# Optimal Controls for Zakai Equations

• Makiko Nisio
Chapter
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 72)

## Abstract

This chapter is an application of previous one. In Sect. , we have introduced the (controlled) Zakai equation, which is a stochastic linear parabolic equation with a Brownian adapted control process. By using the results in Chap. , we will study control problems for Zakai equations related to partially observable diffusions. The control problem for partially observable diffusions turns out to be a completely observable control problem on a Hilbert space, by using the unnormalized conditional probability density given by the Zakai equation (cf. Bensoussan A, Stochastic control of partially observable systems. Cambridge University Press, Cambridge/New York, 1992; Lions, J Commun PDE 8:1101–1134, 1983, I, II; Gozzi and Świech, J Funct Analy 172:466–510, 2000). Section 6.1 is devoted to the analysis of controlled Zakai equations. In Sect. 6.2, we formulate control problems for a system governed by Zakai equations, in the same way as in Chap. . When a control process $$\gamma (\cdot )$$ is chosen, the cost on a time internal [T 0, t] is given by $$\int _{T_{0}}^{t}r(u^{\gamma (\cdot )}(s),\gamma (s))\,ds + F(u^{\gamma (\cdot )}(t))$$, where $$u^{\gamma (\cdot )}(\cdot )$$ is the response of $$\gamma (\cdot )$$. By taking a suitable control process, we want to minimize (or maximize) the expectation of the cost. In Sect. 6.3 we formulate the DPP via the semigroup constructed from the value function, whose generator is related to the HJB equation on a Hilbert space. The viscosity solution of HJB equation is introduced following Gozzi and Świech (J Funct Analy 172:466–510, 2000) in Sect. 6.4. Example 6.1 makes explicitly the connection between controlled Zakai equations and control of partially observable diffusions.

## References

1. [A03]
2. [BC09]
A. Bain, D. Crisan, Fundamentals of Stochastic Filtering (Springer, New York/ London, 2009)
3. [Be52]
R. Bellman, On the theory of dynamic programming. Proc. Nat. Sci. U.S.A. 38, 716–719 (1952)
4. [Be57]
R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)
5. [Be75]
V.E. Benes, Composition and invariance methods for solvinig some stochastic control problems. Adv. Appl. Prob. 7, 299–329 (1975)
6. [BSW80]
V.E. Venes, L.A. Shepp, H.S. Witsenhaussen, Some solvable stochastic control problems. Stochastics 4, 39–83 (1980)
7. [Be92]
A. Bensoussan, Stochastic Control of Partially Observable Systems (Cambridge University Press, Cambridge/New York, 1992)
8. [BN90]
A. Bensoussan, M. Nisio, Nonlinear semigroup arising in the control of diffusions with partial observation. Stoch. Stoch. Rep. 30, 1–45 (1990)
9. [BP99]
T.R. Bieleckii, S.R. Pliska, Risk sensitive dynamic asset management. Appl. Math. Optim. 39, 337–366 (1999)
10. [BS73]
F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)
11. [BM07]
R. Buckdahn, J. Ma, Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45, 2224–2256 (2007)
12. [Br06]
S. Brendle, Portfolio selection under incomplete information. Stoch. Proc. Appl. 116, 701–723 (2006)
13. [CPY09]
M.H. Chsng, T. Pang, J. Yong, Optimal stopping problem for stochastic differential equations with random cofficients. SIAM J. Control Optim. 48, 941–971 (2009)
14. [CI90]
M.G. Crandall, H. Ishii, The maximum principle for semicontinuous functions. Differ. Integral Equ. 3, 1001–1014 (1990)
15. [CIL92]
M.G. Crandall, H. Ishii, P.L. Lions, A user’s guide to viscosity solutions. Bull. AMS NS 27, 1–67 (1992)
16. [DaPZ92]
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimenmensions (Cambridge University Press, Cambridge, 1992)
17. [ElKPQ97]
N. El Karoui, S. Peng, M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)
18. [ElKQ95]
N. El Karoui, M.C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market. SIAM. J. Control Optim. 33, 29–66 (1950)
19. [E83]
L.C. Evans, Classical solutions of Hamilton-Jacobi-Bellman equation for uniformly elliptic operators. Trans. AMS 275, 245–255 (1983)
20. [E98]
L.C. Evans, Partial Differential Equations. GSM19 (AMS, Providence, 1998)Google Scholar
21. [FH11]
W.H. Fleming, D. Hernandez-Hernandez, On the value of stochastic differential games. Commun. Stoch. Anal. 5, 341–351 (2011)
22. [FKSh10]
W.H. Fleming, H. Kaise, S.J. Sheu, Max-plus stochastic control and risk sensitivity. Appl. Math. Optim 62, 81–144 (2010)
23. [FMcE95]
W.H. Fleming, W.H. McEneaney, Risk sensitive control on infinite time horizon. SIAM J. Control Optim. 33, 1881–1915 (1995)
24. [FR75]
W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, Berlin/New York, 1975)
25. [FSh99]
W.H. Fleming, S.J. Sheu, Optimal long term growth rate of expected utility of wealth. Ann. Appl. Prob. 9, 871–903 (1999)
26. [FSh00]
W.H. Fleming, S.J. Sheu, Risk sensitive control and an optimal investment model. Math. Finance 10, 197–213 (2000)
27. [FSh02]
W.H. Fleming, S.J. Sheu, Risk sensitive control and an optimal investment model II. Ann. Appl. Prob. 12, 730–767 (2002)
28. [FS06]
W.H. Fleming, H.M. Soner, Controlled MarkovProcesses and Viscosity Solutions, 2nd edn. (Springer, New York 2006)Google Scholar
29. [FSo89]
W.H. Fleming, P.E. Souganidis, On the existence of value function of two-plsyer, zero-sum stochastic differential games. Indiana Math. J. 38, 293–314 (1989)
30. [FKK72]
M. Fujisaki, G. Kallianpur, H. Kunita, Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9, 19–40 (1972)
31. [GŚ00]
F. Gozzi, A. Świech, Hamilton-Jacobi-bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation. J. Funct. Analy. 172, 466–510 (2000)
32. [HP81]
J.M. Harrison, S.R. Pliska, Martingales and stochastic integrals in the theory of continuous ytading. Stoch. Proc. Appl. 11, 215–260 (1981)
33. [HP83]
J.M. Harrison, S.R. Pliska, Stochastic calculus model of continuous trading; complete markets. Stoch. Proc. Appl. 15, 313–316 (1983)
34. [HNSh10]
H. Hata, H. Nagai, S.I. Sheu, Asymptotics of the probability minimizing a down-side risk. Ann. Appl. Prob. 20, 52–89 (2010)
35. [HS10]
H. Hata, J. Sekine, Explicit solution to a certain nonELQG risk-sensitive stochastic control problem. Appl. Math. Optim. 62, 341–380 (2010)
36. [IW81]
N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam/New York, 1981)
37. [Is92]
H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces. J. Funct. Anal. 105, 301–341 (1992)
38. [I42]
K. Itô, Differential equations determining Markov processes. Zenkoku Shijo Sugaku Danwakai 244, 1352–1400 (1942). (In Japanese)Google Scholar
39. [I51]
K. Itô, On Stochastic Differential Equations. Memoirs of the American Mathematical Society, vol. 4 (AMS, New York City, 1951)Google Scholar
40. [JLL90]
P. Jaillet, D. Lamberton, B. Lapeyer, Variational inequalities and the pricing of American options. Acta Appl. Math. 21, 263–289 (1990)
41. [KS91]
I. Karatzas, S.E. Sheve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)
42. [KS98]
I. Karatzas, S.E. Sheve, Methods of Mathematical Finance (Springer, New York, 1998)
43. [Ko04]
S. Koike, A Biginner’s Guide to the Theory of Viscisity Solutions. MSJ Memoirs, vol. 13 (JMS, Tokyo, 2004)Google Scholar
44. [KN02]
K. Kuroda, H. Nagai, Risk sensitive portfolio optimization and infinite time horizon. Stoch. Stoch. Rep. 73, 309–331 (2002)
45. [Kr09]
N.V. Krylov, Controlled Diffusion Processes, 2nd edn. (Springer, Berlin, 2009)
46. [Ku67]
H.J. Kushner, Dynamical equations for optimal nonlinear filtering. J. Differ. Equ. 3, 179–190 (1967)
47. [La83]
S. Lang, Real Analysis, 2nd edn. (Addison-Wesley, New York, 1983)
48. [L83]
P.L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I. J. Commun. PDE. 8, 1101–1134 (1983)
49. [L88]
P.L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I. The case of bounded stochastic evolution. Acta Math. 161, 243–278 (1988)
50. [L89]
P.L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part II. Optimal control of Zakai equation, in Stochastic Partial Differential Equations and Applications II, ed. by G. Da Prato, L. Tubaro. Lecture Notes in Mathematics, vol. 1390 (Springer, Berlin/Heidelberg, 1989), pp. 147–170. Part III. Uniqueness of viscosity solutions for general second order equations. J. Funct. Anal. 86, 1–18 (1989)Google Scholar
51. [LN83]
P.L. Lions, M. Nisio, A uniqueness result for the semigroup associated with HJB equations. Proc. Jpn. Acad. 58, 273–276 (1983)
52. [LS01]
R.S. Liptser, A.N. Shiryayev, Statistics of Random Processes I, II, 2nd edn. (Springer, New York/Berlin, 2001)
53. [Ma00]
C. Martini, American option prices as unique viscosity solutions to degenerate HJB equations, Rapport de rech, INRIA, 2000Google Scholar
54. [Me71]
R.C. Merton, Optimal consumption and portfolio rules in continuous time. J. Econ. Theory 3, 373–413 (1971)
55. [Me73]
R.C. Merton, Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)
56. [M88]
M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces (Scuola Superiore, Pisa, 1988)
57. [My66]
P.A. Meyer, Probability and Potentials (Blaisdell, Waltham, 1966)
58. [Mo10]
H. Morimoto, Stochastic Control and Mathematical Modeling, Applications in Economics (Cambridge University Press, Cambridge/New York, 2010)
59. [Na03]
H. Nagai, Optimal strategies for risk sensitive portfolio optimization problems for general factor models. SIAM J. Control Optim. 41, 1779–1800 (2003)
60. [N76]
M. Nisio, On a nonlinear semigroup attached to stochastic optimal control. Pull. RIMS 12, 513–537 (1976)
61. [N78]
M. Nisio, On nonlinear semigroups for Markov processes associated with optimal stopping. Appl. Math. Optim. 4, 143–169 (1978)
62. [N81]
M. Nisio, Lecture on Stochastic Control Theory. ISI Lecture Notes, vol. 9 (McMillan India, Delhi 1981)Google Scholar
63. [N88]
M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations. Nagoya Math. J. 110, 163–184 (1988)
64. [P79]
E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, 127–167 (1979)
65. [P93]
E. Pardoux, Stochastic partial differential equations, a review. Bull. Sc. Math. 117, 29–47 (1993)
66. [PP90]
E. Pardoux, S. Peng, Adapted solution of backward stoshastic equation. Syst. Control Lett. 14, 55–61 (1990)
67. [Pe92]
S. Peng, Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30, 284–304 (1992)
68. [Ph02]
H. Pham, Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl. Math. Optim. 46, 55–78 (2002)
69. [R90]
B.L. Rozovskii, Stochastic Evolution Systems (Kluwer Academic, Dordrecht/Boston, 1990)
70. [S08]
A.N. Shiryayev, Optimal Stopping Rules, 2nd edn. (Springer, Berlin/Heidelberg 2008)Google Scholar
71. [St11]
L. Stettner, Penalty method for finite horizon stopping problems. SIAM J. Control Optim. 49, 1078–1099 (2011)
72. [SV79]
D.V. Strook, S.R.S. Varadhan, Multidimensional Diffusion Processes (Springer, Berlin/New York, 1979)Google Scholar
73. [W71]
J.C. Willems, Least squares stationary optimal control and the algebraic Ricatti equation. IEEE. Trans. Auto. Control 16, 621–635 (1971)
74. [Wo68]
W.M. Wonham, on a matrix Ricatti equation of stochastic control. SIAM J. Control Optim. 6, 681–697 (1968)Google Scholar
75. [Y80]
K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin/New York, 1980)
76. [YZ99]
J. Yong, X.Y. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations (Springer, New York, 1999)