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Erschienen in:
Stochastic Differential Equations Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

5. Stochastic Parabolic Equations

verfasst von : Makiko Nisio

Erschienen in: Stochastic Control Theory

Verlag: Springer Japan

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Abstract

This chapter is devoted to stochastic evolution equations in Hilbert spaces, in particular stochastic parabolic type equations of the form
$$\displaystyle{du(t,x,\omega ) = \mathcal{L}(t,\omega )u(t,x,\omega )\,dt + \mathcal{M}(t,\omega )u(t,x,\omega )\,dW_{Q}(t),}$$
where \(\mathcal{L}\) and \(\mathcal{M}\) are second-order elliptic and first-order differential operators and W Q is a colored Wiener process (see Example 5.1).
These equations are generalization of finite-dimensional SDEs and appear in the study of random phenomena in natural sciences and the unnormalized conditional probability of finite-dimensional diffusion processes (see Sect. 5.5), related to filtering equations derived in Fujisaki et al. (Osaka J Math 9:19–40, 1972) and Kushner (J Differ Equ 3:179–190, 1967).
In Sect. 5.1 we collect basic definitions and results for Hilbert space-valued processes; in particular, for continuous martingales, quadratic variations and correlation operators are treated. Stochastic integrals are introduced in Sect. 5.2. Section 5.3 is devoted to the study of stochastic parabolic equations from the viewpoint of Hilbert space-valued SDEs, following Rozovskii (Stochastic evolution systems. Kluwer Academic, Dordrecht/Boston, 1990). By using the results presented, we also consider a semilinear stochastic parabolic equation with Lipschitz nonlinearity in Sect. 5.3.4. Section 5.4 deals with Itô’s formula and in Sect. 5.5 Zakai equations related to filtering problems are given.

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Vorheriges Kapitel Stochastic Differential Games
Nächstes Kapitel Optimal Controls for Zakai Equations
Literatur
[A03]
R.A. Adams, Sobolev Spaces, 2nd edn. (Academic, Amsterdam/Boston, 2003)MATH
[BC09]
A. Bain, D. Crisan, Fundamentals of Stochastic Filtering (Springer, New York/ London, 2009)CrossRefMATH
[Be52]
R. Bellman, On the theory of dynamic programming. Proc. Nat. Sci. U.S.A. 38, 716–719 (1952)CrossRefMATH
[Be57]
R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)MATH
[Be75]
V.E. Benes, Composition and invariance methods for solvinig some stochastic control problems. Adv. Appl. Prob. 7, 299–329 (1975)CrossRefMATHMathSciNet
[BSW80]
V.E. Venes, L.A. Shepp, H.S. Witsenhaussen, Some solvable stochastic control problems. Stochastics 4, 39–83 (1980)CrossRefMathSciNet
[Be92]
A. Bensoussan, Stochastic Control of Partially Observable Systems (Cambridge University Press, Cambridge/New York, 1992)CrossRefMATH
[BN90]
A. Bensoussan, M. Nisio, Nonlinear semigroup arising in the control of diffusions with partial observation. Stoch. Stoch. Rep. 30, 1–45 (1990)CrossRefMATHMathSciNet
[BP99]
T.R. Bieleckii, S.R. Pliska, Risk sensitive dynamic asset management. Appl. Math. Optim. 39, 337–366 (1999)CrossRefMathSciNet
[BS73]
F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)CrossRef
[BM07]
R. Buckdahn, J. Ma, Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45, 2224–2256 (2007)CrossRefMATHMathSciNet
[Br06]
[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)CrossRefMathSciNet
[CI90]
M.G. Crandall, H. Ishii, The maximum principle for semicontinuous functions. Differ. Integral Equ. 3, 1001–1014 (1990)MATHMathSciNet
[CIL92]
[DaPZ92]
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimenmensions (Cambridge University Press, Cambridge, 1992)CrossRef
[ElKPQ97]
[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)CrossRef
[E83]
L.C. Evans, Classical solutions of Hamilton-Jacobi-Bellman equation for uniformly elliptic operators. Trans. AMS 275, 245–255 (1983)CrossRef
[E98]
L.C. Evans, Partial Differential Equations. GSM19 (AMS, Providence, 1998)
[FH11]
W.H. Fleming, D. Hernandez-Hernandez, On the value of stochastic differential games. Commun. Stoch. Anal. 5, 341–351 (2011)MathSciNet
[FKSh10]
[FMcE95]
[FR75]
W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, Berlin/New York, 1975)CrossRefMATH
[FSh99]
[FSh00]
[FSh02]
[FS06]
W.H. Fleming, H.M. Soner, Controlled MarkovProcesses and Viscosity Solutions, 2nd edn. (Springer, New York 2006)
[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)CrossRefMATHMathSciNet
[FKK72]
M. Fujisaki, G. Kallianpur, H. Kunita, Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9, 19–40 (1972)MATHMathSciNet
[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)CrossRefMATH
[HP81]
J.M. Harrison, S.R. Pliska, Martingales and stochastic integrals in the theory of continuous ytading. Stoch. Proc. Appl. 11, 215–260 (1981)CrossRefMATHMathSciNet
[HP83]
J.M. Harrison, S.R. Pliska, Stochastic calculus model of continuous trading; complete markets. Stoch. Proc. Appl. 15, 313–316 (1983)CrossRefMATHMathSciNet
[HNSh10]
H. Hata, H. Nagai, S.I. Sheu, Asymptotics of the probability minimizing a down-side risk. Ann. Appl. Prob. 20, 52–89 (2010)CrossRefMATHMathSciNet
[HS10]
H. Hata, J. Sekine, Explicit solution to a certain nonELQG risk-sensitive stochastic control problem. Appl. Math. Optim. 62, 341–380 (2010)CrossRefMATHMathSciNet
[IW81]
N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam/New York, 1981)MATH
[Is92]
H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces. J. Funct. Anal. 105, 301–341 (1992)CrossRefMATHMathSciNet
[I42]
K. Itô, Differential equations determining Markov processes. Zenkoku Shijo Sugaku Danwakai 244, 1352–1400 (1942). (In Japanese)
[I51]
K. Itô, On Stochastic Differential Equations. Memoirs of the American Mathematical Society, vol. 4 (AMS, New York City, 1951)
[JLL90]
P. Jaillet, D. Lamberton, B. Lapeyer, Variational inequalities and the pricing of American options. Acta Appl. Math. 21, 263–289 (1990)CrossRefMATHMathSciNet
[KS91]
I. Karatzas, S.E. Sheve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)MATH
[KS98]
[Ko04]
S. Koike, A Biginner’s Guide to the Theory of Viscisity Solutions. MSJ Memoirs, vol. 13 (JMS, Tokyo, 2004)
[KN02]
K. Kuroda, H. Nagai, Risk sensitive portfolio optimization and infinite time horizon. Stoch. Stoch. Rep. 73, 309–331 (2002)CrossRefMATHMathSciNet
[Kr09]
N.V. Krylov, Controlled Diffusion Processes, 2nd edn. (Springer, Berlin, 2009)MATH
[Ku67]
[La83]
S. Lang, Real Analysis, 2nd edn. (Addison-Wesley, New York, 1983)MATH
[L83]
P.L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I. J. Commun. PDE. 8, 1101–1134 (1983)CrossRefMATH
[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)MATH
[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)
[LN83]
P.L. Lions, M. Nisio, A uniqueness result for the semigroup associated with HJB equations. Proc. Jpn. Acad. 58, 273–276 (1983)CrossRefMathSciNet
[LS01]
R.S. Liptser, A.N. Shiryayev, Statistics of Random Processes I, II, 2nd edn. (Springer, New York/Berlin, 2001)CrossRef
[Ma00]
C. Martini, American option prices as unique viscosity solutions to degenerate HJB equations, Rapport de rech, INRIA, 2000
[Me71]
[Me73]
[M88]
M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces (Scuola Superiore, Pisa, 1988)MATH
[My66]
P.A. Meyer, Probability and Potentials (Blaisdell, Waltham, 1966)MATH
[Mo10]
H. Morimoto, Stochastic Control and Mathematical Modeling, Applications in Economics (Cambridge University Press, Cambridge/New York, 2010)CrossRefMATH
[Na03]
H. Nagai, Optimal strategies for risk sensitive portfolio optimization problems for general factor models. SIAM J. Control Optim. 41, 1779–1800 (2003)CrossRefMATHMathSciNet
[N76]
[N78]
M. Nisio, On nonlinear semigroups for Markov processes associated with optimal stopping. Appl. Math. Optim. 4, 143–169 (1978)CrossRefMATHMathSciNet
[N81]
M. Nisio, Lecture on Stochastic Control Theory. ISI Lecture Notes, vol. 9 (McMillan India, Delhi 1981)
[N88]
M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations. Nagoya Math. J. 110, 163–184 (1988)MATHMathSciNet
[P79]
[P93]
E. Pardoux, Stochastic partial differential equations, a review. Bull. Sc. Math. 117, 29–47 (1993)MATHMathSciNet
[PP90]
[Pe92]
[Ph02]
H. Pham, Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl. Math. Optim. 46, 55–78 (2002)CrossRefMATHMathSciNet
[R90]
B.L. Rozovskii, Stochastic Evolution Systems (Kluwer Academic, Dordrecht/Boston, 1990)CrossRefMATH
[S08]
A.N. Shiryayev, Optimal Stopping Rules, 2nd edn. (Springer, Berlin/Heidelberg 2008)
[St11]
[SV79]
D.V. Strook, S.R.S. Varadhan, Multidimensional Diffusion Processes (Springer, Berlin/New York, 1979)
[W71]
J.C. Willems, Least squares stationary optimal control and the algebraic Ricatti equation. IEEE. Trans. Auto. Control 16, 621–635 (1971)CrossRefMathSciNet
[Wo68]
W.M. Wonham, on a matrix Ricatti equation of stochastic control. SIAM J. Control Optim. 6, 681–697 (1968)
[Y80]
[YZ99]
J. Yong, X.Y. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations (Springer, New York, 1999)MATH
Metadaten
Titel
Stochastic Parabolic Equations
verfasst von
Makiko Nisio
Copyright-Jahr
2015
Verlag
Springer Japan
Buch
Stochastic Control Theory

Print ISBN: 978-4-431-55122-5

Electronic ISBN: 978-4-431-55123-2

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-4-431-55123-2_5