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SUMMARY:Asymptotic behavior of the error between two different Euler schem
es for the Lévy driven SDEs
DTSTART;VALUE=DATE-TIME:20180711T050000Z
DTEND;VALUE=DATE-TIME:20180711T053000Z
DTSTAMP;VALUE=DATE-TIME:20200403T050455Z
UID:indico-contribution-1748@indico.math.cnrs.fr
DESCRIPTION:Speakers: Thi Bao Tram NGO (PhD student)\nWe study the Multi-l
evel Monte Carlo method introduced by Giles [3] and its applications to fi
nance which is significantly more efficient than the classical Monte Carlo
method. This method for the stochastic differential equations driven by o
nly Brownian Motion had been studied by Ben Alaya and Kebaier [2]. Here\,
we consider the stochastic differential equation driven by a pure jump Lé
vy process. When the Lévy process have a Brownian component\, the speed o
f convergence of the multilevel was recently studied by Dereich and Li [4]
. \n \nNow\, we prove the stable law convergence theorem in the spirit
of Jacod [1]. More precisely\, we consider the SDE of form \n\\begin{equ
ation} \nX_t=x_0+\\int_0^t f(X_{s-})dY_s\, (1) \n\\end{equation} \nw
ith $f\\in\\mathcal{C}^3$ and $Y$ is a Lévy process with the triplet $(b\
,0\,F)$ and look at the asymptotic behavior of the normalized error proces
s $u_{n\,m}(X^n-X^{nm})$ 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 suit
able forms of the rate $u_{n\,m}$. \n \n[1] Jean Jacod. The Euler schem
e for Lévy driven stochastic differential equations: Limit theorems. The
Annals of Probability\, 2004\, Vol.32\, No.3A\, 1830-1872. \n \n[2] Moh
amed Ben Alaya and Ahmed Kebaier. Central limit theorem for the multilevel
Monte Carlo Euler method. Ann.Appl. Probab. 25(1): 211-234\, 2015. \n
\n[3] Michael B.Giles. Multilevel Monte Carlo path simulation\, Oper. Res.
\, 56(3): 607-617\, 2008. \n \n[4] Steffen Dereich and Sangmeng Li. Mul
tilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adap
tive Euler schemes. Ann. Appl. Probab.\, 26(1): 136-185\, 2016.\n\nhttps:/
/indico.math.cnrs.fr/event/3023/contributions/1748/
LOCATION:Ho Chi Minh City University of Science
URL:https://indico.math.cnrs.fr/event/3023/contributions/1748/
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