Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space H with a non-linear diffusion coefficient σ ( X ) and a generic unbounded operator A in the drift term. When the gain...

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Vydáno v:	Applied mathematics & optimization Ročník 73; číslo 2; s. 271 - 312
Hlavní autoři:	Chiarolla, Maria B., De Angelis, Tiziano
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2016 Springer Nature B.V
Témata:	Approximation Calculus of Variations and Optimal Control; Optimization Control Diffusion Drift Hilbert space Inequalities Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Numerical and Computational Physics Optimization Partial differential equations Pricing Simulation Stochastic models Studies Systems Theory Theoretical Viscosity Optimal stopping Infinite-dimensional stochastic analysis 60G40 49J40 Degenerate variational inequalities 35R15 Parabolic partial differential equations
ISSN:	0095-4616, 1432-0606
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Abstract	A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space H with a non-linear diffusion coefficient σ ( X ) and a generic unbounded operator A in the drift term. When the gain function Θ is time-dependent and fulfils mild regularity assumptions, the value function U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient σ ( X ) is specified, the solution of the variational problem is found in a suitable Banach space V fully characterized in terms of a Gaussian measure μ . This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982 ), of well-known results on optimal stopping theory and variational inequalities in R n . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space ... with a non-linear diffusion coefficient ... and a generic unbounded operator A in the drift term. When the gain function ... is time-dependent and fulfils mild regularity assumptions, the value function ... of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient ... is specified, the solution of the variational problem is found in a suitable Banach space ... fully characterized in terms of a Gaussian measure ... This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982 ), of well-known results on optimal stopping theory and variational inequalities in ... These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model. A finite horizon optimal stopping problem for an infinite dimensional diffusion X is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space H with a non-linear diffusion coefficient σ ( X ) and a generic unbounded operator A in the drift term. When the gain function Θ is time-dependent and fulfils mild regularity assumptions, the value function U of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient σ ( X ) is specified, the solution of the variational problem is found in a suitable Banach space V fully characterized in terms of a Gaussian measure μ . This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions (Application of variational inequalities in stochastic control, 1982 ), of well-known results on optimal stopping theory and variational inequalities in R n . These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.
Author	Chiarolla, Maria B. De Angelis, Tiziano
Author_xml	– sequence: 1 givenname: Maria B. surname: Chiarolla fullname: Chiarolla, Maria B. organization: Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza (MEMOTEF), Università di Roma ‘La Sapienza’ – sequence: 2 givenname: Tiziano surname: De Angelis fullname: De Angelis, Tiziano email: tiziano.deangelis@manchester.ac.uk organization: School of Mathematics, University of Manchester
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References	Da PratoGZabczykJSecond Order Partial Differential Equations in Hilbert Spaces2004CambridgeCambridge University Press1012.35001 MenaldiJLOn the optimal stopping time problem for degenerate diffusionsSIAM J. Control Optim.198018669772159292810.1137/03180520462.93045 BarbuVMarinelliCVariational inequalities in Hilbert spaces with measures and optimal stopping problemsAppl. Math. Optim.200857237262238610510.1007/s00245-007-9021-x1144.49006 Da PratoGZabczykJStochastic Equations in Infinite Dimensions1992CambridgeCambridge University Press10.1017/CBO97805116662230761.60052 Menaldi, J.L.: On Degenerate Variational and Quasi-Variational Inequalities of parabolic Type. Analysis and Optimization of Systems, Lecture notes in Control and Information Sciences, vol. 28, pp. 338–356 (1980) LionsPLViscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutionsActa Math.19883–424327897179710.1007/BF023922990757.93082 KrylovNVControlled Diffusion Processes2009BerlinSpringer1171.93004 Chow, P.L., Menaldi, J.L.: Variational Inequalities for the Control of Stochastic Partial Differential Equations. Stochastic Partial Differential Equations and Applications II, Lecture Notes in Mathematics, Springer, Berlin, pp. 42–52 (1989) El Karoui, N.: Les Aspects Probabilistes du Contrôle Stochastique. In: 9th Saint Flour Probability Summer School, Lecture Notes in Math. 876, Springer, Berlin, pp. 73–238 (1979) Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25, Springer, New York Zabczyk, J.: Stopping Problems on Polish Spaces. Ann. Univ. Mariae Curie-Sklodowska, 51 Vol. 1.18 pp. 181–199 (1997) StroockDVaradhanSRSOn degenerate elliptic-parabolic operators of second order and their associated diffusionsComm. Pure Appl. Math.19722565171338781210.1002/cpa.31602506030344.35041 ChiarollaMBDe AngelisTAnalytical pricing of American put options on a zero coupon bond in the Heath-Jarrow-Morton modelStoch. Process.2015125678707329329910.1016/j.spa.2014.09.0211307.91173 DieudonnéJFoundations of Modern Analysis1969LondonAcademic Press0176.00502 Da PratoGAn Introduction to Infinite-Dimensional Analysis2006BerlinSpringer10.1007/3-540-29021-41109.46001 Bogachev, V.I.: Gaussian Measures. American Mathematical Society (1997) LionsPLViscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equationsJ. Funct. Anal.1989861118101393110.1016/0022-1236(89)90062-10757.93084 KelomeDŚwiȩchAViscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equationAppl. Math. Optim.200347253278197438910.1007/s00245-003-0764-81025.49019 PazyASemigroups of Linear Operator and Applications to Partial Differential Equations1983New YorkSpringer10.1007/978-1-4612-5561-10516.47023 AdamsRASobolev Spaces1975LondonAcademic Press0314.46030 ShiryaevANOptimal Stopping Rules1978BerlinSpringer0391.60002 MaZMRöcknerMIntroduction to the theory of (non-symmetric) dirichlet forms1992BerlinSpringer10.1007/978-3-642-77739-40826.31001 ŚwiȩchA“Unbounded” second order partial differential equations in infinite-dimensional Hilbert spacesComm. Partial Differ. Equ.19941911–121999203613011800812.35154 Zabczyk, J.: Stopping Problems in Stochastic Control. In: Proceedings of the International Congress of Mathematicians, vol. 1–2, PWN, Warsaw, pp. 1425–1437 (1984) Ga̧tarekDŚwiȩchAOptimal stopping in Hilbert spaces and pricing of American optionsMath. Methods Oper. Res.199950135147171117710.1007/s0018600500400991.91033 ZabczykJBellman’s inclusions and excessive measuresProbab. Math. Statist.200121110112218697240992.60077 Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2010) MarcozziMDOn the approximation of infinite dimensional optimal stopping problems with application to mathematical financeJ. Sci Comput.200834287307237762110.1007/s10915-007-9168-21133.91502 BensoussanALionsJLApplications of Variational Inequalities in Stochastic Control1982AmsterdamNorth-Holland0478.49002 Lions, P.L.: Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai’s equation. Lecture Notes in Math., 1390, Springer, Berlin, pp. 147–170 (1989) De Angelis, T.: Pricing American Bond Options under HJM: An Infinite Dimensional Variational Inequality. Ph.D thesis (2012) BarbuVSritharanSSOptimal stopping-time problem for stochastic Navier-Stokes equations and infinite-dimensional variational inequalitiesNonlinear Anal.20066410181024219680910.1016/j.na.2005.05.0541091.60005 9302_CR20 JL Menaldi (9302_CR24) 1980; 18 MB Chiarolla (9302_CR7) 2015; 125 9302_CR25 J Zabczyk (9302_CR32) 2001; 21 9302_CR8 A Świȩch (9302_CR29) 1994; 19 9302_CR6 9302_CR5 D Stroock (9302_CR28) 1972; 25 PL Lions (9302_CR21) 1989; 86 A Bensoussan (9302_CR4) 1982 9302_CR31 9302_CR12 9302_CR14 9302_CR15 V Barbu (9302_CR3) 2006; 64 RA Adams (9302_CR1) 1975 AN Shiryaev (9302_CR27) 1978 G Prato Da (9302_CR9) 2006 D Ga̧tarek (9302_CR16) 1999; 50 G Da Prato (9302_CR10) 1992 J Dieudonné (9302_CR13) 1969 NV Krylov (9302_CR18) 2009 V Barbu (9302_CR2) 2008; 57 G Da Prato (9302_CR11) 2004 PL Lions (9302_CR19) 1988; 3–4 MD Marcozzi (9302_CR23) 2008; 34 D Kelome (9302_CR17) 2003; 47 ZM Ma (9302_CR22) 1992 A Pazy (9302_CR26) 1983 9302_CR30
References_xml	– reference: Da PratoGAn Introduction to Infinite-Dimensional Analysis2006BerlinSpringer10.1007/3-540-29021-41109.46001 – reference: BarbuVSritharanSSOptimal stopping-time problem for stochastic Navier-Stokes equations and infinite-dimensional variational inequalitiesNonlinear Anal.20066410181024219680910.1016/j.na.2005.05.0541091.60005 – reference: BensoussanALionsJLApplications of Variational Inequalities in Stochastic Control1982AmsterdamNorth-Holland0478.49002 – reference: MarcozziMDOn the approximation of infinite dimensional optimal stopping problems with application to mathematical financeJ. Sci Comput.200834287307237762110.1007/s10915-007-9168-21133.91502 – reference: ZabczykJBellman’s inclusions and excessive measuresProbab. Math. Statist.200121110112218697240992.60077 – reference: Da PratoGZabczykJStochastic Equations in Infinite Dimensions1992CambridgeCambridge University Press10.1017/CBO97805116662230761.60052 – reference: LionsPLViscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutionsActa Math.19883–424327897179710.1007/BF023922990757.93082 – reference: Bogachev, V.I.: Gaussian Measures. American Mathematical Society (1997) – reference: Zabczyk, J.: Stopping Problems on Polish Spaces. Ann. Univ. Mariae Curie-Sklodowska, 51 Vol. 1.18 pp. 181–199 (1997) – reference: AdamsRASobolev Spaces1975LondonAcademic Press0314.46030 – reference: Zabczyk, J.: Stopping Problems in Stochastic Control. In: Proceedings of the International Congress of Mathematicians, vol. 1–2, PWN, Warsaw, pp. 1425–1437 (1984) – reference: Da PratoGZabczykJSecond Order Partial Differential Equations in Hilbert Spaces2004CambridgeCambridge University Press1012.35001 – reference: Chow, P.L., Menaldi, J.L.: Variational Inequalities for the Control of Stochastic Partial Differential Equations. Stochastic Partial Differential Equations and Applications II, Lecture Notes in Mathematics, Springer, Berlin, pp. 42–52 (1989) – reference: Lions, P.L.: Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai’s equation. Lecture Notes in Math., 1390, Springer, Berlin, pp. 147–170 (1989) – reference: StroockDVaradhanSRSOn degenerate elliptic-parabolic operators of second order and their associated diffusionsComm. Pure Appl. Math.19722565171338781210.1002/cpa.31602506030344.35041 – reference: BarbuVMarinelliCVariational inequalities in Hilbert spaces with measures and optimal stopping problemsAppl. Math. Optim.200857237262238610510.1007/s00245-007-9021-x1144.49006 – reference: Ga̧tarekDŚwiȩchAOptimal stopping in Hilbert spaces and pricing of American optionsMath. Methods Oper. Res.199950135147171117710.1007/s0018600500400991.91033 – reference: LionsPLViscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equationsJ. Funct. Anal.1989861118101393110.1016/0022-1236(89)90062-10757.93084 – reference: KrylovNVControlled Diffusion Processes2009BerlinSpringer1171.93004 – reference: MaZMRöcknerMIntroduction to the theory of (non-symmetric) dirichlet forms1992BerlinSpringer10.1007/978-3-642-77739-40826.31001 – reference: PazyASemigroups of Linear Operator and Applications to Partial Differential Equations1983New YorkSpringer10.1007/978-1-4612-5561-10516.47023 – reference: El Karoui, N.: Les Aspects Probabilistes du Contrôle Stochastique. In: 9th Saint Flour Probability Summer School, Lecture Notes in Math. 876, Springer, Berlin, pp. 73–238 (1979) – reference: Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2010) – reference: MenaldiJLOn the optimal stopping time problem for degenerate diffusionsSIAM J. Control Optim.198018669772159292810.1137/03180520462.93045 – reference: Menaldi, J.L.: On Degenerate Variational and Quasi-Variational Inequalities of parabolic Type. Analysis and Optimization of Systems, Lecture notes in Control and Information Sciences, vol. 28, pp. 338–356 (1980) – reference: DieudonnéJFoundations of Modern Analysis1969LondonAcademic Press0176.00502 – reference: Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25, Springer, New York – reference: KelomeDŚwiȩchAViscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equationAppl. Math. Optim.200347253278197438910.1007/s00245-003-0764-81025.49019 – reference: ChiarollaMBDe AngelisTAnalytical pricing of American put options on a zero coupon bond in the Heath-Jarrow-Morton modelStoch. Process.2015125678707329329910.1016/j.spa.2014.09.0211307.91173 – reference: ŚwiȩchA“Unbounded” second order partial differential equations in infinite-dimensional Hilbert spacesComm. Partial Differ. Equ.19941911–121999203613011800812.35154 – reference: De Angelis, T.: Pricing American Bond Options under HJM: An Infinite Dimensional Variational Inequality. 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SubjectTerms	Approximation Calculus of Variations and Optimal Control; Optimization Control Diffusion Drift Hilbert space Inequalities Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Numerical and Computational Physics Optimization Partial differential equations Pricing Simulation Stochastic models Studies Systems Theory Theoretical Viscosity
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