Abstract
We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.
Similar content being viewed by others
Notes
Nevertheless there are examples where a finite-dimensional Markovian representation can be obtained. We will study this kind of situation in Sect. 4.2, giving sufficient conditions for a finite-dimensional Markovian representation.
For example this process could appear in the cost functional of a control problem.
References
Barndorff-Nielsen, O., Benth, F., Veraart, A.: Modelling energy spot prices by Lévy semistationary processes. CREATES Research Paper (2010)
Bauer, H., Rieder, U.: Stochastic control problems with delay. Math. Methods Oper. Res. 62, 411–427 (2005)
Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and control of infinite dimensional systems. Systems & Control: Foundations & Applications, 2nd edn, xxviii+575 pp. Birkhäuser Boston, Inc., Boston, MA (2007)
Bernhart, M., Tankov, P., Warin, X.: A finite dimensional approximation for pricing moving average options. SIAM J. Financ. Math. 2, 989–1013 (2011)
Hallulli, V.B., Vargiolu, T.: Robustness of the Hobson-Rogers model with respect to the offset function. Seminar on Stochastic Analysis, Random Fields and Applications V, pp. 469–492, Progr. Probab., 59, Birkhäuser, Basel (2008)
Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext, xiv+599 pp. Springer, New York (2011)
Da Prato, G., Zabczyck, J.: Stochastic Equations in Infinite Dimension, Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1992)
Davies, E.B.: One-Parameter Semigroups. Academic Press, London (1980)
Elsanosi, I., Øksendal, B., Sulem, A.: Some solvable stochastic control problems with delay. Stoch. Int. J. Probab. Stoch. Process. 71, 69–89 (2000)
Engel, K., Nagel, R.: One-Paramater Semigroups for Linear Evolution Equations. Springer, Berlin (2000)
Federico, S.: A stochastic control problem with delay arising in a pension fund model. Financ. Stoch. 15, 412–459 (2011)
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. SIAM J. Control Optim. 48, 4910–4937 (2010)
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. SIAM J. Control Optim. 49, 2378–2414 (2011)
Federico, S., Øksendal, B.: Optimal stopping of stochastic differential equations with delay driven by Lévy noise. Potential Anal. 34, 181–198 (2011)
Filipović, D.: Invariant manifolds for weak solutions to stochastic equations. Probab. Theory Relat. Fields 118, 323–341 (2000)
Filipović, D., Teichmann, J.: Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197, 398–432 (2003)
Fischer, M., Nappo, G.: Time discretisation and rate of convergence for the optimal control of continuous-time stochastic systems with delay. Appl. Math. Optim. 57, 177–206 (2008)
Fleming, W., Soner, H.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)
Foschi, P., Pascucci, A.: Path dependent volatility. Decis. Econ. Financ. 31, 13–32 (2008)
Fuhrman, M., Masiero, F., Tessitore, G.: Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 48, 4624–4651 (2010)
Gapeev, P.V., Reiß, M.: An optimal stopping problem in a diffusion-type model with delay. Stat. Probab. Lett. 76, 601–608 (2006)
Gawarecki, L., Mandrekar, V.: Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Probability and its Applications (New York), xvi+291 pp. Springer, Heidelberg (2011)
Gozzi, F., Marinelli, C.: Stochastic optimal control of delay equations arising in advertising models. Stochastic Partial Differential Equations and Applications—VII, pp. 133–148, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL (2006)
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, 291–321 (2009)
Hobson, D.G., Rogers, L.C.: Complete models with stochastic volatility. Math. Financ. 8, 27–48 (1998)
Kolmanovskii, V.B., Maizenberg, T.L.: Optimal control of stochastic systems with aftereffect. Avtom. Telemeh. 1, 47–61 (1973)
Kushner, H.J.: Numerical approximations for nonlinear stochastic systems with delays. Stoch. Int. J. Probab. Stoch. Process. 77, 211–240 (2005)
Larssen, B., Risebro, N.H.: When are HJB-equations in stochastic control of delay systems finite dimensional? Stoch. Anal. Appl. 21, 643–671 (2003)
Mohammed, S-E.A. (1-SIL). Stochastic differential systems with memory: theory, examples and applications. Stochastic Analysis and Related Topics, VI (Geilo, 1996), pp. 1–77, Progr. Probab., 42, Birkhäuser Boston, Boston, MA (1998)
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.) Proceedings of the Conference on Optimal Control and Partial Differential Equations, Paris, December 2000, pp. 64–79. IOS Press, Amsterdam (2001)
Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer. Lecture Notes in Mathematics (2007)
Reiss, M., Fischer, M.: Discretisation of stochastic control problems for continuous time dynamics with delay. J. Comput. Appl. Math. 205, 969–981 (2007)
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn, xiv+602 pp. Springer-Verlag, Berlin (1999)
Vinter, R., Kwong, R.: The infinite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J. Control Optim. 19, 139–153 (1981)
Acknowledgments
The authors are grateful to two anonymous Referees whose suggestions allowed to improve the first version of the paper. The authors also thank Mauro Rosestolato for very fruitful discussions and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Federico, S., Tankov, P. Finite-Dimensional Representations for Controlled Diffusions with Delay. Appl Math Optim 71, 165–194 (2015). https://doi.org/10.1007/s00245-014-9256-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-014-9256-2
Keywords
- Stochastic delay differential equation (SDDE)
- Markovian representation
- Laguerre polynomials
- Stochastic control
- Optimal stopping