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Functionals of diffusion processes as stochastic integrals

Published online by Cambridge University Press:  24 October 2008

M. H. A. Davis
M. H. A. Davis
Affiliation:
Department of Computing and Control, Imperial College, 180 Queen's Gate, London SW7 2BZ
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Extract

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Then

is a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representation

For Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 1 , January 1980 , pp. 157 - 166
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Clark, J. M. C.The representation of functionals of Brownian motion by stochastic integrals. Annals of Math Stat. 41 (1970), 1282–1295.CrossRefGoogle Scholar
(2)Friedman, A.Stochastic differential equations, vol. 1, Academic Press (New York, 1975).Google Scholar
(3)Haussmann, U. G.Functionals of Ito processes as stochastic integrals. SIAM J. Control Optimization 16 (1978), 252–269.CrossRefGoogle Scholar
(4)Haussmann, U. G.On the integral representation of functionals of Ito processes. Stochastics 3 (1979), 17–28.CrossRefGoogle Scholar
(5)Loéve, M.Probability theory, 3rd ed., D. van (Princeton, 1963).Google Scholar