Euler scheme and tempered distributions

Julien Guyon

Stochastic Processes and their Applications, 2006, vol. 116, issue 6, 877-904

Abstract: Given a smooth -valued diffusion starting at point x, we study how fast the Euler scheme with time step 1/n converges in law to the random variable . To be precise, we look for the class of test functions f for which the approximate expectation converges with speed 1/n to . When f is smooth with polynomially growing derivatives or, under a uniform hypoellipticity condition for X, when f is only measurable and bounded, it is known that there exists a constant C1f(x) such that If X is uniformly elliptic, we expand this result to the case when f is a tempered distribution. In such a case, (resp. ) has to be understood as (resp. ) where p(t,x,[dot operator]) (resp. pn(t,x,[dot operator])) is the density of (resp. ). In particular, (1) is valid when f is a measurable function with polynomial growth, a Dirac mass or any derivative of a Dirac mass. We even show that (1) remains valid when f is a measurable function with exponential growth. Actually our results are symmetric in the two space variables x and y of the transition density and we prove that for a function and an O(1/n2) remainder rn which are shown to have gaussian tails and whose dependence on t is made precise. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.

Keywords: Stochastic; differential; equation; Euler; scheme; Rate; of; convergence; Tempered; distributions (search for similar items in EconPapers)
Date: 2006
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (18)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(05)00168-7
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:116:y:2006:i:6:p:877-904

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().