Transition density estimation for stochastic differential equations via forward-reverse representations

Probability density estimation in stochastic environmental models using reverse representations

“…In a more special setting the notion of reverse or adjoint diffusion is due to Thomson (Thomson 1987). The here introduced reverse system can be seen as a generalization of Thomson's approach and is derived in (Milstein, Schoenmakers, and Spokoiny 2002) in a more transparent and more rigorous way. We first introduce a reversed time variables = T + t − s and define the functions…”

“…Remark. In (Milstein, Schoenmakers, and Spokoiny 2002) a forward-reverse projection estimator is also studied. Although the projection estimators has in some sense additional advantages, we only consider applications of (9) in this article, and refer to (Milstein, Schoenmakers, and Spokoiny 2002) for details with respect to projection estimators.…”

“…The most efficient implementation is, however, obtained if the forward realizations and the reverse system realizations are combined. (Milstein, Schoenmakers, and Spokoiny 2002) called this the Forward Reverse Estimator.…”

“…Recently, (Milstein, Schoenmakers, and Spokoiny 2002) introduced the concept of reverse time diffusion. The classical Monte Carlo estimate is based on forward realizations of the original stochastic model.…”

“…The classical Monte Carlo estimate is based on forward realizations of the original stochastic model. (Milstein, Schoenmakers, and Spokoiny 2002) derived a reverse system from the original model and showed that the classical Monte Carlo estimate can also be based on realization of this reverse system. For many applications it is more efficient to use realizations of the reverse system instead of the original model.…”