42 research outputs found
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 Skorokhod
topology of the Hilbert valued stochastic convolution integrals of the type
to a process driven
by a L\'evy process . In Banach spaces we introduce strong, weak and product
modes of -convergence, prove a criterion for the -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 of the Stratonovich stochastic
differential equation , . These
solutions have the form , where
and is
the skew Brownian motion with skewness parameter starting at
. In this paper we show how an addition of small external additive
noise restores uniqueness. In the limit as ,
we recover heterogeneous diffusion corresponding to the physically symmetric
case .Comment: 15 pages, 1 figur