arXiv subjects
Publications and source records attributed to F. Proske.
Using key tools such as It\^o formula for general semi-martingales, moments estimates for L\'{e}vy-type stochastic integrals and properties of regular varying functions we find conditions under which solutions of stochastic differential equation with jumps are almost sure asymptotically equivalent nonrandom function with $t\to \infty$.
We investigate the occupation measures and local times of Takagi-type functions with roughness parameter $\gamma$, which are H\"older continuous with exponent $H=\frac{\log\gamma}{\log(1/2)}.$ Analytical insight is obtained by embedding these functions into a dynamical system related to the baker transform, whose global attractor is the graph of the Takagi function. The associated stable manifolds support Sinai-Bowen-Ruelle (SBR) measures, which we identify with the laws of certain symmetric Bernoulli convolutions. Dually, where duality is induced by time reversal, we derive a representation of the Takagi-type curves centred around the stable fibres in terms of Bernoulli convolutions, thereby relating SBR measures to occupation measures. While Bernoulli convolutions belong to the first Rademacher chaos, we show that the occupation measure is naturally represented as a functional in the second Rademacher chaos within the framework of non-Gaussian Malliavin calculus. Using a Fourier-analytic criterion together with variants of Weyl's equidistribution theorem, we prove that Takagi-type curves admit square-integrable local times for $\gamma=2^{-1/m}, \, m\geq 7,$ and that the same conclusion holds for drifted Takagi curves for almost every $\gamma\in(1/2,1)$.
In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper can be regarded as an infinite-dimensional generalization of those of A. Y. Veretennikov [42] in the case of certain stochastic delay equations with irregular drift coefficients. The approach proposed in this work rests on Malliavin calculus and arguments of a "local time variational calculus", which may also be used to study other types of stochastic equations as e.g. functional It\^{o}-stochastic differential equations in connection with path-dependent Kolmogorov equations [15].
We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.