We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $ρ$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $ρ\ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.
In this paper we study a type of stochastic McKean-Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler-Maruyama approximation existence of its weak solutions is proved. And then we observe pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler-Maruyama approximation has an optimal strong convergence rate.
In this paper we study the strong convergence for the Euler-Maruyama approximation of a class of stochastic differential equations whose both drift and diffusion coefficients are possibly discontinuous.
In this paper, the discrete parameter expansion is adopted to investigate the estimation of heat kernel for Euler-Maruyama scheme of SDEs driven by α-stable noise, which implies krylov's estimate and khasminskii's estimate. As an application, the convergence rate of Euler-Maruyama scheme of a class of multidimensional SDEs with singular drift( in aid of Zvonkin's transformation) is obtained.
We study the strong rates of the Euler-Maruyama approximation for one dimensional stochastic differential equations whose drift coefficient may be neither continuous nor one-sided Lipschitz and diffusion coefficient is Hölder continuous. Especially, we show that the strong rate of the Euler-Maruyama approximation is 1/2 for a large class of equations whose drift is not continuous. We also provide the strong rate for equations whose drift is Hölder continuous and diffusion is nonconstant
We introduce a semi-implicit Euler-Maruyama approximation which preservers the non-colliding property for some class of non-colliding particle systems such as Dyson Brownian motions, Dyson-Ornstein-Uhlenbeck processes and Brownian particles systems with nearest neighbour repulsion, and study its rates of convergence in both $L^p$-norm and path-wise sense.
In this paper, we are concerned with convergence rate of Euler-Maruyama scheme for stochastic differential equations with rough coefficients. The key contributions lie in (i), by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler-Maruyama scheme for a class of stochastic differential equations, which allow the drifts to be Dini-continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler-Maruyama scheme for a range of degenerate stochastic differential equations, whe…
In this paper, we show the convergence rate of Euler-Maruyama scheme for non-degenerate SDEs with Dini continuous coefficients, by the aid of the regularity of the solution to the associated Kolmogorov equation. We obtain the same conclusions using a simple and clever way to simplify the proof and weaken the conditions in \cite{BHY} by the properties of Dini continuous and Taylor expansion.
We consider SDEs with bounded and $α$-Hölder continuous drift, with $α\in (0,1)$, driven by multiplicative noise. We show that under sufficient conditions on the diffusion matrix, which guarantee the existence of a unique strong solution, the weak rate of convergence for the Euler-Maruyama scheme is almost $(1+α)/2$. The present paper forms part of the author's master's thesis.
We study in this article the strong rate of convergence of the Euler-Maruyama scheme and associated with the jump-type equation introduced in Li and Mytnik. We obtain the strong rate of convergence under similar assumptions for strong existence and pathwise uniqueness. Models of this type can be considered as a generalization of the CIR (Cox-Ingersoll-Ross) process with jumps.
The aim of this paper is to study weak and strong convergence of the Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation $\mathrm{d} X_t=σ(X_t) \mathrm{d} W_t$ with non-sticky condition. For proving this, we first prove that the Euler--Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation $\mathrm{d} X_t=|X_t|^α \mathrm{d} W_t$, $α\in (0,1/2)$ with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.
This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.
Regime-switching processes contain two components: continuous component and discrete component, which can be used to describe a continuous dynamical system in a random environment. Such processes have many different properties than general diffusion processes, and much more difficulties are needed to be overcome due to the intensive interaction between continuous and discrete component. We give conditions for the existence and uniqueness of invariant measures for state-dependent regime-switching diffusion processes by constructing a new Markov chain to control the evolution of the state-depend…
In this paper, we consider a numerical approximation of the stochastic differential equation (SDE) $$X_{t}=x_{0}+ \int_{0}^{t} b(s, X_{s}) \mathrm{d}s + L_{t},~x_{0} \in \mathbb{R}^{d},~t \in [0,T],$$ where the drift coefficient $b:[0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ is Hölder continuous in both time and space variables and the noise $L=(L_t)_{0 \leq t \leq T}$ is a $d$-dimensional Lévy process. We provide the rate of convergence for the Euler-Maruyama approximation when $L$ is a Wiener process or a truncated symmetric $α$-stable process with $α\in (1,2)$. Our technique is based on the…
Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional Lévy process $(L_t)_{t \geq 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples…
Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework
该研究证明在 $G$-框架下，当延迟或数值步长足够小时，$G$-SDDE、辅助 $G$-SDE 及其 EM 方法在 $p$ 阶矩实际指数稳定性上彼此等价，从而支持通过模拟检验稳定性。
We give bounds for the total variation distance between the solutions to two stochastic differential equations starting at the same point and with close coefficients, which applies in particular to the distance between an exact solution and its Euler-Maruyama scheme in small time. We show that for small $t$, the total variation distance is of order $t^{r/(2r+1)}$ if the noise coefficient $σ$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in \mathbb{N}$ and if the drift is $C^1$ with bounded derivatives, using multi-step Richardson-Romberg extrapolation. We do not require the drift to be b…
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q -- L $ρ$ drift coefficient when the condition d $ρ$ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 -- (d $ρ$ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove …
We consider the Euler-Maruyama approximation for multi-dimensional stochastic differential equations with irregular coefficients. We provide the rate of strong convergence where the possibly discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is Hölder continuous.
In this paper, we study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.
We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by $π$ and $π_η$ respectively ($η$ is the step size of the EM scheme). We construct an empirical measure $Π_η$ of the EM scheme as a statistic of $π_η$, and use Stein's method developed in \citet{FSX19} to prove a central limit theorem of $Π_η$. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting $η^{-1/2}(Π_η(.)-π(.))$ into a martingale differ…
The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process
We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence …
Classical approximation results for stochastic differential equations analyze the $L^p$-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate $0.5$ with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.
We will introduce Euler-Maruyama approximations given by an orthogonal system in $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler-Maruyama approximation base…
The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method are proved and the convergence of the numerical invariant measure to the underlying one is shown. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control.
In this article, we consider numerical schemes for polynomial diffusions on the unit ball, which are solutions of stochastic differential equations with a diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a semi-implicit Euler--Maruyama scheme with the projection onto the unit ball and provide the $L^{2}$-rate of convergence. The main idea to consider the numerical scheme is the transformation argument introduced by Swart for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.
The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asy…
Tamed-adaptive Euler-Maruyama approximation for SDEs with superlinearly growing and piecewise continuous drift, superlinearly growing and locally Hölder continuous diffusion
An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder's continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.
The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $α$-Hölder drift in the recent literature the rate $α/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $α>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.
We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+σ(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $σ: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal…
