Conditional expansions and their applications. (English) Zbl 1075.60515

Summary: We consider a conditional limit theorem and conditional asymptotic expansions. Our discussion will be based on the Malliavin calculus. First, we treat a problem of lifting limit theorems to their conditional counterparts. Next, we provide asymptotic expansions in a general setting including the so-called small \(\sigma \)-models. In order to give a basis to the asymptotic expansion scheme for perturbed jump systems, we will build an extension to the Watanabe theory in part. Finally, we derive the asymptotic expansions (double Edgeworth expansions) of conditional expectations.


60F99 Limit theorems in probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60J60 Diffusion processes
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