Abstract
Our purpose is to study an optimal ergodic control problem where the state of the system is given by a diffusion process with jumps in the whole space. The corresponding dynamic programming (or Hamilton-Jacobi-Bellman) equation is a quasi-linear integro-differential equation of second order. A key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only locally bounded and Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic HJB equation is established.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
A. Bensoussan, Perturbation methods in optimal control, Wiley, New York, 1988.
A. Bensoussan and J.L. Lions, Applications of variationals inequalities in stochastic control, North-Holland, Amsterdam, 1982.
A. Bensoussan and J.L. Lions, Impulse control and quasi-variational inequalities, Gauthier-Villars, Paris, 1984.
V.S. Borkar, Topics in Controlled Markov Chains, Pitman Research Notes in Mathematics Series No 240, Longman, Essex, 1991.
M.G. Crandall, H. Ishii and P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc, 27 (1992), 1–67.
S.N. Ethier and T.G. Kurtz, Markov processes, Wiley, New York, 1986.
W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions, Springer-Verlag, New York, 1992.
M.G. Garroni and J.L. Menaldi Green functions for second order integral-differential problems, Pitman Research Notes in Mathematics Series No 275, Longman, Essex, 1992.
I.I. Gikhman and A.V. Skorokhod, Stochastic differential equations, Springer-Verlag, Berlin, 1972.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Second Edition, Springer-Verlag, New York, 1983.
F. Gimbert and P.L. Lions, Existence and regularity results for solutions of second order elliptic integro-differential operators, Ricerche di Matematica, 33 (1984), 315–358
R.Z. Khasminskii, (Hasminskii) Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, The Netherlands, 1980.
N.V. Krylov, Nonlinear elliptic and parabolic equations of second order, Reidel, Dordrecht, 1987
O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and quasilinear elliptic equations, Academinc Press, New York, 1968.
P.L. Lions, A remark on Bony Maximum principle, Procedings Am. Math. Soc, 88 (1982), 503–508.
J.L. Menaldi, On the stopping time problem for degenerate diffusions, SIAM J. Control Optim., 18 (1980), 697–721.
J.L. Menaldi, Optimal impulse control problems for degenerate diffusions with jumps, Acta Appl. Math., 8 (1987), 165–198.
J.L. Menaldi and M. Robin, Ergodic control of reflected diffusions with jumps, Appl. Math. Optim., 35 (1997), 117–137.
J.L. Menaldi and M. Robin, Invariant Measure for Diffusions with Jumps, Appl. Math. Optim., to appear.
M.H. Protter and H.F. Weinberger, Maximum principles in differential equations, Second edition, Springer-Verlag, New York, 1984.
M. Robin, Long-term average cost control problems for continuous time Markov processes: A survey, Acta Appl. Math., 1 (1983), 281–299.
D.W. Stroock and S.R. Varadhan, Multidimensional diffusion process, Springer-Verlag, Berlin, 1979.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Menaldi, JL., Robin, M. (1999). On Optimal Ergodic Control of Diffusions with Jumps. In: McEneaney, W.M., Yin, G.G., Zhang, Q. (eds) Stochastic Analysis, Control, Optimization and Applications. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1784-8_26
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1784-8_26
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7281-6
Online ISBN: 978-1-4612-1784-8
eBook Packages: Springer Book Archive