Class Of Stochastic Differential Equations Research Articles

On dynamic deviation measures and continuous-time portfolio optimization

In this paper, we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency, we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of stochastic differential equations. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure, we formulate a dynamic mean-deviation portfolio optimization problem in a jump-diffusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

The “No Sampling Parameter Estimation (NSPE)” algorithm for stochastic differential equations

The parameter estimation problem in stochastic differential equations (SDEs) has gained much attention in recent years due to their increased applications in the field of pharmaceuticals, ecosystem modeling, and medical data like EKG, blood pressure, and sugar levels. The predictive power of SDEs lie in the choice of parameter values that describe the real data effectively. The classical SDE parameter estimation methods are largely based on likelihood estimation, which is computationally expensive. In this work, we propose a relatively simplified approach based on deterministic nonlinear optimization method which does not require sampling. The results from the suggested No Sampling Parameter Estimation (NSPE) algorithm for selected examples are compared with the data, results from deterministic ODE (ordinary differential equation) model and traditional methods of maximum likelihood estimation (MLE) and generalized method of moments (GMM) for SDEs. The NSPE algorithm is more accurate and reduces computation time significantly when compared to the traditional methods.

Stochastic shadowing analysis of a class of stochastic differential equations

Stochastic shadowing analysis of a class of stochastic differential equations

Fractional stochastic differential equation with discontinuous diffusion

ABSTRACTIn this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.

A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations

A closed-form analytical approximation is derived for the response transition probability density function (PDF) of a certain class of stochastic differential equations with constant drift and nonlinear diffusion coefficients. This is done by resorting to a recently developed Wiener path integral based technique (WPI) in conjunction with a Cauchy–Schwarz inequality treatment of the problem. The derived approximation can be used, due to its analytical nature, as a direct SDE response PDF estimate that requires zero computational effort for its determination. Further, it facilitates an error quantification analysis, which yields an a priori estimate of the anticipated accuracy obtained by applying the approximate methodology. The reliability of the approximation is demonstrated via several engineering mechanics/dynamics related numerical examples pertaining to the stochastic beam bending problem, as well as to the response determination of stochastically excited nonlinear oscillators.

Asymptotical boundedness and stability for stochastic differential equations with delay driven by G-Brownian motion

The purpose of this paper is to discuss a class of stochastic differential equations with delay driven by G -Brownian motion ( G -SDDEs, in short). The asymptotical boundedness and exponential stability of the equations are obtained by means of G -Lyapunov function. An example is proposed to illustrate the theory obtained.

Explosive solutions for stochastic differential equations driven by Lévy processes

The explosive solutions for a class of stochastic differential equations driven by Lévy processes have been considered via Lyapunov method. How the jumping noise affects the explosion has been studied.

Analysis of the Closure Approximation for a Class of Stochastic Differential Equations

AbstractMotivated by the numerical study of spin-boson dynamics in quantum open systems, we present a convergence analysis of the closure approximation for a class of stochastic differential equations. We show that the naive Monte Carlo simulation of the system by direct temporal discretization is not feasible through variance analysis and numerical experiments. We also show that the Wiener chaos expansion exhibits very slow convergence and high computational cost. Though efficient and accurate, the rationale of the moment closure approach remains mysterious. We rigorously prove that the low moments in the moment closure approximation of the considered model are of exponential convergence to the exact result. It is further extended to more general nonlinear problems and applied to the original spin-boson model with similar structure.

On Jump-Diffusive Driving Noise Sources

We study some linear and nonlinear shot noise models where the jumps are drawn from a compound Poisson process with jump sizes following an Erlang-m distribution. We show that the associated Master equation can be written as a spatial mth order partial differential equation without integral term. This differential form is valid for state-dependent Poisson rates and we use it to characterize, via a mean-field approach, the collective dynamics of a large population of pure jump processes interacting via their Poisson rates. We explicitly show that for an appropriate class of interactions, the speed of a tight collective traveling wave behavior can be triggered by the jump size parameter m. As a second application we consider an exceptional class of stochastic differential equations with nonlinear drift, Poisson shot noise and an additional White Gaussian Noise term, for which explicit solutions to the associated Master equation are derived.

Strong convergence for the Euler–Maruyama approximation of stochastic differential equations with discontinuous coefficients

In this paper we study the strong convergence for the Euler–Maruyama approximation of a class of stochastic differential equations whose both drift and diffusion coefficients are possibly discontinuous.

Existence and regularity of a linear nonlocal Fokker–Planck equation with growing drift

The nonlocal Fokker–Planck equations for a class of stochastic differential equations with non-Gaussian α-stable Lévy motion in Euclidean space are studied. The existence and uniqueness of weak solution are obtained with rough drift. The solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded. Moreover, the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity.

Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process

We study exponential integrability properties of the Cox--Ingersoll--Ross (CIR) process and its Euler--Maruyama discretizations with various types of truncation and reflection at $0$. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler--Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.

A powered Gronwall-type inequality and applications to stochastic differential equations

In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.

Rough differential equations with unbounded drift term

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply strong completeness and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin–Wentzell-type large deviation result.

Pointwise weak existence for diffusions associated with degenerate elliptic forms and 2-admissible weights

Using analysis for 2-admissible functions in weighted Sobolev spaces and stochastic calculus for possibly degenerate symmetric elliptic forms, we construct weak solutions to a wide class of stochastic differential equations starting from an explicitly specified subset in Euclidean space. The solutions have typically unbounded and discontinuous drift but may still in some cases start from all points of $\mathbb{R}^d$ and thus in particular from those where the drift terms are infinite. As a consequence of our approach we are able to provide new non-explosion criteria for the unique strong solutions of \cite{Zh}.

A class of stochastic differential equations with pathwise unique solutions

We propose a new method viz., using stochastic partial differential equations to study the pathwise uniqueness of stochastic (ordinary) differential equations. We prove the existence and pathwise uniqueness of a class of stochastic differential equations with coefficients in suitable Hermite-Sobolev class using our approach.

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

From Rough Path Estimates to Multilevel Monte Carlo

New classes of stochastic differential equations can now be studied using rough path theory (see, e.g., [T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, Springer, Berlin, 2007] or [P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014]). In this paper we investigate, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense. Our focus lies on numerical implementations, and more specifically on the savings possible via multilevel methods. Our analysis relies on a subtle combination of pathwise estimates, Gaussian concentration, and multilevel ideas. Numerical examples are given which both illustrate and confirm our findings.

The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments

In this paper we consider stochastic differential equations with piecewise constant arguments. These equations describe hybrid dynamical systems, that is, combinations of continuous and discrete systems. Our aim is to establish the mean square convergence of the Euler–Maruyama approximate solution under quite general conditions, that is, the global Lipschitz condition and the linear growth condition, which guarantee the existence and uniqueness of the true solution. Then we show that the initial equation is exponentially stable in mean square if and only if, for some sufficiently small step-size Δ, the Euler–Maruyama method is exponentially stable in mean square. The stability study does not involve the Lyapunov functions nor functionals. It should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time-dependent delays which can be found in the literature. However, in the convergence analysis of the Euler–Maruyama method of these equations, it is required that the delay functions satisfy Lipschitz continuity condition. In the present paper, the delay functions do not satisfy that condition, so it represents an extension of results from the papers Mao (2003, 2007), using the technique similar to that from the first paper.

Nonlinear Lyapunov criteria for stochastic explosive solutions

This paper deals with the problem of explosive solutions for a class of stochastic differential equations via Lyapunov’s method. We give some sufficient conditions on the existence of explosive solutions for stochastic differential equations driven by Brownian motions.