Multidimensional Stochastic Differential Equations Research Articles

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

There is not abstract.

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.

On Λ-Fleming–Viot processes with general frequency-dependent selection

AbstractWe construct a multitype constant-size population model allowing for general selective interactions as well as extreme reproductive events. Our multidimensional model aims for the generality of adaptive dynamics and the tractability of population genetics. It generalises the idea of Krone and Neuhauser [39] and González Casanova and Spanò [29], who represented the selection by allowing individuals to sample several potential parents in the previous generation before choosing the ‘strongest’ one, by allowing individuals to use any rule to choose their parent. The type of the newborn can even not be one of the types of the potential parents, which allows modelling mutations. Via a large population limit, we obtain a generalisation of$\Lambda$-Fleming–Viot processes, with a diffusion term and a general frequency-dependent selection, which allows for non-transitive interactions between the different types present in the population. We provide some properties of these processes related to extinction and fixation events, and give conditions for them to be realised as unique strong solutions of multidimensional stochastic differential equations with jumps. Finally, we illustrate the generality of our model with applications to some classical biological interactions. This framework provides a natural bridge between two of the most prominent modelling frameworks of biological evolution: population genetics and eco-evolutionary models.

General multilevel Monte Carlo methods for pricing discretely monitored Asian options

We describe general multilevel Monte Carlo methods that estimate the price of an Asian option monitored at m fixed dates. For a variety of processes that can be simulated exactly, we prove that, for the same computational cost, our method yields an unbiased estimator with variance lower than the variance of the standard Monte Carlo estimator by a factor of order m. We show how to combine our approach with the Milstein scheme for processes driven by scalar stochastic differential equations, and with the Euler scheme for processes driven by multidimensional stochastic differential equations. Numerical experiments confirm that our method outperforms the conventional Monte Carlo algorithm by a factor proportional to m.

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.

Homeomorphism flows for SDEs driven by G-Brownian motion with non-Lipschitz coefficients

In this paper, we shall prove the homeomorphism property of solutions for multidimensional stochastic differential equations (SDEs) driven by [Formula: see text]-Brownian motion with non-Lipschitz coefficients with respect to the initial values.