Summary
In this paper, martingale measures, introduced by J.B. Walsh, are investigated. We prove, with techniques of stochastic calculus, that each continuous orthogonal martingale measure is the time-changed image martingale measure of a white noise.
We also exhibit a representation theorem for certain vector martingale measures as stochastic integrals of orthogonal martingale measures. Thus we can study the following martingale problem:
whereL is a second order differential operator andq a predictable random measure-valued process. We prove that this problem is bound to a stochastic differential equation with a term integral with respect to a martingale measure.
Article PDF
Similar content being viewed by others
References
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel. Paris: Hermann 1976
El Karoui, N., Lepeltier, J.P.: Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson. Z. Wahrscheinlichkeitstheor. Verw. Geb.39, 111–133 (1977)
El Karoui, N., Huu Nguyen, D., Jeanblanc-Picqué, M.: Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics20, 169–221 (1987)
El Karoui, N., Jeanblanc-Picqué: Partially observable diffusions with control in the observation process. Prépublication du Laboratoire de Probabilités de l'Université Paris VI.
Funaki, T.: A certain class of diffusion processes associated with non linear parabolic equations. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 331–348 (1984)
Grigelionis, B.: On the representation of integer valued measures by means of stochastic integrals with respect to Poisson measures. Litov. Mat. Sb.11, 93–108 (1971)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam/Kodansha: North Holland Mathematical Library 1981
Jacod, J., Mémin, J.: Weak and strong solutions of stochastic differential equations: existence and stability. In: Proc. Durham Symp. 1980 (Lect. Notes Math., vol. 851) Berlin Heidelberg New York: Springer 1981
Méléard, S., Roelly-Coppoletta, S.: Systèmes de particules et mesures martingales: un théorème de propagation du chaos. Séminaire de Probabilités XXII (Lect. Notes Math. vol. 1321, pp. 438–448) Berlin Heidelberg New York: Springer 1988
Méléard, S., Roelly-Coppoletta, S.: A generalized equation for a continuous measure branching process. To appear in: Proceedings “Stochastic PDE's and Applications.” Trento 1988 Berlin Heidelberg New York: Springer
Méléard, S., Roelly-Coppoletta, S.: Discontinuous measure-valued branching processes and generalized stochastic equations. Prépublication no. 9, du Laboratoire de Probabilités de l'Université Paris VI.
Neveu, J.: Processus aléatoires gaussiens: Séminaire de Mathématiques supérieures-1968, Presses de l'Université de Montréal
Skorohod, A.V.: Studies in the theory of random processes. Reading, Mass.: Addison Wesley 1965
Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d'été de Probabilités de Saint-Flour XIV-1984. Berlin Heidelberg New York: Springer 1986
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Karoui, N.E., Méléard, S. Martingale measures and stochastic calculus. Probab. Th. Rel. Fields 84, 83–101 (1990). https://doi.org/10.1007/BF01288560
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01288560