is described.     

Three-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss three-stage stochastic Runge–Kutta (SRK) methods with strong order 1.0 for a strong solution of Stratonovich stochastic differential equations (SDEs). Higher deterministic order is considered. Two methods, a three-stage explicit (E3) method and a three-stage semi-implicit (SI3) method, are constructed in this paper. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of several standard test problems.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

Simplified order 4.0 weak Taylor schemes for additive noise

Following Kloeden and Platen [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992] Taylor schemes are considered here as the starting point to obtain simplified Taylor schemes replacing the multiple integrals by simpler variables. The conditions that a group of variables has to fulfill so that the new scheme reaches weak-order 4.0 in the additive noise case are given explicitly, as well as the way to find such groups. For a particular selection, a pair of stochastic schemes with order 4.0 in the weak sense, correcting the one proposed in [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992], is obtained.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

The role of coefficients of a general SPDE on the stability and convergence of a finite difference method

In this paper for the approximate solution of stochastic partial differential equations (SPDEs) of Itô-type, the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed. The finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported. We prove the consistency and stability of these methods and investigate the influence of the multiplier (particularly multiplier of the random noise) in mean square stability. From stochastic version of Lax-Richtmyer the convergence of these methods under some conditions are established. Numerical experiments are included to show the efficiency of the methods.     

Forward,backward and symmetric stochastic integration

Summary We define three types of non causal stochastic integrals: forward, backward and symmetric. Our approach consists in approximating the integrator. Two optics are considered: the first one is based on traditional usual stochastic calculus and the second one on Wiener distributions.     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Stochastic conservation laws: Weak-in-time formulation and strong entropy condition

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng and Nualart [8] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.