Abstract
We study an optimal stopping problem for a stochastic differential equation with delay driven by a Lévy noise. Approaching the problem by its infinite-dimensional representation, we derive conditions yielding an explicit solution to the problem. Applications to the American put option problem are shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chojnowska-Michalik, A.: Representation theorem for general stochastic delay equations. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 26(7), 635–642 (1978)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge, UK (1996)
David, D.: Optimal control of stochastic delayed systems with jumps (2008) (preprint)
Diekmann, O., Van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag, New York (1995)
Elsanousi, I., Øksendal, B., Sulem, A.: Some solvable stochastic control problems with delay. Stoch. Stoch. Rep. 71, 69–89 (2000)
Federico, S.: A stochastic control problem with delay arising in a pension fund model. Finance Stoch. (2009) (to appear)
Federico, S., Goldys, B., Gozzi, F.: HJB equations for the optimal control of differential equations with delays and state constraints, I: regularity of viscosity solutions (2009) (submitted, ArXiv preprint)
Federico, S., Goldys, B., Gozzi, F.: HJB equations for the optimal control of differential equations with delays and state constraints, II: verification and optimal feedbacks (2009) (ArXiv, preprint)
Gapeev, P., Reiss, M.: Optimal stopping problem in a diffusion-type model with delay. Stoch. Probab. Lett. 76(6), 601–608 (2006)
Gozzi, F., Marinelli, C.: Stochastic optimal control of delay equations arising in advertising models. In: Da Prato, G., et al. (eds.) Stochastic Partial Differential Equations and Applications VII, Papers of the 7th Meeting, Levico Terme, Italy, 5–10 January 2004. Lecture Notes in Pure and Applied Mathematics, vol. 245, pp. 133–148. Chapman & Hall/CRC, Boca Raton, FL (2004)
Gozzi, F., Marinelli, C., Savin, S.: On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects. J. Optim. Theory Appl. 142(2), 291–321 (2009)
Kolmanovski, V.B., Shaikhet, L.E.: Control of Systems with Aftereffect. American Mathematical Society (1996)
Larssen, B.: Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Rep. 74(3–4), 651–673 (2002)
Larssen, B., Risebro, N.H.: When are HJB-equations for control problems with stochastic delay equations finite-dimensional? Stoch. Anal. Appl. 21(3), 643–671 (2003)
Mordecki, E.: Optimal stopping and perpetual options for Lévy processes. Finance Stoch. 6, 473–493 (2002)
Øksendal, B., Sulem, A.: A maximum principle for optimal control of stochastic systems with delay with applications to finance. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and PDE, Essays in Honour of Alain Bensoussan, pp. 64–79. IOS Press, Amsterdam (2001)
Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer (2007)
Øksendal, B., Zhang, T.: Optimal control with partial information for stochastic Volterra equations. International Journal of Stochastic Analysis, University of Oslo (2008) (to appear)
Peskir, G., Shirayev, A.: Optimal Stopping and Free-Boundary Problems. Birkhauser, Cambridge, MA (2006)
Peszat, S., Zabczyk, J.: Stochastic partial differential equtions with Lévy noise. Encyclopedia of Mathematics and its Applications, vol. 113. Cambridge University Press, Cambridge, UK (2008)
Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin Heidelberg New York (2003)
Vinter, R.B., Kwong, R.H.: The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J. ControlOptim. 19(1), 139–153 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087].
Rights and permissions
About this article
Cite this article
Federico, S., Øksendal, B.K. Optimal Stopping of Stochastic Differential Equations with Delay Driven by Lévy Noise. Potential Anal 34, 181–198 (2011). https://doi.org/10.1007/s11118-010-9187-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-010-9187-8