Abstract
Fourier analysis provides one of the well known methods by which local behaviour of Gaussian processes, especially their occupation densities, can be investigated. Berman [3] initiated on approach which proved to be rather successful also in the more general area of Gaussian random fields and random fields with independent increments (see Geman, Horowitz [6] for a survey, Ehm [4]). The observation basic to this approach is comprised in the statement: the Fourier transform of the occupation measure of a real valued function is square integrable if and only if it posesses a square integrable density which then serves as a “local time” or “occupation density”. It is therefore, at least in principle, quite general.
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References
Baumgärtel, H. Analytic perturbation theory for matrices and operators. Birkhäuser: Basel, Boston (1985).
Bellman, R.: Introduction to matrix analysis. McGraw-Hill: New York (1970).
Berman, S.M.: Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 (1969), 277–300.
Ehm, W.: Sample function properties of multi-parameter stable processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 56 (1981), 195–228.
Fredholm, I.: Sur une classe d’équations fonctionnelles. Acta Math. 27 (1903), 365–390.
Geman, D., Horowitz, J. Occupation densities. Ann. Probab. 8 (1980), 1–67.
Imkeller, P. Occupation densities for stochastic integral processes in the second Wiener chaos. Preprint, Univ. of B.C. (1990).
Jeulin,Th.,Yor,M. (eds). Grossissement de filtrations: exemples et applications. Séminaire de Calcul Stochastique, Paris 1982/83. LNM 1118. Springer: Berlin, Heidelberg, New York (1985).
Jörgens, K. Linear integral operators. Pitman: Boston, London (1982).
Kato, T.: Perturbation theory for linear operators. Springer: Berlin, Heidelberg, New York (1966).
Nualart, D.: Noncausal stochastic integrals and calculus. LNM 1516. Springer: Berlin, Heidelberg, New York (1988).
Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields 78 (1988), 535–581.
Nualart, D., Pardoux, E. Boundary value problems for stochastic differential equation. Preprint (1990).
Nualart, D., Pardoux, E.: Second order stochastic differential equations with Dirichlet boundary conditions. Preprint (1990).
Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin calculus. Probab. Th. Rel. Fields 73 (1986), 255–280.
Ocone, D., Pardoux, E.: Linear stochastic differential equations with boundary conditions. Probab. Th. Rel. Fields, to appear. (1990).
Pietsch, A.Eigenvalues and s-numbers. Cambridge University Press: Cambridge, London (1987).
Reed, M., Simon, B.: Methods of modern mathematical physics. IV: Analysis of operators. Acad. Press: New York (1978).
Rellich, F.: Perturbation theory of eigenvalue problems. Gordon, Breach: New York, London (1969).
Simon, B.: Trace ideals and their applications. London Math. Soc. Lecture Notes Series 35. Cambridge University Press: Cambridge, London (1979).
Skorohod, A.V.: On a generalization of a stochastic integral. Theor. Prob. Appl. 20 (1975), 219–233.
Smithies, F.: Integral equations. Cambridge University Press: Cambridge, London (1965).
Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Springer: Berlin, Heidelberg, New York (1984).
Zakai, M.: The Malliavin calculus. Acta Appl. Math. 3 (1985), 175–207.
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Imkeller, P. (1991). On the Existence of Occupation Densitites of Stochastic Integral Processes via Operator Theory. In: Çinlar, E., Fitzsimmons, P.J., Williams, R.J. (eds) Seminar on Stochastic Processes, 1990. Progress in Probability, vol 24. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-0562-0_10
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DOI: https://doi.org/10.1007/978-1-4684-0562-0_10
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