Summary
A theory of stochastic differential equations driven by predictable processes in Stratonovich sense is developed. These driving processes include a large class of discontinuous semimartingales. The theory of stochastic differential equations driven by continuous semimartingales in Stratonovich sense is extended without involving Lebesgue-Stieltjes integrals as done by Meyer. Moreover, a change of variables formula without extra terms involving the jumps of the processes holds for this theory. Results on approximation of driving processes are preserved.
References
Benes, V.E., Shepp, L.A., Witsenhausen, H.S.: Some solvable stochastic control problems. Stochastics4, 134–160 (1980)
Doleans-Dade, C.: On the existence and unicity of solutions of stochastic integral equations, Z. Wahrscheinlichkeitstheor. Verw. Geb.36, 93–101 (1976)
Ferreyra, G.: A Wong-Zakai type theorem for certain discontinuous semimartingales. J. Theor. Probab.2, 313–323 (1989)
Ferreyra, G.: The optimal control problem for the Vidale-Wolfe advertising model revisited. (preprint 1988)
Gihman, I.I., Skorohod, A.V.: Stochastic differential equations. New York Berlin Heidelberg: Springer 1972
Harrison, J.M.: Brownian models of queueing networks with heterogeneous customer populations. Proceedings of the IMA workshop on stochastic differential systems. Berlin Heidelberg New York: Springer (in press)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. New York Amsterdam: North Holland 1981
Karatzas, I., Shreve, S.: Connections between optimal stopping and singular stochastic control; I. Monotone follower problems. SIAM J. Control Optimization22, 856–877 (1984)
Kushner, H.J.: Jump-diffusion approximation for ordinary differential equations with wide-band random right hand sides. SIAM J. Control Optimization17, 729–744 (1979)
Marcus, S.: Modelling and approximation of stochastic differential equations driven by semimartingales. Stochastics4, 223–245 (1981)
Meyer, P.A.: Un cours sur les integrales stochastiques. In: Meyer, P.A. (ed.) Seminaire de probabilities X. (Lect. Notes Math., vol. 551, pp. 245–400) Berlin Heidelberg New York: Springer 1976
Picard, J.: Approximation of stochastic differential equations and application of the stochastic calculus of variations to the rate of convergence. (preprint 1987)
Protter, P.: Approximation of solutions of stochastic differential equations driven by semimartingales. Ann. Probab.13, 716–743 (1985)
Taksar, M.I.: Average optimal singular control and a related stopping problem. Math. Oper. Res.10, 63–81 (1985)
Sussmann, H.J.: On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab.6, 19–41 (1987)
Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat.36, 1560–1564 (1965)
Zakai, M.: Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheor. Verw. Geb.12, 87–97 (1969)
Author information
Authors and Affiliations
Additional information
Research partially supported by the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, USA; by the AFOSR under contract #AFOSR-85-0315, the ARO under contract #DAAG 29-84-K-0082, and #DAAL 03-86-K-0171.
Rights and permissions
About this article
Cite this article
Ferreyra, G. Approximation of stochastic equations driven by predictable processes. Probab. Th. Rel. Fields 83, 391–403 (1989). https://doi.org/10.1007/BF00964371
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00964371
Keywords
- Differential Equation
- Stochastic Process
- Probability Theory
- Large Class
- Mathematical Biology