9-11 July 2018
Ho Chi Minh City University of Science
Asia/Ho_Chi_Minh timezone

Asymptotic behavior of the error between two different Euler schemes for the Lévy driven SDEs

11 Jul 2018, 12:00
30m
Ho Chi Minh City University of Science

Ho Chi Minh City University of Science

227 Nguyễn Văn Cừ, Phường 4, T.P. Hồ Chí Minh

Speaker

Ms Thi Bao Tram NGO (PhD student)

Description

We study the Multi-level Monte Carlo method introduced by Giles [3] and its applications to finance which is significantly more efficient than the classical Monte Carlo method. This method for the stochastic differential equations driven by only Brownian Motion had been studied by Ben Alaya and Kebaier [2]. Here, we consider the stochastic differential equation driven by a pure jump Lévy process. When the Lévy process have a Brownian component, the speed of convergence of the multilevel was recently studied by Dereich and Li [4].

Now, we prove the stable law convergence theorem in the spirit of Jacod [1]. More precisely, we consider the SDE of form
\begin{equation}
X_t=x_0+\int_0^t f(X_{s-})dY_s, (1)
\end{equation}
with $f\in\mathcal{C}^3$ and $Y$ is a Lévy process with the triplet $(b,0,F)$ and look at the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are two different Euler approximations with step sizes $1/n$ and $1/nm$ respectively. The rate $u_{n,m}$ is an appropriate rate going to infinity such that the normalized error converges to non-trivial limit. Under some different assumptions on the properties of the Lévy process $Y$ in $(1)$, we found different suitable forms of the rate $u_{n,m}$.

[1] Jean Jacod. The Euler scheme for Lévy driven stochastic differential equations: Limit theorems. The Annals of Probability, 2004, Vol.32, No.3A, 1830-1872.

[2] Mohamed Ben Alaya and Ahmed Kebaier. Central limit theorem for the multilevel Monte Carlo Euler method. Ann.Appl. Probab. 25(1): 211-234, 2015.

[3] Michael B.Giles. Multilevel Monte Carlo path simulation, Oper. Res., 56(3): 607-617, 2008.

[4] Steffen Dereich and Sangmeng Li. Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes. Ann. Appl. Probab., 26(1): 136-185, 2016.

Primary authors

Prof. Ahmed KEBAIER (Associate Professor) Prof. Mohamed BEN ALAYA (Professor) Ms Thi Bao Tram NGO (PhD student)

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