Original Stochastic Differential Equations Research Articles

Pathwise Convergent Approximation for the Fractional SDEs

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of overdamped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker--Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker--Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

In this paper, an averaging principle for multidimensional, time dependent, stochastic differential equations (SDEs) driven by fractional Brownian motion and standard Brownian motion was established. We combined the pathwise approach with the Itô stochastic calculus to handle both types of integrals involved and proved that the original SDEs can be approximated by averaged SDEs in the manner of mean square convergence.

A strong order $1/2$ method for multidimensional SDEs with discontinuous drift

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.

On the construction of non-affine jump-diffusion models

ABSTRACTWe describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

Stability of the solution of stochastic differential equation driven by time-changed Lévy noise

This paper studies stabilities of the solution of stochastic differential equations (SDE) driven by time-changed Lévy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for the solution of time-changed SDE to be stable in different senses will be established. The connection between stability of the solution to time-changed SDE and that to corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Lévy noise, where the time-change processes are the inverse of general Lévy subordinators. These results are an important generalization of the results of Q. Wu (2016).

Accuracy of Maximum Likelihood Parameter Estimators for Heston Stochastic Volatility SDE

We study approximate maximum likelihood estimators (MLEs) for the parameters of the widely used Heston Stock price and volatility stochastic differential equations (SDEs). We compute explicit closed form estimators maximizing the discretized log-likelihood of $$N$$ observations recorded at times $$T,2T, \ldots , NT$$ . We compute the asymptotic biases of these parameter estimators for $$T$$ fixed and $$N \rightarrow \infty $$ , as well as the rate at which these biases vanish when $$T \rightarrow 0$$ . We determine asymptotically consistent explicit modifications of these MLEs. For the Heston volatility SDE, we identify a canonical form determined by two canonical parameters $$0 < \omega < 1$$ and $$\zeta > 1/2$$ which are explicit functions of the original SDE parameters. We analyze theoretically the asymptotic distribution of the MLEs and of their consistent modifications, and we outline their concrete speeds of convergence by numerical simulations. We clarify in terms of $$\zeta $$ the precise dichotomy between asymptotic normality and attraction by stable like distributions with heavy tails. We illustrate numerical model fitting for Heston SDEs by two concrete examples, one for daily data and one for intraday data, both with moderate values of $$N$$ .

Terminal-Dependent Statistical Inference for the FBSDEs Models

The original stochastic differential equations (OSDEs) and forward-backward stochastic differential equations (FBSDEs) are often used to model complex dynamic process that arise in financial, ecological, and many other areas. The main difference between OSDEs and FBSDEs is that the latter is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. It is interesting but challenging to estimate FBSDE parameters from noisy data and the terminal condition. However, to the best of our knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. We proposed a nonparametric terminal control variables estimation method to address this problem. The reason why we use the terminal control variables is that the newly proposed inference procedures inherit the terminal-dependent characteristic. Through this new proposed method, the estimators of the functional coefficients of the FBSDEs model are obtained. The asymptotic properties of the estimators are also discussed. Simulation studies show that the proposed method gives satisfying estimates for the FBSDE parameters from noisy data and the terminal condition. A simulation is performed to test the feasibility of our method.

Causal interpretation of stochastic differential equations

We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a L\'evy process, the postintervention distribution is identifiable from the generator of the SDE.

Terminal-Dependent Statistical Inferences for FBSDE

Backward stochastic differential equation (BSDE) has been well studied and widely applied in mathematical finance. The main difference from the original stochastic differential equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which plays key roles in certain financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inference for such model has not been explored in the existing literature. This article proposes two terminal-dependent estimation methods via terminal control variable the integral form models of forward-backward stochastic differential equation (FBSDE). We take these measures because the resulting models contain terminal condition as model variable, and therefore, the corresponding estimators inherit the terminal-dependent characteristic. In this article, the FBSDE is first rewritten as regression versions and then two semi-parametric estimation approaches are proposed. Because of the control variable and integral form, our regression versions are more complex than the classical ones, and the inference methods are somewhat different from which designed for the OSDE. Even so, the statistical properties of the terminal-dependent methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.

Terminal-Dependent Statistical Inference for the Integral Form of FBSDE

Backward Stochastic Differential Equation (BSDE) has been well studied and widely applied. The main difference from the Original Stochastic Differential Equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. This paper is concerned with the statistical inference for the integral form of Forward-Backward Stochastic Differential Equation (FBSDE). The reason why I use its integral form rather than the differential form is that the newly proposed inference procedure inherits the terminal-dependent characteristic. In this paper the FBSDE is first rewritten as a regression version, and then a semiparametric estimation procedure is proposed. Because of the integral form, the newly proposed regression version is more complex than the classical one, and thus the inference methods are somewhat different from those designed for the OSDE. Even so, the statistical properties of the new method are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors of the proposed estimators.

Stochastic processes, slaves and supersymmetry

We extend the work of Tănase-Nicola and Kurchan on the structure of diffusion processes and the associated supersymmetry algebra by examining the responses of a simple statistical system to external disturbances of various kinds. We consider both the stochastic differential equations (SDEs) for the process and the associated diffusion equation. The influence of the disturbances can be understood by augmenting the original SDE with an equation for slave variables. The evolution of the slave variables describes the behaviour of line elements carried along in the stochastic flow. These line elements, together with the associated surface and volume elements constructed from them, provide the basis of the supersymmetry properties of the theory. For ease of visualization, and in order to emphasize a helpful electromagnetic analogy, we work in three dimensions. The results are all generalizable to higher dimensions and can be specialized to one and two dimensions. The electromagnetic analogy is a useful starting point for calculating asymptotic results at low temperature that can be compared with direct numerical evaluations. We also examine the problems that arise in a direct numerical simulation of the stochastic equation together with the slave equations. We pay special attention to the dependence of the slave variable statistics on temperature. We identify in specific models the critical temperature below which the slave variable distribution ceases to have a variance and consider the effect on estimates of susceptibilities.

Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity

General autonomous stochastic differential equations (SDEs) driven by one-dimensional Brownian motion in the Stratonovich sense with a conserved quantity $I(y)$ are considered. Relying on this conserved quantity, an equivalent “skew gradient” (SG) form of original SDEs is constructed. With the aim of constructing conserved numerical methods, direct discrete gradient approaches which approximate directly the SG system lead to two conserved methods. In indirect discrete gradient approaches, we first split the SG system into subsystems which preserve the original conserved quantity, then apply direct discrete gradient approaches to subsystems to produce two composition schemes. The mean-square convergence of order 1 for these four methods depends on the assumptions that the coefficients of SDEs are Lipschitz continuous with bounded second moments along the solution of SDEs, are twice continuous differentiable, and have derivatives that are uniformly bounded. Three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods.

A multiscale probabilistic collocation method for subsurface flow in heterogeneous media

Owing to the spatial variability of the media properties, uncertainty quantification for subsurface flow and solute transport usually requires high‐resolution simulations. In this work, a multiscale probabilistic collocation method (MSPCM) is developed for solving such problems in a computationally efficient manner. The subsurface flow problem is cast in a stochastic framework, and probabilistic collocation strategy is used to represent the original stochastic differential equation. The resulting equations are a set of decoupled deterministic equations with respect to collocation points. A multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh. Coarse‐scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points. The coarse‐scale solution is then obtained by solving a modified coarse formulation that takes into account the fine‐scale heterogeneity. The fine‐scale solution is reconstructed after the coarse‐scale solution is available. Since the PCM and multiscale finite element method are implemented at different levels, the MSPCM inherits their respective advantages, in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid. The performance of the proposed method is demonstrated with numerical examples. The capability of MSPCM in reproducing the probability density functions (PDFs) of head and velocity is investigated. The numerical results show that the MSPCM with proper coarsening level is able to capture small‐scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity.

Quadratic forms for Feynman–Kac semigroups

Some problems in a stochastic setting often involve the need to evaluate the Feynman–Kac formula that follows from models described in terms of stochastic differential equations. Equivalent representations in terms of partial differential equations are also of interest, and these establish the well-known connection between probabilistic and deterministic formulations of these problems. In this Letter, this connection is studied in terms of the quadratic form associated with the Feynman–Kac semigroup. The probability measures that naturally arise in this approach, and thus define how Brownian motion is killed at a specified rate while exiting a set, are interpreted as a random time change of the original stochastic differential equation. Furthermore, since random time changes alter the diffusion coefficients in stochastic differential equations while Girsanov-type measure transformations alter their drift coefficients, their simultaneous use should lead to more tractable solutions for some classes of problems. For example, the minimization of some quadratic forms leads to solutions that satisfy certain partial differential equations and, therefore, the techniques discussed provide a variational approach for finding these solutions.

Estimation in Stochastic Differential Equations with a State Dependent Diffusion Term

Abstract The most widely used methods for estimating parameters in stochastic differential equations, where the discrete time observations are corrupted with white noise, are based on (extended) Kalman filtering techniques for calculating the one-step prediction errors. Such methods cannot be used directly for estimation of parameters in stochastic differential equations (SDE) with a state dependent diffusion term. This paper describes a general transformation for some univariate SDEs that eliminates the state dependent diffusion term using the Ito formula. By employing this transformation the methods based on the Kalman filter techniques may still be used. Using simulation it is demonstrated that the approach leads to practically the same results as if a second order filter was used on the original SDE. As a case study a SDE describing the short term interest rate is studied.

Exact convergence rate of the Euler-Maruyama scheme, with application to sampling design

We consider regular partitions of the interval [0,T] into n subintervals based on a sampling density h. If xt is the solution of the original stochastic differential equation and is the approximate solution using the Euler-Maruyama scheme corresponding to a time discretization partition π, we find the limit as a functional of the sampling density h and the parameters of the stochastic differential equation. We then find the best sampling density and thus the best way to discretize regularly the interval for integrated mean square approximation error. The linear case is also considered in detail. We also apply the same idea to obtain optimal asymptotically efficient schemes.

Envelope correlation function of the narrow-band non-Gaussian process

The stationary response of oscillators possessing linear damping and non-linear stiffness to the white Gaussian noise excitation is considered. The original stochastic differential equation (SDE) is used to establish relations between the system parameters and the envelope correlation function. The approximation approach is based on the moment approximation and replacement of the envelope generating SDE by the new one with analytical solution of the corresponding Fokker-Planck equation. The usefulness of the method is illustrated by application to the randomly driven Duffing oscillator.