A multilevel Monte Carlo algorithm for SDEs with jumps
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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. ...
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Vihola, Matti (Institute for Operations Research and the Management Sciences, 2018)Multilevel Monte Carlo (MLMC) and recently proposed unbiased estimators are closely related. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes ...
Ylinen, Juha (University of Jyväskylä, 2015)
Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting Geiss, Christel; Steinicke, Alexander (Shandong Daxue, 2018)We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116: 1358–1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346: 345–358, 2008) hold under more simplified ...
Geiss, Christel; Labart, Céline; Luoto, Antti (Cambridge University Press (CUP), 2020)Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk from the underlying Brownian motion B by Skorokhod embedding, one can show -convergence of ...
Briand, Philippe; Geiss, Christel; Geiss, Stefan; Labart, Céline (International Statistical Institute, 2021)In this paper, we study in the Markovian case the rate of convergence in Wasserstein distance when the solution to a BSDE is approximated by a solution to a BSDE driven by a scaled random walk as introduced in Briand, ...