Discrete-time simulation of Stochastic Volterra equations

Abstract

We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multilevel Monte-Carlo method. By using and adapting some results from Zhang (2008), together with the Garsia–Rodemich–Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multilevel) Monte-Carlo method, and compute their complexity. We finally provide some numerical simulation results.

Introduction

We study the discrete-time approximation problem for stochastic Volterra equations of the form

$$X_t=X_0+{\int }_0^t{K}_1(t,s)b(s,X_s)ds+{\int }_0^t{K}_2(t,s)\sigma (s,X_s)d{W}_s,t\in [0,T],$$

by means of the Euler scheme, the Milstein scheme and the corresponding Multilevel Monte-Carlo method. In the above equation, $X$ is an ${\mathbb{R}}^d$-valued process, $W$ is a $d$-dimensional standard Brownian motion, $K_1,K_2$ are (possibly singular) kernels, and $b,\sigma$ are coefficient functions whose properties will be detailed below.

As natural extension of (deterministic) Volterra equations, the stochastic Volterra equation is motivated by the physics of heat transfer (see for instance the introductory example of the book of Gripenberg, Londen and Staffans [17] with a random source term), the physics of dissipative dynamics and anomalous diffusions (see for instance Jakšić and Pillet [20], resp. Lutz [25]), and has been studied since the works of Berger and Mizel [6] and Protter [27] in the non-singular kernels and Lipschitz coefficients case. Let us also mention the recent rough volatility modelling in mathematical finance, which leads to some affine Volterra equations, see e.g. El Euch and Rosenbaum [10], and Abi Jaber, Larsson, and Pulido [2].

The main objective of the paper is to study the discrete-time simulation problem for the stochastic Volterra equation (1). Observe that when $K_1\equiv K_2\equiv I_d$, the Volterra equation degenerates into a standard SDE, and the corresponding Monte-Carlo simulation problem has been tremendously studied during the last decades. In general, the simulation of SDEs is based on discrete-time schemes, and to estimate the expectation of a functional of an SDE by Monte-Carlo method, one has two kinds of error: the discretization error and the statistical error. The statistical error is proportional to $\frac{1}{\sqrt{N}}$, where $N$ is the number of simulated copies of the SDE, by an application of the Central Limit Theorem. The discretization error depends essentially on the time step $\Delta t$. For the most simple Euler scheme, a (weak) convergence rate of the discretization error has been initially obtained by Talay and Tubaro [29]. Since then, many works have been devoted to study various schemes under different conditions. For an overview on this subject, let us refer to Kloeden and Platen [23], Graham and Talay [16], and also Jourdain and Kohatsu-Higa [21] for a recent review. To reduce the discretization error, one needs to use finer discretization, which increases the computational complexity for the simulation of the process, and hence increases the statistical error given a fixed total computation effort. Then one needs to make a trade-off between the two errors to minimize the total error.

To improve the usual trade-off between the two errors, Giles [14] introduced the so-called Multilevel Monte-Carlo (MLMC) method, which has been applied and improved in various situations, and has generated a stream of literature, see e.g. Giles and Szpruch [15], Alaya and Kebaier [4], etc. The main idea of the MLMC method is to consider different levels of the time discretization, and rewrite the finest discrete scheme as a telescopic sum of differences between consecutive levels, and then to choose the number of simulations at each level in an optimal way. Let us mention that MLMC has already been studied in the setting of SDEs driven by fractional Brownian motions (denoted later by fBm): first in Kloeden, Neuenkirch and Pavani [22] with Hurst exponent $H>\frac{1}{2}$ and additive fractional noise, and then extensions to rough SDEs in Bayer, Friz, Riedel, and Schoenmakers [3]. This latter article corresponds to a Hurst exponent $H\in (\frac{1}{4},\frac{1}{2})$, which is still far from the observed roughness of the volatility ($H\approx 0.1$, see Gatheral, Jaisson and Rosenbaum [12]). The advantage of the Volterra approach compared to integration w.r.t. fBm is that one can achieve very low path regularities, while an equivalent approach through rough paths would be restricted, in practice (although not theoretically), to $H>\frac{1}{4}$ [3].

In this paper, we will study the discretization error of the Euler scheme and the Milstein scheme for the stochastic Volterra equation (1) with any Hölder regularity (Hurst exponent) between $0$ and $1$, and then adapt the MLMC technique in our context. For the stochastic Volterra equation in a more general form, the corresponding Euler scheme has already been studied by Zhang [31], where the main results state that, for the uniform discretization scheme with time step $\Delta t=2^{-n}T$, the discretization error is bounded by $C{2}^{-n\eta }$, for some constant $\eta >0$ (which is not given explicitly but might be found in the proof). In this paper, we let ${\left(X_t^n\right)}_{0\le t\le T}$ denote the solution of the Euler scheme with a general (not necessary uniform) discretization ${\pi }_n$, and adapt the techniques in [31] to our context to obtain an explicit convergence rate of $\mathbb{E}\left[{|X_t-X_t^n|}^p\right]$ for each fixed $t\in [0,T]$ and $p\ge 1$. Then, in place of the argument with Kolmogorov’s continuity criterium used in [31], we apply the technique based on the Garsia–Rodemich–Rumsey lemma to obtain an explicit rate for the supremum norm error $\mathbb{E}\left[{sup}_{0\le t\le T}{|X_t-X_t^n|}^p\right]$. Our new technique provides a better convergence rate than the one in [31], in particular when not all moments of the initial condition are integrable, and the discretization ${\pi }_n$ could be arbitrary rather than the special uniform discretization of size $T{2}^{-n}$ that is required in the technical proof of [31] (see also Remark 2.3). Next, we introduce and extend our techniques and results to a higher order scheme, the Milstein scheme, in order to improve the convergence rate. We then study the MLMC method based on the Euler scheme, and compare their computational cost for a given theoretical error. These different methods are also tested with various numerical examples. We would like also to mention the recent paper [24] which appeared at the same time as ours, where the authors study both Euler and Milstein scheme of Volterra equation (1) with special kernel $K_1(t,s)≔{(t-s)}^{-\alpha }$ and $K_2(t,s)≔{(t-s)}^{-\beta }$. The paper establishes a convergence rate result on $\mathbb{E}\left[{|X_t-X_t^n|}^p\right]$ for every fixed $t\in [0,T]$, which is essentially the same as ours.

The rest of the paper is organized as follows. In Section 2, we state some conditions on $K_1,K_2,b$ and $\sigma$ that we require for the Euler and Milstein schemes. We then present these two schemes and the corresponding convergence rate results in Theorem 2.2, Theorem 2.4. In the third part of this section, we detail the Multilevel Monte-Carlo method to approximate quantities of the form $\mathbb{E}\left[f\left(X_{\cdot }\right)\right]$ and provide some complexity analysis for a given error. Then, in Section 3, we provide some numerical examples for these simulation methods. Finally, Section 4 gathers the proofs of Theorem 2.2, Theorem 2.4.