Abstract
We consider multidimensional stochastic differential equations of the form
dX t =b(t,X t )dt+dW t ,0≤t≤R,R<∞,(1)
with arbitrary initial (probability) distribution μ on ℝ d , d≥1. The first aim of this paper is to give handy-to- verify analytic (i.e. nonstochastic) conditions for the existence of a weak solution of (1), where the drift b will be allowed to have singularities. These investigations are illustrated by various examples. We first concentrate on (a uniform form of) the Novikov condition
E μ exp1 2∫ 0 R |b(s,X s )| 2 ds<∞(2)
and then investigate further sufficient conditions for the applicability of the Girsanov- Maruyama theorem which are not covered by (2). The outcoming results improve some of those of H. J. Engelbert and W. Schmidt [Math. Nachr. 119, 97-115] (for time-independent drifts b(x)) and N. I. Portenko [Theory Probab. Appl. 20, 27-37 (1975); translation from Teor. Veroyatn. Primen. 20, 29-39] (for time-dependent drifts b(t,x)). One of the examples involves a drift which is singular on a dense set in ℝ d but nevertheless satisfies (2).
The second aim of this paper is to discuss some general properties and applications of (2). For instance, we investigate whether the factor 1/2 in the Novikov condition (2) “can be replaced” by 1/2±ε (ε>0). Furthermore, we give several equivalence characterizations of (2) [being connected to the well-known R. Z. Khas'minskij lemma, ibid. 4, 309-318 (1960) resp. ibid. 4, 332-341]. Finally, it is shown that under the Novikov condition (2), the diffusion process with drift b has finite relative entropy with respect to Wiener measure (and thus finite“energy”).
Stummer, W