On the Ito formula in a Banach Space

2000 ◽  
Vol 7 (1) ◽  
pp. 155-168
Author(s):  
B. Mamporia
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Wiener Process ◽  
Separable Banach Space ◽  
Stochastic Integral ◽  
Covariance Operator ◽  
Itô Formula ◽  
Ito Formula ◽  
Special Construction ◽  
Frechet Derivatives

Abstract If (Wt ) t∈[ 0, 1] is a Wiener process in an arbitrary separable Banach space X, ψ : [0, 1] × X → Y is a continuous function with values in another separable Banach space, and ψ has continuous Frechet derivatives , and , then the Ito formula is obtained for ψ(t, Wt ). The method is based on the concept of covariance operator and a special construction of the Ito stochastic integral.

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 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

scholarly journals The Topological Support of Gaussian Measure in Banach Space

1975 ◽  
Vol 57 ◽  
pp. 59-63 ◽  
Author(s):  
N. N. Vakhania
Keyword(s):  
Banach Space ◽  
Probability Measure ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Covariance Operator ◽  
Probability Measures ◽  
Gaussian Measures ◽  
Linear Spaces

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

Itô stochastic integral. Itô formula. Tanaka formula

2009 ◽  
pp. 193-213
Author(s):  
Dmytro Gusak ◽  
Alexander Kukush ◽  
Alexey Kulik ◽  
Yuliya Mishura ◽  
Andrey Pilipenko
Keyword(s):  
Stochastic Integral ◽  
Itô Formula ◽  
Ito Formula ◽  
Tanaka Formula

On the Existence and Uniqueness of a Solution to a Stochastic Differential Equation in a Banach Space

2004 ◽  
Vol 11 (3) ◽  
pp. 515-526
Author(s):  
B. Mamporia
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Covariance Operator ◽  
Sufficient Condition ◽  
Existence Of A Solution ◽  
Arbitrary Banach Space ◽  
Uniqueness Of A Solution

Abstarct A sufficient condition is given for the existence of a solution to a stochastic differential equation in an arbitrary Banach space. The method is based on the concept of covariance operator and a special construction of the Itô stochastic integral in an arbitrary Banach space.

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