Marcus versus Stratonovich for Systems with Jump Noise

The famous It\^o-Stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise. In physics literature, this dilemma is often resolved in favour of the Stratonovich prescription because of its two characteristic properties valid for systems driven by Brownian motion: (i) it allows physicists to treat stochastic integrals in the same way as conventional integrals, and (ii) it appears naturally as a result of a small correlation time limit procedure. On the other hand, the Marcus prescription [IEEE Trans. Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to retain (i) and (ii) for systems driven by a Poisson process, L\'evy flights or more general jump processes. In present communication we present an in-depth comparison of the It\^o, Stratonovich, and Marcus equations for systems with multiplicative jump noise. By the examples of areal-valued linear system and a complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we compare solutions obtained with the three prescriptions.Comment: 14 pages, 4 figure

Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces

We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type $\int_0^t F_\gamma(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a L\'evy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in probability of stochastically continuous c\`adl\`ag processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish criteria for the convergence in probability of L\'evy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein--Uhlenbeck processes with diagonalisable generators.Comment: 34 pages, 1 figur

Stochastic selection problem for a Stratonovich SDE with power non-linearity

In our paper [Bernoulli 26(2), 2020, 1381--1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^{\alpha}\circ dB$, $\alpha\in (0,1)$. These solutions have the form $X_t^\theta=F(B^\theta_t)$, where $F(x)=\frac{1}{1-\alpha}|x|^{1/(1-\alpha)}\text{sign}\, x$ and $B^\theta$ is the skew Brownian motion with skewness parameter $\theta\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta=0$.Comment: 15 pages, 1 figur