Proceedings of the 2018 Winter Simulation Conference M. Rabe, A. A. Juan, N. Mustafee, A. Skoogh, S. Jain, and B. Johansson, eds. SIMULATION-BASED ASSESSMENT OF THE STATIONARY TAIL DISTRIBUTION OF A STOCHASTIC DIFFERENTIAL EQUATION Krzysztof Bisewski Daan Crommelin Centrum Wiskunde & Informatica Science Park 123 1098 XG, Amsterdam, NETHERLANDS Michel Mandjes University of Amsterdam Science Park 105 1098 XG, Amsterdam, NETHERLANDS ABSTRACT A commonly used approach to analyzing stochastic differential equations (SDEs) relies on performing Monte Carlo simulation with a discrete-time counterpart. In this paper we study the impact of such a time-discretization when assessing the stationary tail distribution. For a family of semi-implicit Euler discretization schemes with time-step h > 0, we quantify the relative error due to the discretization, as a function of h and the exceedance level x. By studying the existence of certain (polynomial and exponential) moments, using a sequence of prototypical examples, we demonstrate that this error may tend to 0 or ∞. The results show that the original shape of the tail can be heavily affected by the discretization. The cases studied indicate that one has to be very careful when estimating the stationary tail distribution using Euler discretization schemes. 1 INTRODUCTION Let (X t ) t ≥0 solve the stochastic differential equation (SDE) dX t = f (X t ) dt + g(X t ) d W t , (1) with an initial condition X 0 ∼ ξ . The functions f : R → R and g : R → R are called drift and volatility respectively, while ξ follows an arbitrary, tight (possibly degenerate) probability law concentrated on R. SDEs are used in a variety of application areas, e.g. chemistry (Yang et al. 2006) and climate science (Majda et al. 2009). Under some conditions on f and g, X t converges to a stationary distribution as t → ∞. Let μ 0 be the corresponding stationary (or invariant, ergodic) measure, that is, the unique probability measure such that X 0 ∼ μ 0 implies X t ∼ μ 0 for all t > 0. In the following, we abbreviate ¯ μ 0 (x) := μ 0 ((x, ∞)). We are interested in determining the shape of the tail of μ 0 , i.e., the way ¯ μ 0 (x) decays to 0 as x → ∞. Besides the one-dimensional case, no explicit expressions for ¯ μ 0 (x) are available, thus motivating the use of simulation-based methods. Ideally, one would sample a path of (X t ) t ≥0 (in continuous time, that is), and estimate ¯ μ 0 (x) by the fraction of time it spends above level x in a time interval [0, T ] (which, by the ergodic theorem, converges to ¯ μ 0 (x) as T → ∞). It is evidently impossible to sample a continuous and infinitely long path of a process (X t ) t ≥0 on a computer, explaining the need for time-discretization and truncation. Discretization schemes are not exact and may intrinsically change the dynamics of the original continuous-time process defined through (1). As a consequence, the stationary measure pertaining to the discretized process will generally differ from μ 0 (or might not even exist!) A few relevant references on this topic are Roberts and Tweedie (1996), Stramer and Tweedie (1999), and Mattingly et al. (2002). In this paper we study the effect of discretization on the shape of the tail of the stationary distribution. In order to illustrate the problem that might occur, we use the Ornstein-Uhlenbeck (OU) process as an example. Let (X t ) t ≥0 solve dX t = −X t dt + √ 2d W t ; it can be shown that μ 0 ∼ N(0, 1). The forward Euler 1742 978-1-5386-6572-5/18/$31.00 ©2018 IEEE