We introduce and study some backward Kolmogorov equations associated to filtering problems. In the stochastic framework, SDEs for measure-valued processes arise naturally (Zakai Kushner–Stratonovich equation). The have been intensively studies, assuming that admit a density then by exploiting calculus in Hilbert spaces. Our approach differs from this since we do not assume existence of work dir...

The purpose of this paper is to establish the convergence in law of the sequence of “midpoint” Riemann sums for a stochastic process of the form f ′(W ), where W is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-ofvariable formula in law with a second order correction term which is an Itô integral of f ′′(W ) with respect to...

Let B be a fractional Brownian motion with Hurst parameter H = 1/6. It is known that the symmetric Stratonovich-style Riemann sums for ∫ g(B(s)) dB(s) do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correcti...

Recently, the supersymmetry method was extended from Gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the Hubbard–Stratonovich transformation. Here, we complete this extension by including arbitrary orthogonally and unitary–symplectically invariant matrix ensembles. The results are equivalent to, but the approach is different from the superbosonization formul...

Rigorous justification of the Hubbard-Stratonovich transformation for the Pruisken-Schäfer type of parameterisations of real hyperbolic O(m,n)−invariant domains remains a challenging problem. We show that a naive choice of the volume element invalidates the transformation, and put forward a conjecture about the correct form which ensures the desired structure. The conjecture is supported by com...