In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by Lévy processes. The underlying numerical schemes are based on jump-adapted Euler schemes. We prove stable convergence of an idealised scheme. Further, we deduce limit theorems for certain classes of functionals depending on the whole trajectory of the process. In particular, we allow dependence on marginals, integral averages and the supremum of the process. The idealised scheme is related to two practically implementable schemes and corresponding central limit theorems are given. In all cases, we obtain errors of order $N^{-1/2}(\log N)^{1/2}$ in the computational time $N$ which is the same order as obtained in the classical set-up analysed by Giles [Oper. Res. 56 (2008) 607-617]. Finally, we use the central limit theorems to optimise the parameters of the multilevel scheme.
In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme a…
Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] …
In this paper, we present a generic methodology for the efficient numerical approximation of the density function of the McKean-Vlasov SDEs. The weak error analysis for the projected process motivates us to combine the iterative Multilevel Monte Carlo method for McKean-Vlasov SDEs \cite{szpruch2019} with non-interacting kernels and projection estimation of particle densities \cite{belomestny2018projected}. By exploiting smoothness of the coefficients for McKean-Vlasov SDEs, in the best case scenario (i.e $C^{\infty}$ for the coefficients), we obtain the complexity of order $O(ε^{-2}|\logε|^4)$…
In this paper, we show the strong well-posedness of Mckean-Vlasov SDEs with non-Lipschitz coefficients. Moreover, propagation of chaos and the convergence rate for Euler's scheme of Mckean-Vlasov SDEs are also obtained.
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.
The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean \cite{mckean1966class}. The interacting diffusions pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs, but provide a great challenge for Monte Carlo simulations. This is due to the nonlinear dependence of the bias on the statistical error arising through the approximation of the law of the process. This and the fact that particles/diffusions are not independent render classical variance reduction t…
The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a non-degenerate main part driven by a spherically-symmetric stable process.
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.
Let $μ\in \mathcal{P}_2(\mathbb R^d)$, where $\mathcal{P}_2(\mathbb R^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $Φ:\mathcal P_2(\mathbb R^d)\rightarrow \mathbb R $. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $Φ(μ)$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws $μ^N = \frac1N \sum_{i=1}^{N}δ_{X_i}$, where a) $(X_i)_{i=1}^N$ is a sequence of i.i.d samples from $μ$ or b) $(X_i)_{i=1}^N$ is a system of interacting particles (diffusions)…
In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.
The paper studies the rate of convergence of the weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. In addition, the rate robustness to the approximation of the increments of the driving process is studied and approximate Euler scheme considered. A rate of convergence is derived for an approximate jump-adapted scheme as well.
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.
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(Δt)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $ε$ from $O(ε^{-3})$ to $O(ε^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(Δt^{1/2})$ requires simulation, or approximation, of Lévy areas. In this paper, through the construct…
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.
We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional Lévy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum nor…
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 article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac13<H<\frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac12$. The current contribution generalizes the modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue …
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.
Rio gave a concise bound for the central limit theorem in the Vaserstein distances, which is a ratio between some higher moments and some powers of the variance. As a corollary, it gives an estimate for the normal approximation of the small jumps of the Lévy processes, and Fournier applied that to the Euler approximation of Lévy-driven stochastic differential equations. However both results are restricted to the one-dimensional case. It will be shown in this article that, following Davie's idea, one can generalise Rio's result to multidimensional case, and the number of moments required is ind…
In this paper, we are interested in the time discrete approximation of Ef(X(T)) when X is the solution of a stochastic differential equation with a diffusion coefficient function of the form |x|^a. We propose a symmetrized version of the Euler scheme, applied to X. The symmetrized version is very easy to simulate on a computer. For smooth functions f, we prove the Feynman Kac representation u(t,x) = E_{t,x} f(X(T)), for u(t,x) solving the associated Kolmogorov PDE and we obtain the upper-bounds on the spatial derivatives of u up to the order four. Then we show that the weak error of our symmet…
This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607-617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267-307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the o…
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…
The paper estimates the rate of convergence of the weak Euler approximation for the solutions of SDEs with Hoelder continuous coefficients driven by point and martingale measures. The equation considered has a non-degenerate main part whose jump intensity measure is absolutely continuous with respect to the Levy measure of a spherically-symmetric stable process. It includes the nondegenerate diffusions and SDEs driven by Levy processes.
We describe an Euler scheme to approximate solutions of Lévy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (2012). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate $\mathcal{O}(n^{-1/2})$. The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the S…
In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian Motion with Hurst parameter H>1/2 and obtain some (functional) convergences properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.
In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.
We consider upper bounds for the approximation error E|g(X)-g(\hat X)|^p, where X and \hat X are random variables such that \hat X is an approximation of X in the L_p-norm, and the function g belongs to certain function classes, which contain e.g. functions of bounded variation. We apply the results to the approximations of a solution of a stochastic differential equation at time T by the Euler and Milstein schemes. For the Euler scheme we provide also a lower bound.
In this paper, we consider scalar stochastic differential equations (SDEs) with a superlinearly growing and piecewise continuous drift coefficient. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate 1/2 for all $p\in [1, +\infty)$ is recovered for the tamed Euler scheme. A numerical example is provided to support our conclusion.
SDE driven by an $α$-stable process, $α\in \lbrack 1,2),$ with Lipshitz continuous coefficient and $β$-Hölder drift is considered. The existence and uniqueness of a strong solution is proved when $β>1-α/2$ by showing that it is $L_{p}$-limit of Euler approximations. The $L_{p}$-error (rate of convergence) is obtained for a nondegenerate truncated and nontruncated driving process. The rate in the case of Lipshitz continuous coefficients is derived as well.
In this paper, we establish the theory of chaos propagation and propose an Euler-Maruyama scheme for McKean-Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst exponent $H \in (0,1)$. Meanwhile, upper bounds for errors in the Euler method is obtained. A numerical example is demonstrated to verify the theoretical results.
Firstly, we investigate Euler-Maruyama approximation for solutions of stochastic differential equations (SDEs) driven by a symmetric α stable process under Komatsu condition for coefficients. The approximation implies naturally the existence of strong solutions. Secondly, we study the stability of solutions under Komatsu condition, and also discuss it under Belfadli-Ouknine condition.
In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler-Maruyama scheme for such a class of SDEs. We prove convergence of order $1/2$. Finally, we present numerical examples.
