In this paper we present two methods for computing filtered estimates for moments of integrals and stochastic integrals of continuous-time nonlinear systems. The first method utilizes recursive stochastic partial differential equations. The second method utilizes conditional moment generating functions. An application of these methods leads to the discovery of new classes of finitedimensional f...

We study stochastic convolutions providing by fundamental solutions of a class of integrodifferential equations which interpolate the heat and the wave equations. We give sufficient condition for the existence of function–valued convolutions in terms of the covariance kernel of a noise given by spatially homogeneous Wiener process.

The normal inverse Gaussian (NIG) distribution is a recent flexible closed form distribution that may be applied as a model of heavy-tailed processes. The NIG distribution is completely specified by four real valued parameters that have natural interpretations in terms of the shape of the resulting probability density function. By choosing the parameters appropriately, one can describe a wide r...

The random point field which describes the position distribution of the system of ideal boson gas in a state of Bose-Einstein condensation is obtained through the thermodynamic limit. The resulting point field is given by convolution of two independent point fields: the so called boson process whose generating functional is represented by inverse of the Fredholm determinant for an operator rela...

Discrete multiplicative turbulent cascades are described using a formalism involving infinitely divisible random measures. This permits to consider the continuous limit of a cascade developed on a continuum of scales, and to provide the stochastic equations defining such processes, involving infinitely divisible stochastic integrals. Causal evolution laws are also given. This gives the first ge...

In the paper, by virtue of convolution theorem for Laplace transforms, with aid three monotonicity rules ratios two functions, definite integrals, and in terms majorization, light other analytic techniques, author presents decreasing properties defined four polygamma functions.

Measures with values in non-Archimedean fields, which are quasi-invariant and descending at infinity on topological vector spaces over non-Archimedean fields, are studied in this paper. Moreover, their characteristic functionals are considered. In particular, measures having convolution properties like classical Gaussian measures are investigated in the paper. Applications of such measures to p...

A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic skew product flows, Nonlinearity 24 (2011) 1361–1367, gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous ...