May 24, 2001 - approximation Y converges with weak order 0 if there exists a constant K such that for each polynomial g we have. jE g X T ,E g Y T j K ;. 1.3.
Rate of Weak Convergence of the Euler Approximation for Di usion Processes with Jumps Kestutis Kubilius 1
and Eckhard Platen2
May 24, 2001
Abstract. The paper estimates the speed of convergence of the Euler approximation for di usion processes with jump component which have Holder continuous coecients.
Key words: Di usion processes, stochastic di erential equations, Poisson jump measure, Euler approximation, simulation algorithm.
AMS Classi cation: 60H10, 65C05
Lithuanian Academy of Sciences Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania 2 University of Technology Sydney, School of Finance & Economics and School of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia 1
1 Introduction We consider the d-dimensional It^o process X = fX (t); t 2 [0; T ]g with jump component, see Ikeda & Watanabe (1989) or Gikhman & Skorokhod (1982), which is de ned as the weak solution of the stochastic equation
X (t) = X (0) + +
Z t
0 Z tZ 0
,
a(s; X (s,)) ds +
Z t 0
b(s; X (s,)) dWs
c(s; X (s,); ) q(d; ds);
(1.1)
t 2 [0; T ]. Here W denotes an m-dimensional standard Wiener process, q is a Poisson martingale measure and we have , = 0 if there exists a constant K such that for each polynomial g we have
jE (g(X (T ))) , E (g(Y (T )))j < K ;
(1.3)
where > 0 is the maximum step size of the time discretization and K might depend on g but not on . Milstein (1978, 1985) was one of the rst who studied the order of weak convergence of discrete time approximations for di usions. His result for the Euler approximation relates it to rst weak order convergence, that is = 1. In Platen 2
(1999) and Kloeden & Platen (1999) an extensive list of papers is given that deal with the discrete time approximation of It^o processes by Euler and higher order schemes. Talay investigated in Talay (1984, 1986) a class of second weak order, = 2, approximations for the di usion case. For the Euler approximation Y of the It^o Process with jump component X in (1.1) rst weak order convergence, that is = 1, as has been shown in Mikulevicius & Platen (1988) in the case when the coecient functions a, b and c are four times continuous di erentiable with respect to x. In practical situations the coecient functions a, b and c and the function g do sometimes not have the smoothness properties assumed in the above cited papers. Our aim is to prove still some weak order of convergence for the Euler approximation under Holder conditions on a, b and c. This means we will generalize the result in Mikulevicius & Platen (1991) to the case of a di usion process with Poisson jump component. The paper is organized in the following way. Within the Sections 2, 3 and 4 we will de ne the It^o process with jump component, the basic time discretization and the Euler approximation. Section 5 contains the main theorem, which is proved in the remaining part of the paper. For unexplained notations we refer the reader to Ikeda & Watanabe (1989) or Jacod & Shiryaev (1987).
2 It^o Process with Jump Component Let ( ; F ; P ) be a complete probability space with ltrations F = (Ft)t2[0;T ] and F~ = (F~t)t2[0;T ], satisfying the usual conditions. The ltration F will relate to the It^o process X and F~ to the Euler approximation Y . We consider the d-dimensional It^o process X = fX (t); t 2 [0; T ]g with jump component which is de ned as a weak solution of the stochastic equation (1.1) where X (0) denotes the F 0-measurable initial value. The Wiener process W is F -adapted. Furthermore, we write the Poisson martingale measure as
q(d; ds) = p(d; ds) , (d) ds:
(2.1)
Here p(d; ds) is an F -adapted, integer-valued Poisson jump measure with dual predictable projection (d) ds and
E (p(d; ds)) = (d) ds:
(2.2)
The function a = (ai)di=1 denotes the d-dimensional drift vector, b = (bi;j )d;m i;j =1 the d m-di usion matrix and c = (ci)di=1 the d-dimensional jump vector. We assume that a and b are continuous functions and c satis es the condition lim x!y s!t
Z
,
jc(s; x; ) , c(t; y; )j2 (d) = 0 3
(2.3)
for (t; y) 2 [0; T ]
