Multidimensional Stochastic Differential Equations Research Articles

On the comparison theorem for multi-dimensional stochastic differential equations with jumps

In this paper, we give a necessary and sufficient condition under which the comparison theorem holds for multi-dimensional stochastic differential equations (SDEs) with jumps and for matrix-valued SDEs with jumps.

Nonnegative compartment dynamical system modelling with stochastic differential equations

Compartment models are widely used in various physical sciences and adequately describe many biological phenomena. Elements such as blood, gut, liver and lean tissue are characterized as homogeneous compartments, within which the drug resides for a time, later to transit to another compartment, perhaps recycling or eventually vanishing. We address the issue of compartment dynamical system modelling using multidimensional stochastic differential equations, rather than the classical approach based on the continuous-time Markov chain. Pure-jump processes are employed as perturbation to the deterministic compartmental dynamical system. Unlike with the Brownian motion, noise can be incorporated into both outflows and inter-compartmental flows without violating nonnegativity of the compartmental system, under mild technical conditions. The proposed formulation is easy to simulate, shares various essential properties with the corresponding deterministic ordinary differential equation, such as asymptotic behaviors in mean, steady states and average residence times, and can be made as close to the corresponding diffusion approximation as desired. Asymptotic mean-square stability of the steady state is proved to hold under some assumptions on the model structure. Numerical results are provided to illustrate the effectiveness of our formulation.

Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H>1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann--Stieltjes integral.

L p solutions of finite and infinite time interval BSDEs with non-Lipschitz coefficients

In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) in under weaker assumptions on the coefficients, considering both a finite and an infinite time interval. We establish a general existence and uniqueness result of solutions in to finite and infinite time interval BSDEs with non-Lipschitz coefficients, which includes the corresponding results in Pardoux and Peng [Syst. Control Lett. 14 (1990), pp. 55–61], Mao [Stoch. Proc. Appl. 58 (1995), pp. 281–292], Chen and Wang [Chin. Sci. Bull. 42 (1997), pp. 2379–2383], Constantin [Analele Universităţii din Timişoara, Seria Matematică-Informatică XXXIX (2001), pp. 15–22], Wang and Wang [Chin. J. Appl. Probab. Statist. 19 (2003), pp. 245–251], Chen and Wang [J. Austral. Math. Soc. (Ser. A) 69 (2000), pp. 187–211] and Wang and Huang [Stat. Probability Lett. 79 (2009), pp. 1438–1443] as its particular cases.

Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with lévy noise

We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein–Uhlenbeck process with Levy noise, i.e., for the solution of a linear stochastic differential equation with Levy noise.

Error Exponents for Neyman-Pearson Detection of a Continuous-Time Gaussian Markov Process From Regular or Irregular Samples

This paper addresses the detection of a stochastic process in noise from a finite sample under various sampling schemes. We consider two hypotheses. The noise only hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The signal plus noise hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants and corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by error exponents that are completely characterized. It turns out that they are related to the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks.

Pathwise uniqueness of multi-dimensional stochastic differential equations with Hölder diffusion coefficients

We extend Yamada-Watababe’s criterion [J. Math. Kyoto Univ., 1971, 11: 553–563] on the pathwise uniqueness of one-dimensional stochastic differential equations to a special class of multi-dimensional stochastic differential equations.

Multi-dimensional backward stochastic differential equations with one reflecting lower barrier of Itô diffusion type

This paper investigates a class of multi-dimensional stochastic differential equations with one reflecting lower barrier (RBSDEs in short), where the random obstacle is described as an Itô diffusion type of stochastic differential equation. The existence and uniqueness results for adapted solutions to such RBSDEs are established based on a penalization scheme and some higher moment estimates for solutions to penalized BSDEs under the Lipschitz condition and a higher moment condition on the coefficients. Finally, two examples are given to illustrate our theory and their applications.

Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities

Consider a multidimensional stochastic differential equation of the form \(X_{t}=x+\int_{0}^{t}b(X_{s-})\,ds+\int_{0}^{t}f(X_{s-})\,dZ_{s}\), where (Zs)s≥0 is a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to the time step for the difference of these densities.

On Existence and Uniqueness of Stationary Distributions for Stochastic Delay Differential Equations with Positivity Constraints

Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering. There is interest in understanding what effect noise has on the behavior of such models. Here we consider a multidimensional stochastic delay differential equation with normal reflection as a noisy analogue of a deterministic system with delayed feedback and positivity constraints. We obtain sufficient conditions for existence and uniqueness of stationary distributions for such equations. The results are applied to an example from Internet rate control and a simple biochemical reaction system.

Exact scenario simulation for selected multi-dimensional stochastic processes

Accurate scenario simulation methods for solutions of multi - dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for in- stance, in finance. Various discrete time simulation methods have been devel- oped over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and nu- merical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi- dimensional solutions of stochastic differential equations including Ornstein- Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.

EXISTENCE OF STRONG SOLUTIONS AND UNIQUENESS IN LAW FOR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

In this paper we establish the existence of pathwise solutions and the uniqueness in law for multidimensional stochastic differential equations driven by a multi-dimensional fractional Brownian motion with Hurst parameter H > 1/2.

Participating life insurance policies: an accurate and efficient parallel software for COTS clusters

In this paper we discuss the development of a parallel software for the numerical simulation of Participating Life Insurance Policies in distributed environments. The main computational kernels in the mathematical models for the solution of the problem are multidimensional integrals and stochastic differential equations. The former is solved by means of Monte Carlo method combined with the Antithetic Variates variance reduction technique, while differential equations are approximated via a fully implicit, positivity-preserving, Euler method. The parallelization strategy we adopted relies on the parallelization of Monte Carlo algorithm. We implemented and tested the software on a PC Linux cluster.

ANTICIPATING MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH REFLECTIONS

In this paper, the strong solutions of multidimensional stochastic differential equations with reflecting boundary and with a non-adapted initial random variable are established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of the strong solutions. The results extend the work [8] of Lions and Sznitman with a non-adapted initial random variable.

Laplace approximation of transition densities posed as Brownian expectations

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.

Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.

Automated Generation of Reduced Stochastic Weather Models I: Simultaneous Dimension and Model Reduction for Time Series Analysis

We present a method for simultaneous dimension reduction, model fitting, and metastability analysis of high-dimensional time series. The approach is based on the combination of hidden Markov models (HMMs) with localized principal component analysis (PCA) (which is used to identify the essential dimensions in the form of empirical orthogonal functions (EOFs) for each of the hidden states) and fitting of multidimensional stochastic differential equations (SDEs). This means that the analyzed data is clustered according to differences in essential dimensions and SDE models specific to each of the hidden states. We derive explicit estimators for PCA-SDE model parameters in the case of fixed sequences of HMM states and employ the expectation-maximization algorithm for numerical optimization of HMM-PCA-SDE parameters. We demonstrate the performance of the method by application to historical temperature data in Europe during 1976–2002. In a comparison with the standard SARMA (seasonal autoregressive moving average model) technique for time series analysis, the HMM-PCA-SDE method exhibits better numerical performance and efficiency, especially on high-dimensional data sets and for a 20-dimensional reduced state space. We also compare the results of both models w.r.t. errors of one-day temperature predictions.

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Levy process with a Levy measure f θ(z)dz, and we admit the case ∫ f θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models.

Freidlin–Wentzell's large deviations for homeomorphism flows of non-Lipschitz SDEs

We prove a Freidlin–Wentzell large deviation principle for multi-dimensional stochastic differential equations with non-Lipschitz coefficients and apply it to the Brownian motion on the diffeomorphism group of the disc constructed recently by Airault, Malliavin and Thalmaier.

Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients

In this paper, we study the homeomorphic property of solutions of multi-dimensional stochastic differential equations with non-Lipschitz coefficients.