A multilevel Monte Carlo algorithm for SDEs with jumps

Abstract

The multilevel Monte Carlo algorithm is an extension of the traditional Monte Carlo algorithm. It is a numerical method, which allows us to approximate the expected value of a random variable X. We use some appropriate discretization method to obtain approximations X_1, X_2, ... ,X_L of X such that each approximation is made with a finer grid. The more accuracy we want from our approximation, the more the computational cost grows. The multilevel method exploits evaluation at multiple levels of refining discretizations allowing us to achieve a better accuracy with lower cost.

In this thesis we use the Euler scheme to approximate the solution of the stochastic differential equation, and then we use the multilevel Monte Carlo algorithm to estimate the expected value of the solution. We prove that the mean squared error of the estimator is O(h^2) with computational complexity O(h^(-2)(log h)^2) with stochastic differential equations driven by a Brownian motion. Lastly, we prove that with computational complexity O(n), when the driving process is a Lévy process without Brownian component the error is O(n^(-1/2)) and with the Brownian component O(n^(-1/2) (log n)^(3/2)).

As a background theory we introduce the basic concepts of probability and of stochastic processes, namely the Brownian motion, the Poisson processes and Lévy processes. We formulate the famous Lévy-Itô decomposition, which allows us to represent a Lévy process as the combination of a jump process and a Brownian motion. Additionally, we consider stochastic integration with respect to the Brownian motion, a martingale and the Poisson random measure. We use these to formulate the stochastic differential equations in two cases, driven by a Brownian motion or by a Lévy process.