1,720,995 research outputs found
A note on the invariance under change of measure for stochastic test function and distribution spaces
No full textIn this note we prove that the spaces .S/, .S/ and G, G are invariant under a certain
class of translations of the underlying Brownian motion. This problem arises naturally in
dealing with anticipating stochastic differential equations, in particular when the Girsanov
theorem is involved. The proofs are based on a Bayes formula for second-quantization
operators that was derived by Lanconelli [Lanconelli, A., 2006a. Bayes' formula for second
quantization operators. Stoch. Dyn. 6 (2), 245253] and on the properties of the translation
operators
A new probabilistic representation for the solution of the heat equation
No full textWe obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida-Malliavin derivatives of the random variables involved, exhibits many useful properties that are to some extents opposite to some peculiar features of the Wick produc
Using Malliavin calculus to solve a chemical diffusion master equation
Full text linkWe propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in [4] for modelling the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space
Bayes' formula for second quantization operators
Full text linkThe Bayes’ formula provides the relationship between conditional expectations with
respect to absolutely continuous measures. The conditional expectation is in the context
of the Wiener space — an example of second quantization operator. In this note we
obtain a formula that generalizes the above-mentioned Bayes’ rule to general second
quantization operators
Translated Brownian motions and associated Wick products
No full textThe concept of Wick product is strongly related to the underlying
Brownian motion we have fixed on the probability space. Via the Girsanov’s
theorem we construct a family of new Brownian motions, obtained as
translations of the original one, and to each of them we associate a Wick
product. This produces a family of Wick products, named gamma-Wick products,
parameterized by the performed translations. We aim to describe this family of
products. We also define a new family of stochastic integrals, which are related
in a natural way to the gamma-Wick products
Computing conditional expectation of multidimensional diffusion processes
No full textWe study multidimensional diffusion processes and give an explicit representation for their conditional
expectation. Starting from the solution formula for one dimensional stochastic differential equations
found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of
multidimensional diffusions without resorting to the Markov property of the process and therefore
without requiring an explicit expression for the semi group associated to it
A note about the invariance of the basic reproduction number for stochastically perturbed SIS models
Full text linkIn Gray et al. [A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (3) (2011) 876–902] a susceptible-infected-susceptible (SIS) stochastic differential equation (SDE), obtained via a suitable random perturbation of the disease transmission coefficient in the classic SIS model, has been studied. Such random perturbation enters via an informal manipulation of stochastic differentials and leads to an Itô's-type SDE. The authors identify a stochastic reproduction number, which differs from the standard one for the presence of those additional parameters that describe the employed random perturbation, and show that, similarly to the deterministic case, the stochastic reproduction number rules the asymptotic behavior of the solution. Aiming to make that random perturbation rigorous, we suggest an alternative approach based on a Wong–Zakai approximation argument thus arriving at a different stochastic model corresponding to the Stratonovich version of the Itô equation analyzed in Gray et al. Rather surprisingly, the asymptotic behavior of this alternative model turns out to be governed by the same reproduction number as the deterministic SIS equation. In other words, the random perturbation does not modify the threshold for extinction and persistence of the disease
Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations
Full text linkWe extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero
The Ornstein-Uhlenbeck process and a related Malliavin calculus
No full textWe consider two different Brownian motions, B and Ba; each of
them produces a Wiener-Itˆo chaos representation and therefore defines a
Malliavin derivative, D and Da, and a Skorohod integral, and a, respectively.
Our aim is to rewrite the differential operators Da and a in terms of
D and
A remark on the renormalized square of the solution of the stochastic heat equation and on its associated evolution
No full textWe consider the renormalized square of the solution of the stochastic heat equation and obtain for this process a new dynamic involving the second quantization of an unbounded operator. This is achieved by a version of the Itô formula which we derive by a simple application of the Wick chain rul
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