Stochastic integral with respect to a generalized wiener process in a conuclear space

2005 ◽  
pp. 159-168 ◽  
Author(s):  
Tomasz Bojdecki ◽  
Jacek Jakubowski
Keyword(s):  
Wiener Process ◽  
Stochastic Integral ◽  
Generalized Wiener Process

Stochastic Integrals

2019 ◽  
pp. 43-66
Author(s):  
Tomas Björk
Keyword(s):  
Wiener Process ◽  
Stochastic Integral ◽  
Stochastic Integrals ◽  
Itô Formula ◽  
Martingale Theory ◽  
Ito Formula

We introduce the Wiener process, the Itô stochastic integral, and derive the Itô formula. The connection with martingale theory is discussed, and there are several worked-out examples

INTEGRAL REPRESENTATIONS OF CYLINDRICAL LOCAL MARTINGALES IN EVERY SEPARABLE BANACH SPACE

2007 ◽  
Vol 10 (03) ◽  
pp. 365-379 ◽  
Author(s):  
MARTIN ONDREJÁT
Keyword(s):  
Banach Space ◽  
Wiener Process ◽  
Separable Banach Space ◽  
Stochastic Integral ◽  
Integral Representations ◽  
Local Martingale ◽  
Sufficient Condition ◽  
General Sufficient Condition

A general sufficient condition for a continuous cylindrical local martingale on a separable Banach space to be a stochastic integral with respect to a Wiener process is proven.

scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

scholarly journals ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS

2002 ◽  
Vol 45 (2) ◽  
pp. 301-325
Author(s):  
James S. Groves
Keyword(s):  
Banach Space ◽  
Wiener Process ◽  
Separable Banach Space ◽  
Stochastic Integral ◽  
Continuity Condition ◽  
Factorization Condition ◽  
Spectral Representations ◽  
Densely Defined ◽  
Uniformly Bounded

AbstractFor Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through $\ell^2$ as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation$$ \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, $$with values in a separable Banach space, is studied. The operator $\sLa$ is closed and densely defined. A weak solution $(\bm{Z}_t,\bm{B}_t)$, where $\bm{Z}_t$ is centred, Gaussian and stationary, while $\bm{B}_t$ is a Q-Wiener process, is given when $\ri\sLa$ and $\ri\sLa^*$ generate $C_0$ groups and the resolvent of $\sLa$ is uniformly bounded on the imaginary axis. Both $\bm{Z}_t$ and $\bm{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process.AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10

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