UNIVERSITY
OF WROCŁAW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
44.2 44.1 43.2 43.1 42.2 42.1 41.2
41.1 40.2 40.1 39.2 39.1 38.2 38.1
37.2 37.1 36.2 36.1 35.2 35.1 34.2
34.1 33.2 33.1 32.2 32.1 31.2 31.1
30.2 30.1 29.2 29.1 28.2 28.1 27.2
27.1 26.2 26.1 25.2 25.1 24.2 24.1
23.2 23.1 22.2 22.1 21.2 21.1 20.2
20.1 19.2 19.1 18.2 18.1 17.2 17.1
16.2 16.1 15 14.2 14.1 13.2 13.1
12.2 12.1 11.2 11.1 10.2 10.1 9.2
9.1 8 7.2 7.1 6.2 6.1 5.2
5.1 4.2 4.1 3.2 3.1 2.2 2.1
1.2 1.1
 
 
WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 41, Fasc. 2,
pages 193 - 215
DOI: 10.37190/0208-4147.41.2.1
Published online 19.8.2021
 

A time-changed stochastic control problem and its maximum principle maximum principle

Erkan Nane
Yinan Ni

Abstract:

This paper studies a time-changed stochastic control problem, where the underlying stochastic process is a Lévy noise time-changed by an inverse subordinator. We establish a maximum principle for the time-changed stochastic control problem. We also prove the existence and uniqueness of the corresponding time-changed backward stochastic differential equation involved in the stochastic control problem. Some examples are provided for illustration.

2010 AMS Mathematics Subject Classification: Primary 93E20, 39A50, 60H05;

Keywords and phrases: optimal stochastic control, time-changed L\'evy process, maximum principle

A. Bensoussan, Maximum principle and dynamic programming approaches to the optimal control of partially observed diffusions, Stochastics 9 (1983), 169-222.

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl. 44 (1973), 384-404.

W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM J. Control 43 (2004), 502-531.

N. C. Framstad, B. Øksendal and A. Sulem, Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, J. Optim. Theory Appl. 124 (2005), 511-512.

J. Janczura and A. Wyłomańska, Subdynamics of financial data from fractional Fokker-Planck equation, Acta Phys. Polon. B 40 (2009), 1341–1351.

M. Hahn, K. Kobayashi and S. Umarov, SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theoret. Probab. 25 (2012), 262-279.

E. Jum and K. Kobayashi, A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion, Probab. Math. Statist. 36 (2016), 201-220.

H. J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems, SIAM J. Control 10 (1972), 550-565.

M. Magdziarz and T. Zorawik, Stochastic representation of a fractional subdiffusion equation. The case of infinitely divisible waiting times, Lévy noise and space-time-dependent coefficients, Proc. Amer. Math. Soc. 144 (2016), 1767-1778.

M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model. Nat. Phenom. 8 (2013), no. 2, 1-16.

E. Nane and Y. Ni, Path stability of stochastic differential equations driven by time-changed L'evy noises, ALEA Latin Amer. J. Probab. Math. Statist. 15 (2018), 479-507.

E. Nane and Y. Ni, Stability of the solution of stochastic differential equation driven by time-changed L'evy noise, Proc. Amer. Math. Soc. 145 (2017), 3085-3104.

E. Nane and Y. Ni, Stochastic solution of fractional Fokker-Planck equations with space-time-dependent coefficients, J. Math. Anal. Appl. 442 (2016), 103-116.

S. G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim. 28 (1990), 966-979.

W. J. Runggaldier, On stochastic control in finance, in: Mathematical Systems Theory in Biology, Communications, Computation, and Finance, Springer, New York, 2003, 317-344.

D. Wilkinson, Stochastic Modelling for Systems Biology, Chapman and Hall/CRC, New York, 2006.

Q. Wu, Stability analysis for a class of nonlinear time-changed systems, Cogent Math. 3 (2016), art. 1228273, 10 pp.

Download:    Abstract    Full text   Go Back