cases this allows us to examine the rate of convergence of semimartingales with reflection to the diffusion process with reflection. One such example was given ...
Lithuanian Mathematical Journal,
Vol. 32, N o . 2, 1992
O N T H E R A T E OF C O N V E R G E N C E OF T H E D I F F U S I O N APPROXIMATIONS K. K u b i I i u s 1. I N T R O D U C T I O N In recent years, several papers with the estimator of the L6vy-Prokhorov distance between distributions of semimartingales and diffusion processes appeared (see, for example, [9]). The results were formulated in terms of triplets of predictable characteristics for the prelimiting and limiting processes. Let X n be a sequence of semimartingales and let t
Xt = Xo + / a ( s , X )
ds -I- Mr,
t/> O,
(1)
0
where M is a local martingale. Suppose h is a H51der function on Skorokhod's space D[0,T](~m). We are interested in estimation of the L6vy-Prokhorov distance between distributions of h ( X n) and h(X). In some cases this allows us to examine the rate of convergence of semimartingales with reflection to the diffusion process with reflection. One such example was given in [15]. One can find some results on weak convergence in a more general situation in [1] and [8]. We would especially like to note that the method used in [9] is unsuitable in our situation. The process defined by (1) is non-Markov, while the limiting processes in [9] were Markovian. The results obtained will be used for a queueing network model.
2. M A I N R E S U L T S Let (f~, 9", P ) be a complete probability space with filtrations IFn -- (9"~)t/>0 satisfying the usual conditions. We say that X n = (X~)t>~o is a special (P, ]F~)-semimartingale if X~' = X~' + A~' + M~n,
t/> O,
where A n is a predictable process of locally finite variation and M n is a local martingale. Let X n be an m-dimensional (P, F~)-semimartingale. It is well known t h a t it can be uniquely represented in the form t
X~ =X'~ +a'~
t
t +
x(pn-IIn)(ds, dx)+ o
I~1~0,
0 I~1>1
where a " is a predictable process of locally finite variation, X ne is a continuous martingale part of X n with quadratic characteristics C n (i.e., C" = ( c(id)n )l 0 : p { ( z , y ) : d(z,y) >/e} ~ e}, i.e., there exist random elements X and Y with L ( X ) = Q1 and L(Y) = Q2 such that ~(Q1, Q2) = X(X, Y). It is easy to see that, for any X , Y 9 V,
~(~(x), ~(v)) ~ 1. For any x 9 D define a mapping fP: D --+ C by f~(x) = xtk + (pt/T - k)(xtk+, -- xt~) for tk ~< t / 0,
t ]
0
where M ~ and M are local martingales, and the function a satisfies inequalities (2), (3). PROPOSITION 3. The following statements are true:
1. Let M ~ and M be continuous local martingales. Then there exists a constant C = C(T, k) such that 7rT(Zn , X) < C r T ((X~, M n), (Xo, M)); 2. Let la(t,x)l p, I~[i)] 0, from Doob's inequality we get
l}
and Nt~ = M n (t A 7-tn). Then
E(sup \ t 7)
,
