4,878 research outputs found
Integral Representation of Generalized Grey Brownian Motion
In this paper we investigate the representation of a class of non Gaussian
processes, namely generalized grey Brownian motion, in terms of a weighted
integral of a stochastic process which is a solution of a certain stochastic
differential equation. In particular the underlying process can be seen as a
non Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing
the representation results of Muravlev as well as Harms and Stefanovits to the
non Gaussian case.Comment: arXiv admin note: text overlap with arXiv:1708.06784,
arXiv:1807.0786
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
Exchangeable pairs on Wiener chaos
In [14], Nourdin and Peccati combined the Malliavin calculus and Stein's
method of normal approximation to associate a rate of convergence to the
celebrated fourth moment theorem [19] of Nualart and Peccati. Their analysis,
known as the Malliavin-Stein method nowadays, has found many applications
towards stochastic geometry, statistical physics and zeros of random
polynomials, to name a few. In this article, we further explore the relation
between these two fields of mathematics. In particular, we construct
exchangeable pairs of Brownian motions and we discover a natural link between
Malliavin operators and these exchangeable pairs. By combining our findings
with E. Meckes' infinitesimal version of exchangeable pairs, we can give
another proof of the quantitative fourth moment theorem. Finally, we extend our
result to the multidimensional case.Comment: 19 pages, submitte
Structural operational semantics for stochastic and weighted transition systems
We introduce weighted GSOS, a general syntactic framework to specify well-behaved transition systems where transitions are equipped with weights coming from a commutative monoid. We prove that weighted bisimilarity is a congruence on systems defined by weighted GSOS specifications. We illustrate the flexibility of the framework by instantiating it to handle some special cases, most notably that of stochastic transition systems. Through examples we provide weighted-GSOS definitions for common stochastic operators in the literature
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