We consider the Merton problem of finding the strategies of investment and consumption in the case where the evolution of risk assets is described by the exponential model and the role of the main process is played by the integral of a certain stationary “physical” white noise generated by the centered Poisson process. It is shown that the optimal controls computed for the limiting case are ε-sufficient controls for the original system.
Similar content being viewed by others
References
P. A. Samuelson, “Rational theory of warrant pricing,” Industr. Manag. Rev., No. 6, 13–31 (1965).
M. M. Leonenko, Yu. S. Mishura, Ya. M. Parkhomenko, and M. I. Yadrenko, Theoretical-Probability and Statistical Methods in Econometrics and Financial Mathematics [in Ukrainian], Informtekhnika, Kyiv (1995).
R. C. Merton, “Optimum consumption and portfolio rules in continuous time model,” J. Econ. Theory, No. 3, 373–413 (1971).
R. Sh. Liptser and A. N. Shiryaev, “Martingales and limit theorems for random processes,” in: Itogi VINITI. Ser. Contemporary Problems in Mathematics. Fundamental Trends [in Russian], Vol. 45, VINITI, Moscow (1989), pp. 159–251.
D. O. Chikin, “Functional limit theorem for stationary processes: a martingale approach,” Teor. Ver. Primen., 14, No. 4, 731–741 (1989).
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes [Russian translation], Vol. 2, Fizmatlit, Moscow (1994).
A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1987).
A. V. Skorokhod, Elements of Probability Theory and the Theory of Random Processes [in Ukrainian], Vyshcha Shkola, Kyiv (1975).
V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, and A. F. Turbin, A Handbook of Probability Theory and Mathematical Statistics [in Russian], Naukova Dumka, Kiev (1978).
B. V. Bondarev and S. M. Kozyr’, “On the estimation of the rate of approach of the solution of an ordinary differential equation perturbed by a physical white noise to the solution of the corresponding Itô equation. I,” Prykl. Statyst. Aktuar. Finans. Mat., No. 2, 63–91 (2006).
A. V. Mel’nikov, S. N. Volkov, and M. L. Nechaev, Mathematics of Financial Obligations [in Russian], GUVShÉ, Moscow (2001).
B. D. Gnedenko, “On one extension of the notion of martingale,” Teor. Ver. Primen., 50, No. 34, 763–767 (2005).
B. V. Bondarev and I. L. Shurko, “Diffusion approximation of random processes with independent increments in probability,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, Fiz.-Mat. Tekhn. Nauk, No. 9, 3–4 (1988).
V. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1968).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 8, pp. 1025–1039, August, 2009.
Rights and permissions
About this article
Cite this article
Bondarev, B.V., Kozyr’, S.M. On the ε-sufficient control in one merton problem with “physical” white noise. Ukr Math J 61, 1215–1232 (2009). https://doi.org/10.1007/s11253-010-0272-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0272-3