Banach space valued stochastic differential equations
Thesis/Dissertation
·
OSTI ID:7112080
Let W(t,[omega]) be a Brownian motion on an abstract Wiener space (i,H,B) corresponding to the canonical normal distribution of H. The well known theorem of Girsanov is proved for such a process with the perturbing term taking values in the Hilbert space H. Consider the stochastic integral equation [xi](t) = x + [integral][sub o][sup t] A(s,[xi](s))dW (s) + [integral][sub o][sup t] [sigma](s,[xi](s))ds where x is in B, A([center dot],[center dot]) and [sigma]([center dot],[center dot]) are bounded continuous functions from O,[infinity] x B to I + L[sub 2](H) and H respectively. Here L[sub 2](H) denotes the collection of Hilbert Schmidt operators on H. Furthermore, suppose for every s [ge] O the restriction of A(s,[center dot]) to H is invertible and A(s,[center dot]) and [sigma](s,[center dot]) are both Frechet differentiable in the directions of H with bounded derivatives. Under suitable conditions, it is proved that for each t [ge] O, the measure generated by the solution [xi](t) of the above stochastic integral equation is differentiable in the directions of H in the sense of Fomin. By adding more conditions on A and [sigma], it is shown that the transition probability associated with the solution of the stochastic differential equation d[xi](t) = A(t,[xi](t))dW (t) + [sigma](t,[xi])dt satisfies the infinite dimensional Kolmogorov's forward equation in the distribution sense.
- Research Organization:
- Louisiana State Univ. and Agricultural and Mechanical Coll., Baton Rouge, LA (United States)
- OSTI ID:
- 7112080
- Country of Publication:
- United States
- Language:
- English
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