Multidimensional Stochastic Differential Equations Research Articles

Smith’s reduction for random processes and application for stochastic differential equations

Abstract The main focus of this work is a result related to multidimensional stochastic differential equations and one-dimensional SDEs (stochastic differential equations), using the Smith’s π-projection approach. This method is a commonly used technique in mathematics to simplify complex systems by projecting them onto lower-dimensional spaces, which can facilitate analysis and resolution. In the context of multidimensional stochastic processes (i.e., systems involving multiple random variables), extending the Smith’s π-projection approach we mention would likely involve defining a multidimensional version of this projection. This could include extending the concepts of projection and dimension reduction to multiple random variables, as well as analyzing how these projections can simplify multidimensional stochastic differential equations. As an application, we will prove the existence of almost periodic solutions to n-dimensional stochastic differential equations using the Smith projection. In fact, we will generalize the result from the article [M.-A. Boudref and A. Berboucha, Existence of almost periodic solutions of stochastic differential equations with periodic coefficients, Random Oper. Stoch. Equ. 25 2017, 1, 57–70] to dimensions n > 1 {n>1} .

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost O(ϵ−2) for the required target accuracy ϵ. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

Correlated noise enhancement of coherence and fidelity in coupled qubits

ABSTRACT It is generally assumed that environmental noise arising from thermal fluctuations are detrimental to preserving coherence and entanglement in a quantum system. In the simplest sense, dephasing and decoherence are tied to energy fluctuations driven by coupling between the system and the normal modes of the bath. Here, we explore the role of noise correlation in an open-loop model quantum communication system whereby the ‘sender’ and the ‘receiver’ are subject to local environments with various degrees of correlation or anti-correlation. We introduce correlation within the spectral density by solving multidimensional stochastic differential equations and introduce these into the Redfield equations of motion for the system density matrix. We find that correlation can enhance both the fidelity and purity of a maximally entangled (Bell) state. Moreover, by comparing the evolution of different initial Bell states, we show that one can effectively probe the correlation between two local environments. These observations may be useful in the design of high-fidelity quantum gates and communication protocols.

Asymptotic properties for the parameter estimation in stochastic (functional) differential equations with Hölder drift

In this paper, by taking Zvonkin's transformation, we investigate parameter estimation for a class of multidimensional stochastic differential equations with small perturbation parameters in diffusion coefficients, where the drift coefficients not only have an unknown parameter θ but also are Hölder continuous. These processes may enhance the applicability of our results to considerable practical models. Due to the irregular drift, the primary challenge is dealing with the mean square error between an accurate and numerical solution. Under these settings, we demonstrate the consistency and asymptotic normality of error concerning the least squares estimator in probability when stepsize δ → 0 and small parameter ε → 0 simultaneously. Moreover, we extend the results to the case of stochastic functional differential equations.

On Runge–Kutta-type methods for solving multidimensional stochastic differential equations

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A new algorithm for computing path integrals and weak approximation of SDEs inspired by large deviations and Malliavin calculus

The paper gives a novel path integral formula inspired by large deviation theory and Malliavin calculus. The proposed finite-dimensional approximation of integrals on path space will be a new higher-order weak approximation of multidimensional stochastic differential equations where the dominant part of the local expansion is governed by Varadhan's geodesic distance and the correction terms are given as Malliavin weights. An optimal truncation of asymptotic expansion is used to reduce computational effort. Kusuoka's estimate is applied to justify the finite-dimensional approximation of path integrals. An efficient simulation method is provided with the algorithm. Numerical results are shown to verify the effectiveness.

Weak convergence of balanced stochastic Runge–Kutta methods for stochastic differential equations

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.

Euler Approximation for A Class of Singular Multi-Dimensional SDEs Driven by an Additive Fractional Noise

We consider a class of multi-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst index . In particular, the drift coefficient blows up at 0. We first prove that this equation has a unique positive solution, and then propose a semi-implicit Euler approximation scheme for the equation, and finally show that it is also positive, and study its rate of convergence.

Positivity and boundedness preserving numerical scheme for the stochastic epidemic model with square-root diffusion term

This work concerns about the numerical solution to the stochastic epidemic model proposed by Cai et al. [2]. The typical features of the model including the positivity and boundedness of the solution and the presence of the square-root diffusion term make this an interesting and challenging work. By modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving numerical scheme, which is proved to have a strong convergence to the true solution over finite time intervals. We also demonstrate that the principle of this method is applicable to a bunch of popular stochastic differential equation (SDE) models, e.g. the mean-reverting square-root process, an important financial model, and the multi-dimensional SDE SIR epidemic model.

On the stability of mean-field stochastic differential equations with irregular expectation functional

In this paper, we consider multidimensional mean-field stochastic differential equations where the coefficients depend on the law in the form of a Lebesgue integral with respect to the measure of the solution. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application for optimal control of diffusions. Namely, we propose a result on the approximation of the solution associated to a relaxed control.

Convergence rate of the EM algorithm for SDEs with low regular drifts

AbstractIn this paper we employ a Gaussian-type heat kernel estimate to establish Krylov’s estimate and Khasminskii’s estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin’s transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz.

Data-Driven Continuous-Time Modeling of Power System Uncertainty Using a Multi-Dimensional Stochastic Differential Equation

This paper proposes a generalized continuous random process modeling framework for imposing any desired probability distribution, autocorrelation and power spectral density of measured data of power system uncertainty by developing a multi-dimension stochastic differential equation (SDE) together with a quadratic output equation. An algorithm for determining appropriate parameters of the proposed SDE model is presented. The effectiveness of derived random modeling scheme is verified by carrying out comparative simulations using actual time series data of wind speed, solar radiation, PJM dynamic regulation, and wind power. The proposed SDE model can be readily integrated in various electrical component models for reliable operation, stability assessment and control of renewable power systems.

Conditional LQ time-inconsistent Markov-switching stochastic optimal control problem for diffusion with jumps

The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.

Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.

Numerical Spline Method for Simulation of Stochastic Differential Equations systems

نقدم في هذا البحث طريقة شرائحية عددية مع وسيطي تجميع لحل نظم من المعادلات التفاضلية العشوائية متعددة الأبعاد. تمت محاكاة عملية وينر العشوائية متعددة الأبعاد المستمرة مع الزمن كعملية منفصلة، ثم دراسة الاستقرار العشوائي بمتوسط المربعات للطريقة عندما تُطَبقْ لحل نظم من المعادلات التفاضلية العشوائية الخطية من البعد الثاني. تبين الدراسة أن الطريقة تكون مستقرة ومتقاربة بمتوسط المربعات من المرتبة الثالثة عندما يتم تطبيقها لحل نظم من المعادلات التفاضلية العشوائية خطية وغير خطية. وقد تم اختبار فعالية الطريقة المقترحة بحل مسألتي اختبار الأولى خطية والثانية غير خطية، وتشير النتائج العددية إلى فعالية وكفاءة الطريقة الشرائحية المقترحة وأن النتائج الحاصلة جديرة بالاهتمام.

Viability for Coupled SDEs Driven by Fractional Brownian Motion

In this paper, we are concerned with a class of coupled multidimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $$H\in (1/2,1)$$ . Using pathwise approach and based on Perov’s fixed point theorem, we prove that the existence and uniqueness of solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some priori estimates, we obtain a viability result for the stochastic systems under investigation. The sufficient and necessary condition is also an alternative global existence result for the fractional differential equations with restrictions on the state. Finally, by direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution for the coupled stochastic systems under investigation evolves in some particular sets.

Viability property for multi-dimensional stochastic differential equation and its applications to comparison theorem

In this paper, we focus on the problem of viability property for the multi-dimensional stochastic differential equations, where the coefficients of the stochastic differential equations may be random and discontinuous in the time variable. First, we establish a necessary and sufficient condition for viability property with respect to a given non-empty closed convex set. And furthermore, with the help of this result, we also study the multi-dimensional comparison theorems about stochastic differential equations.

{{varvec{L}}}^{{varvec{p}}}-Solutions and Comparison Results for L\xe9vy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

We present a unified approach to L^p-solutions (p > 1) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.