A Maximum Principle for SDEs of Mean-Field Type
Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control probl...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Andersson, Daniel [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Applied mathematics & optimization - Springer-Verlag, 1974, 63(2010), 3 vom: 30. Okt., Seite 341-356 |
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| Übergeordnetes Werk: |
volume:63 ; year:2010 ; number:3 ; day:30 ; month:10 ; pages:341-356 |
| Links: |
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| DOI / URN: |
10.1007/s00245-010-9123-8 |
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| Katalog-ID: |
OLC2072642612 |
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| 520 | |a Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. | ||
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10.1007/s00245-010-9123-8 doi (DE-627)OLC2072642612 (DE-He213)s00245-010-9123-8-p DE-627 ger DE-627 rakwb eng 510 VZ Andersson, Daniel verfasserin aut A Maximum Principle for SDEs of Mean-Field Type 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. Stochastic control Maximum principle Mean-field SDE McKean-Vlasov equation Time inconsistent control Djehiche, Boualem aut Enthalten in Applied mathematics & optimization Springer-Verlag, 1974 63(2010), 3 vom: 30. Okt., Seite 341-356 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:63 year:2010 number:3 day:30 month:10 pages:341-356 https://doi.org/10.1007/s00245-010-9123-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_32 GBV_ILN_60 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4307 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4700 AR 63 2010 3 30 10 341-356 |
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10.1007/s00245-010-9123-8 doi (DE-627)OLC2072642612 (DE-He213)s00245-010-9123-8-p DE-627 ger DE-627 rakwb eng 510 VZ Andersson, Daniel verfasserin aut A Maximum Principle for SDEs of Mean-Field Type 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. Stochastic control Maximum principle Mean-field SDE McKean-Vlasov equation Time inconsistent control Djehiche, Boualem aut Enthalten in Applied mathematics & optimization Springer-Verlag, 1974 63(2010), 3 vom: 30. Okt., Seite 341-356 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:63 year:2010 number:3 day:30 month:10 pages:341-356 https://doi.org/10.1007/s00245-010-9123-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_32 GBV_ILN_60 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4307 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4700 AR 63 2010 3 30 10 341-356 |
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10.1007/s00245-010-9123-8 doi (DE-627)OLC2072642612 (DE-He213)s00245-010-9123-8-p DE-627 ger DE-627 rakwb eng 510 VZ Andersson, Daniel verfasserin aut A Maximum Principle for SDEs of Mean-Field Type 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. Stochastic control Maximum principle Mean-field SDE McKean-Vlasov equation Time inconsistent control Djehiche, Boualem aut Enthalten in Applied mathematics & optimization Springer-Verlag, 1974 63(2010), 3 vom: 30. Okt., Seite 341-356 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:63 year:2010 number:3 day:30 month:10 pages:341-356 https://doi.org/10.1007/s00245-010-9123-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_32 GBV_ILN_60 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4307 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4700 AR 63 2010 3 30 10 341-356 |
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10.1007/s00245-010-9123-8 doi (DE-627)OLC2072642612 (DE-He213)s00245-010-9123-8-p DE-627 ger DE-627 rakwb eng 510 VZ Andersson, Daniel verfasserin aut A Maximum Principle for SDEs of Mean-Field Type 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. Stochastic control Maximum principle Mean-field SDE McKean-Vlasov equation Time inconsistent control Djehiche, Boualem aut Enthalten in Applied mathematics & optimization Springer-Verlag, 1974 63(2010), 3 vom: 30. Okt., Seite 341-356 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:63 year:2010 number:3 day:30 month:10 pages:341-356 https://doi.org/10.1007/s00245-010-9123-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_32 GBV_ILN_60 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4307 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4700 AR 63 2010 3 30 10 341-356 |
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10.1007/s00245-010-9123-8 doi (DE-627)OLC2072642612 (DE-He213)s00245-010-9123-8-p DE-627 ger DE-627 rakwb eng 510 VZ Andersson, Daniel verfasserin aut A Maximum Principle for SDEs of Mean-Field Type 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. Stochastic control Maximum principle Mean-field SDE McKean-Vlasov equation Time inconsistent control Djehiche, Boualem aut Enthalten in Applied mathematics & optimization Springer-Verlag, 1974 63(2010), 3 vom: 30. Okt., Seite 341-356 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:63 year:2010 number:3 day:30 month:10 pages:341-356 https://doi.org/10.1007/s00245-010-9123-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_32 GBV_ILN_60 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4116 GBV_ILN_4277 GBV_ILN_4307 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4700 AR 63 2010 3 30 10 341-356 |
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A Maximum Principle for SDEs of Mean-Field Type |
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A Maximum Principle for SDEs of Mean-Field Type |
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Andersson, Daniel |
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a maximum principle for sdes of mean-field type |
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A Maximum Principle for SDEs of Mean-Field Type |
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Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. © Springer Science+Business Media, LLC 2010 |
| abstractGer |
Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. © Springer Science+Business Media, LLC 2010 |
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Abstract We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem. © Springer Science+Business Media, LLC 2010 |
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