scholarly journals A weak stochastic integral in Banach space with application to a linear stochastic differential equation

1983 ◽  
Vol 10 (1) ◽  
pp. 97-125 ◽  
Author(s):  
Nadav Berman ◽  
William L. Root
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Stochastic Differential Equation ◽  
Stochastic Integral ◽  
Linear Stochastic Differential Equation

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.

Weak Solutions for Semi-Martingales

1981 ◽  
Vol 33 (5) ◽  
pp. 1165-1181 ◽  
Author(s):  
J. Pellaumail
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Weak Solution ◽  
Stochastic Differential Equation ◽  
Marginal Distribution ◽  
Uniform Norm ◽  
Section 8 ◽  
Initial Space ◽  
Sample Paths ◽  
The Given

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).Namely, this weak solution is a probability, so-called “rule,” defined on (DH × Ω), DH being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.

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