Schemes For Stochastic Differential Equations Research Articles

Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability issue is solved essentially in Cass et al., (2013). In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size n) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter H>1/3 and its Malliavin derivatives.

Weak approximation schemes for SDEs with super-linearly growing coefficients

We propose a new class of weak approximation schemes for stochastic differential equations with coefficients of suplinearly growth. Both the modified weak Euler schemes and the drift-implicit weak Euler scheme are studied. Under certain non-globally Lipschitz conditions, the proposed schemes are proved to have first-order convergence in the weak sense. Numerical experiments are included to confirm the theoretical results.

A deep learning-based Monte Carlo simulation scheme for stochastic differential equations driven by fractional Brownian motion

Stochastic differential equations (SDEs) are widely used models to describe the evolution of stochastic processes. Among them, SDEs driven by fractional Brownian motion (fBm) have been shown to be capable of describing systems with temporal dependencies. In this paper, we develop a neural network based Monte Carlo methodology in which we can efficiently simulate SDEs that are governed by fBm. Particularly, we focus on large time step simulations. A property of fBm that complicates the development of such Monte Carlo schemes is the long-range temporal correlation. To this end, we build the network based on the encoder–decoder framework and employ the attention mechanism to learn the temporal relationships in the historical paths of such SDEs. In addition, a loss function based on the quantile loss is used, where the quantile levels to be predicted are determined by means of the stochastic collocation method. Experimental results show that this kind of loss function is superior to conventional loss functions in terms of solution accuracy, and the resulting scheme can learn and simulate SDEs driven by fBm accurately and highly efficiently.

Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

Assessment of Stochastic Numerical Schemes for Stochastic Differential Equations with “White Noise” Using Itô’s Integral

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method.

Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients

Abstract We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein’s weak error analysis on the one-step approximation of SDEs, we prove a general result on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates for several numerical schemes of half-order strong convergence, such as tamed and balanced schemes. Numerical examples are presented to verify our theoretical analysis.

Truncated Euler–Maruyama method for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient

In this paper, we propose a truncated Euler–Maruyama scheme for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient. Meanwhile, the convergence rate of the numerical method is established. Numerical example is demonstrated to verify the theoretical results.

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

Approximation of distribution-independent and distribution-dependent stochastic differential equations with singular drifts

This work is concerned with convergence rate of Euler–Maruyama scheme for distribution-independent and distribution-dependent stochastic differential equations, where the drifts involved are Dini continuous and unbounded. Via introducing a new Zvonkin-type’s transformation, we investigate convergence rate of Euler–Maruyama scheme for stochastic differential equations with singular coefficients which allows the drifts to be unbounded. Moreover, via the analysis of interacting particle systems, we show an approximation issue on a class of distribution-dependent stochastic differential equations with singular drifts. More specifically, propagation of chaos and convergence rate of the Euler–Maruyama scheme associated with the interacting particle systems are investigated.

The Carathéodory approximation scheme for stochastic differential equations with G‐Lévy process

This article investigates the Carathéodory approximation scheme for stochastic differential equations (SDEs) in the G‐Lévy process framework. In the context of the G‐Lévy process, the possibility of SDE solutions has been developed. The solution's boundedness and uniqueness have been discussed. The convergence of approximate Carathéodory solutions to the unique solution of the given SDEs has been determined. The G‐Lévy framework is used to obtain exponential estimates for solutions to SDEs. An example has been provided to validate the results.

Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

Efficient numerical valuation of European options under the two-asset Kou jump-diffusion model

We study the L1-approximation of the log-Heston stochastic differential equation at equidistant time points by Euler-type methods. We establish the convergence order 1/2 – ∊ for ∊ > 0 arbitrarily small if the Feller index v of the underlying Cox– Ingersoll–Ross process satisfies v > 1. Thus, we recover the standard convergence order of the Euler scheme for stochastic differential equations with globally Lipschitz coefficients. Moreover, we discuss the v ≥ 1 case and illustrate our findings with several numerical examples.

Continuous Time Limit of the Stochastic Ensemble Kalman Inversion: Strong Convergence Analysis

The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous-time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former, and convergence in moments for the latter. The methods employed can also be applied to the analysis of more general numerical schemes for stochastic differential equations in general.

Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises

The strong convergence rate of the Euler scheme for stochastic differential equations driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter H∈(13,12) and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the 2D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order 2H for the case that the drift coefficient has bounded derivatives up to order three and have strong order H+12 for linear cases.

Positivity-preserving numerical schemes for stochastic differential equations

The solutions of stochastic differential equations arising in biology, finance and so on often have positivity. However, numerical solutions by the standard schemes often fail to satisfy this property. In this paper, we propose positivity-preserving numerical schemes for stochastic differential equations by virtue of Itô’s formula. We also show the convergence result of the proposed scheme and demonstrate their effectiveness by numerical examples.

Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions

We develop a new framework for error analysis on stochastic numerical schemes, with the rough path theory and stochastic backward error analysis. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter H∈(14,12) is (2H−12)− for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the best probable convergence rate of implementable approximations of the Lévy area of fractional Brownian motion. Our result gives a positive answer to the conjecture proposed in [12] for the case H∈(13,12), and reveals for the first time that numerical schemes constructed by a second-order Taylor expansion converge for the case H∈(14,13].

A numerical scheme for stochastic differential equations with distributional drift

In this paper we introduce a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a convergence rate in a suitable L1-norm and, as a by-product, a convergence rate for a numerical scheme applied to SDEs with drift in Lp-spaces with p∈(1,∞).

Curved schemes for stochastic differential equations on, or near, manifolds

Given a stochastic differential equation (SDE) inRnwhose solution is constrained to lie in some manifoldM⊂Rn, we identify a class of numerical schemes for the SDE whose iterates remain close toMto high order. These schemes approximate a geometrically invariant scheme, which gives perfect solutions for any SDE that is diffeomorphic ton-dimensional Brownian motion. Unlike projection-based methods, they may be implemented without explicit knowledge ofM. They can even be implemented if the solution merely remains close toM, without being exactly confined to it. Our approach does not require simulating any iterated Itô integrals beyond those needed to implement the Euler–Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their geometric advantages in a variety of numerical contexts, including Monte Carlo simulation of the Riemannian Langevin equation.

A Koopman framework for rare event simulation in stochastic differential equations

We exploit the relationship between the stochastic Koopman operator and the Kolmogorov backward equation to construct importance sampling schemes for stochastic differential equations. Specifically, we propose using eigenfunctions of the stochastic Koopman operator to approximate the Doob transform for an observable of interest (e.g., associated with a rare event) which in turn yields an approximation of the corresponding zero-variance importance sampling estimator. Our approach is broadly applicable and systematic, treating non-normal systems, non-gradient systems, and systems with oscillatory dynamics or rank-deficient noise in a common framework. In nonlinear settings where the stochastic Koopman eigenfunctions cannot be derived analytically, we use dynamic mode decomposition (DMD) methods to approximate them numerically, but the framework is agnostic to the particular numerical method employed. Numerical experiments demonstrate that even coarse approximations of a few eigenfunctions, where the latter are built from non-rare trajectories, can produce effective importance sampling schemes for rare events.

Numerical Scheme for Solving Stochastic Differential Equations with <i>G</i>-Lévy Process

In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itô formula and G-expectation property, we propose Euler scheme and Milstein scheme which have order-1.0 convergence rate. And two numerical experiments including Ornstein-Uhlenbeck and Black-Scholes cases are given.