Vol. 41, No. 2, 1972

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Stochastic integrals in abstract Wiener space

Hui-Hsiung Kuo

Vol. 41 (1972), No. 2, 469–483

Let W(t,ω) be the Wiener process on an abstract Wiener space (i,H,B) corresponding to the canonical normal distributions on H. Stochastic integrals 0tξ(s,ω)dW(s,ω) and 0t(ζ(s,ω), dW(s,ω)) are defined for non-anticipating transformations ξ with values in (B,B) such that (ξ(i,ω) I)(B) B and ζ with values in H. Suppose X(t,ω) = x0+ 0tξ(s,ω)dW(s,ω) + 0tσ(s,ω)ds, where σ is a non-anticipating transformation with values in H. Let f(t,x) be a continuous function on R × B, continuously twice differentiable in the H-directions with D2f(t,x) ∈ℬ1(H,H) for the x variable and once differentiable for the t variable. Then f(t,X(t,ω)) = f(0,x0) + 0tξ(s,ω)Df(s,X(s,ω))dW(s,ω) + 0t{∂f∕∂s(s,X(s,ω)) + Df(s,X(s,ω))(s,ω)+ 1
2 trace[ξ(s,ω)D2f(s,X(s,ω))ξ(s,ω)]}ds, where ,is the inner product of H. Under certain assumptions on A and σ it is shown that the stochastic integral equation X(t,ω) = x0 + 0tA(X(s,ω))dW(s,ω) + 0tσ(X(s,ω))ds has a unique solution. This solution is a homogeneous strong Markov process.

Mathematical Subject Classification 2000
Primary: 28A40
Secondary: 60H05
Received: 23 November 1970
Published: 1 May 1972
Hui-Hsiung Kuo