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Stochastic integral representation of multiplicative operator functionals of a Wiener process


Author: Mark A. Pinsky
Journal: Trans. Amer. Math. Soc. 167 (1972), 89-104
MSC: Primary 60J60
DOI: https://doi.org/10.1090/S0002-9947-1972-0295433-8
MathSciNet review: 0295433
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Abstract: Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation

$\displaystyle M(t) = I + \sum\limits_{j = 1}^d {\int_0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int_0^t {M(s){B_0}(x(s))ds,} $

where $ {B_0}, \ldots ,{B_d}$ are bounded operator functions on $ {R^d}$. The conditions require that the equation $ T(t)f = E[M(t)f(x(t))]$ define a semigroup on $ {L^2}({R^d})$ whose infinitesimal generator has a domain which contains all linear functions of the coordinates $ ({x_1}, \ldots ,{x_d})$. The proof of this result depends on an a priori representation of the semigroup $ T(t)$ in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with $ {B_0} = 0$. A sufficient condition that M belong to this class is that $ E[M(t)]$ be the identity operator on L and that $ M(t)$ be invertible for each $ t > 0$. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.

References [Enhancements On Off] (What's this?)

  • [1] D. G. Babbitt, Wiener integral representations for certain semigroups which have infinitesimal generators with matrix coefficients, J. Math. Mech. 19 (1970), 1051-1067. MR 0270205 (42:5097)
  • [2] C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 181-194. MR 0283883 (44:1113)
  • [3] K. Itô and S. Watanabe, Transformation of Markov processes by multiplicative functionals, Ann. Inst. Fourier (Grenoble) 15 (1965), fasc. 1, 13-30. MR 32 #1755. MR 0184282 (32:1755)
  • [4] H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209-245. MR 36 #945. MR 0217856 (36:945)
  • [5] P. A. Meyer, Séminaire de probabilités. Vol. I (Université de Strasbourg, 1966-67), Lecture Notes in Math., vol. 39, Springer-Verlag, Berlin, 1967.
  • [6] -, Probability and potentials, Blaisdell, Waltham, Mass., 1966; French ed., Publ. Inst. Math. Univ. Strasbourg, no. 14, Actualités Sci. Indust., no. 1318, Hermann, Paris, 1966. MR 34 #5118; MR 34 #5119.
  • [7] H. P. McKean, Jr., Stochastic integrals, Probability and Math. Statist., no. 5, Academic Press, New York, 1969. MR 40 #947. MR 0247684 (40:947)
  • [8] M. Pinsky, Multiplicative operator functionals of a Markov process, Bull. Amer. Math. Soc. 77 (1971), 377-380. MR 0298769 (45:7818)
  • [9] D. W. Stroock, On certain systems of parabolic equations, Comm. Pure Appl. Math. 23 (1970), 447-457. MR 0272075 (42:6956)
  • [10] H. Tanaka, Note on continuous additive functionals of the 1-dimensional Brownian path, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/63), 251-257. MR 29 #6559. MR 0169307 (29:6559)
  • [11] A. D. Wentsell (A. D. Ventcel'), On continuous additive functionals of a multidimensional Wiener process, Dokl. Akad. Nauk SSSR 142 (1962), 1223-1226=Soviet Math. Dokl. 3 (1962), 264-266. MR 27 #4265. MR 0154316 (27:4265)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0295433-8
Keywords: Multiplicative operator functional, martingale stochastic integral, Hilbert-Schmidt operators, parabolic systems of equations
Article copyright: © Copyright 1972 American Mathematical Society

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