×

zbMATH — the first resource for mathematics

Densité des diffusions en temps petit: Développements asymptotiques. I. (English) Zbl 0546.60079
Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 402-408 (1984).
[For the entire collection see Zbl 0527.00020.]
Consider a stochastic differential equation \(dx_ t=\sigma (t,x_ t)dw_ t+b(t,x_ t)dt\). The author investigates the asymptotic behaviour of the transition density p(s,x,t,y) of the solution \(x_ t\) when t-s tends to zero. This problem is important in the theory of parabolic equations. It has been studied by a number of authors, especially in the homogeneous case. The limit behaviour of the density is tightly connected with the Riemann metrics \(d_ s(x,y)\) defined by the vector field \(x\to a(s,x)^{-1}\) where \(a=\sigma\sigma^*.\)
In the paper the following expansion of the transition density is given: \[ p(s,x,t,y)=(t-s)^{-m/2}\exp (-(d^ 2_ s(x,y)/2(t-s))(\alpha_ 0+(t-s)\alpha_ 1+...+(t-s)^ N\alpha_ N+O((t-s)^{N+1})) \] where m is the dimension of the space. The probabilistic description of the coefficients and an estimate of the remainder are obtained. The article is self-contained. The proofs are based on the Laplace method combined with the Girsanov formula.
Reviewer: Yu.M.Kabanov

MSC:
60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
Full Text: Numdam