Method For SDEs Research Articles

A higher-order approximation method for jump-diffusion SDEs with a discontinuous drift coefficient

We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme, we prove Lp-convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than assumptions applied before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4; in a special case it even has convergence order 1. Order 3/4 is conjectured to be optimal.

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition

The logarithmic truncated EM method with weaker conditions

In 2014, Neuenkirch and Szpruch established the drift-implicit Euler-Maruyama method for a class of SDEs which take values in a given domain. However, expensive computational cost is required for implementation of an implicit numerical method. A competitive positivity preserving explicit numerical method for SDEs which take values in the positive domain is the logarithmic truncated Euler-Maruyama method. However, assumptions for the logarithmic truncated Euler-Maruyama method used in previous work are restrictive which exclude some important SDE models with specific parameters. The main aim of this paper is to use weaker assumptions to establish strong convergence theory for the logarithmic truncated Euler-Maruyama method.

An explicit Euler–Maruyama method for McKean–Vlasov SDEs driven by fractional Brownian motion

In this paper, we establish the theory of propagation of chaos and propose an Euler–Maruyama method for McKean–Vlasov SDEs driven by fractional Brownian motion with Hurst parameter H∈(0,1/2)∪(1/2,1). Meanwhile, upper bounds for errors in the Euler–Maruyama method are obtained. Two numerical examples are demonstrated to verify the theoretical results.

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

Mean-square convergence and stability of compensated stochastic theta methods for jump-diffusion SDEs with super-linearly growing coefficients

Mean-square convergence and stability of compensated stochastic theta methods for jump-diffusion SDEs with super-linearly growing coefficients

The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero

Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.

Split-step theta Milstein methods for SDEs with non-globally Lipschitz diffusion coefficients

The present work aims to analyze mean-square convergence rates of split-step theta Milstein methods with method parameters θ∈[12,1] for stochastic differential equations with non-globally Lipschitz diffusion coefficients. The expected convergence rate of order one is successfully proved under more relaxed conditions, compared to existing relevant results. Numerical examples are also offered to confirm the theoretical findings.

Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

This paper proposes an adaptive timestep construction for an Euler–Maruyama approximation of SDEs with nonglobally Lipschitz drift. It is proved that if the timestep is bounded appropriately, then over a finite time interval the numerical approximation is stable, and the expected number of timesteps is finite. Furthermore, the order of strong convergence is the same as usual, that is, order $\frac{1}{2}$ for SDEs with a nonuniform globally Lipschitz volatility, and order $1$ for Langevin SDEs with unit volatility and a drift with sufficient smoothness. For a class of ergodic SDEs, we also show that the bound for the moments and the strong error of the numerical solution are uniform in $T$, which allow us to introduce the adaptive multilevel Monte Carlo method to compute the expectations with respect to the invariant distribution. The analysis is supported by numerical experiments.

Correction note: A strong order 1/2 method for multidimensional SDEs with discontinuous drift

Correction note: A strong order 1/2 method for multidimensional SDEs with discontinuous drift

Multilevel Monte Carlo method for ergodic SDEs without contractivity

This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon–Nikodym derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to T. Moreover, the variance of the new level estimators increase linearly in T, which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to O(ε−2|log⁡ε|2) from O(ε−3|log⁡ε|) of the standard Monte Carlo method. Numerical experiments support our analysis.

The projected explicit Itô–Taylor methods for stochastic differential equations under locally Lipschitz conditions and polynomial growth conditions

Although the numerical methods to stochastic differential equations with the coefficients of locally Lipschitz and polynomial growth have been discussed commonly by some authors, there are few works on the high strong order numerical methods. In this paper, the mean-square convergence of the general projected explicit Itô–Taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition, suitable local Lipschitz conditions and polynomial growth conditions. Our analysis follows the idea of stochastic C-stability and stochastic B-consistency. After giving the selection strategy of optimal parameters, we prove that the projected explicit Itô–Taylor methods, owning optimal parameters, share the same mean-square convergence orders with non-projected ones. Finally, two numerical experiments are presented to show the effectiveness of theoretical results. • The general projected explicit Itô–Taylor methods for SDEs are constructed. • A simple selection strategy for proper projection parameters is introduced. • The methods run well for SDEs under locally Lipschitz, polynomial growth condition. • The methods hold the same mean-square convergence order as non-projection ones.

The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients

In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients. By the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.

An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.

An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium

In this work we investigate the interplay of almost sure and mean-square stability for linear SDEs and the Monte Carlo method for estimating the second moment of the solution process. In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but asymptotically mean-square unstable, the latter property is determined by rarely occurring trajectories that are sufficiently far away from the origin. The standard Monte Carlo approach for estimating higher moments essentially computes a finite number of trajectories and is bound to miss those rare events. It thus fails to reproduce the correct mean-square dynamics (under reasonable cost). A straightforward application of variance reduction techniques will typically not resolve the situation unless these methods force the rare, exploding trajectories to happen more frequently. Here we propose an appropriately tuned importance sampling technique based on Girsanov’s theorem to deal with the rare event simulation. In addition further variance reduction techniques, such as multilevel Monte Carlo, can be applied to control the variance of the modified Monte Carlo estimators. As an illustrative example we discuss the numerical treatment of the stochastic heat equation with multiplicative noise and present simulation results.

Cheap arbitrary high order methods for single integrand SDEs

For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order $p_d$ we obtain methods converging in the mean-square and weak sense with order $\lfloor p_d/2\rfloor$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.

The linear Steklov method for SDEs with non-globally Lipschitz coefficients: Strong convergence and simulation

We present an explicit numerical method for solving stochastic differential equations with non-globally Lipschitz coefficients. A linear version of the Steklov average under a split-step formulation supports our new solver. The linear Steklov method converges strongly with a standard one-half order. Also, we present numerical evidence that the explicit linear Steklov reproduces almost surely stability solutions with high-accuracy for diverse application models even for stochastic differential systems with super-linear diffusion coefficients.