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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
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(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