ABSTRACT
Chapter 5
The method of the
maxingale problem
The method of nite-dimensional distributions considered in Chap-
ter 4 does not allow us to prove LD convergence in distribution to
idempotent processes other than idempotent processes with indepen-
dent increments. In this chapter we consider a dierent approach,
which is an analogue of the martingale problem approach in weak
convergence theory and consists in identifying the limit deviability
as a solution to a maxingale problem. As in Chapter 4, we con-
sider a net of semimartingales fX
; 2 g dened on respective
stochastic bases (
;F
;F
; P
) with paths in D = D (R
+
;R
d
). We
assume as xed a net fr
; 2 g of real numbers greater than 1
converging to 1 as 2 . It is used as a rate for LD convergences
below, which refer to the Skorohod topology. The limit semimaxin-
gale X is assumed to be \canonical" in that it is dened on D by
X
t
(x) = x
t
; x 2 D ; t 2 R
+
. It will actually be Luzin-continuous
so that we can equivalently consider it as the canonical idempotent
process on C = C (R
+
;R
d
). The next two sections are concerned with
identifying maxingale problems whose solutions are LD accumulution
points of fL(X
); 2 g: Section 5.1 species the maxingale prob-
lem in terms of convergence of stochastic exponentials and assumes
the Cramer condition for the X
, while Section 5.2 considers conver-
gence of the characteristics of the semimaxingales and does without
the Cramer condition. Section 5.3 is devoted to specic LD conver-
gence results. Section 5.4 considers applications to large deviation
convergence of Markov processes.
