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Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets
Large deviations for some fast stochastic volatility models by viscosity methods
1. | Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35133 Padova, Italy, Italy, Italy |
References:
[1] |
O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control,, SIAM J. Control Optim., 40 (): 1159.
doi: 10.1137/S0363012900366741. |
[2] |
O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61.
doi: 10.1007/s00205-003-0266-5. |
[3] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp.
doi: 10.1090/S0065-9266-09-00588-2. |
[4] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[5] |
M. Avellaneda, D. Boyer-Olson, J. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C.R. Math. Acad. Sci. Paris, 336 (2003), 263-266.
doi: 10.1016/S1631-073X(03)00032-3. |
[6] |
M. Arisawa and P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[7] |
S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione, Master thesis, University of Padova, 2012. |
[8] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[9] |
M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265.
doi: 10.1137/090748147. |
[10] |
M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach, Eur. J. Control, 17 (2011), 30-45.
doi: 10.3166/ejc.17.30-45. |
[11] |
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, ().
|
[12] |
G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44.
doi: 10.1007/BF01445155. |
[13] |
F. Camilli, A. Cesaroni and C. Marchi, Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates, Adv. Nonlinear Stud., 11 (2011), 405-428. |
[14] |
F. Da Lio and O. Ley, Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications, SIAM J. Control Optim., 45 (2006), 74-106.
doi: 10.1137/S0363012904440897. |
[15] |
G. Dal Maso and H. Frankowska, Value functions for Boltza problems with discontinuous lagrangian and Hamilton-Jacobi inequality, ESAIM Control Optim. Calc. Var., 5 (2000), 369-393.
doi: 10.1051/cocv:2000114. |
[16] |
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998.
doi: 10.1007/978-1-4612-5320-4. |
[17] |
P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods, Stoch. Process. Appl., 122 (2012), 1947-1987.
doi: 10.1016/j.spa.2011.12.006. |
[18] |
L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.
doi: 10.1017/S0308210500018631. |
[19] |
L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 1-20. |
[20] |
J. Feng, M. Forde and J.-P. Fouque, Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, SIAM J. Financial Math., 1 (2010), 126-141.
doi: 10.1137/090745465. |
[21] |
J. Feng, J.-P. Fouque and R. Kumar, Small time asymptotic for fast mean-reverting stochstic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575.
doi: 10.1214/11-AAP801. |
[22] |
J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/131. |
[23] |
W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions, Springer, New York, 2006. |
[24] |
J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge university press, Cambridge, 2000. |
[25] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), 1648-1665.
doi: 10.1137/S0036139902401550. |
[26] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), 22-42.
doi: 10.1137/030600291. |
[27] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9781139020534. |
[28] |
D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., ().
|
[29] |
Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-13242-5. |
[30] |
H. Kaise and S. Sheu, On the structure of solution of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2006), 284-320.
doi: 10.1214/009117905000000431. |
[31] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston 1990.
doi: 10.1007/978-1-4612-4482-0. |
[32] |
H.J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.
doi: 10.1007/s00245-010-9104-y. |
[33] |
R. Lipster, Large deviations for two scaled diffusions, Probab. Theory Relat. Fields, 106 (1996), 71-104.
doi: 10.1007/s004400050058. |
[34] |
K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion, Appl Math Optim, 67 (2013), 123-161.
doi: 10.1007/s00245-012-9183-z. |
[35] |
A. Takahashi and K. Yamamoto, A remark on a singular perturbation method for option pricing under a stochastic volatility, Asia-Pacific Financial Markets, 16 (2009), 333-345. |
[36] |
A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging, Stochastic Process. Appl., 89 (2000), 69-79.
doi: 10.1016/S0304-4149(00)00013-2. |
[37] |
D. Williams, Probability with Martingales, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511813658. |
show all references
References:
[1] |
O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control,, SIAM J. Control Optim., 40 (): 1159.
doi: 10.1137/S0363012900366741. |
[2] |
O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61.
doi: 10.1007/s00205-003-0266-5. |
[3] |
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010), vi+77 pp.
doi: 10.1090/S0065-9266-09-00588-2. |
[4] |
O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387.
doi: 10.1016/j.jde.2007.05.027. |
[5] |
M. Avellaneda, D. Boyer-Olson, J. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C.R. Math. Acad. Sci. Paris, 336 (2003), 263-266.
doi: 10.1016/S1631-073X(03)00032-3. |
[6] |
M. Arisawa and P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
doi: 10.1080/03605309808821413. |
[7] |
S. Balbinot, Valore Critico Per Hamiltoniane non Coercive e Applicazioni a Problemi di Omogeneizzazione, Master thesis, University of Padova, 2012. |
[8] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[9] |
M. Bardi, A. Cesaroni and L. Manca, Convergence by viscosity methods in multiscale financial models with stochastic volatility, SIAM J. Financial Math., 1 (2010), 230-265.
doi: 10.1137/090748147. |
[10] |
M. Bardi and A. Cesaroni, Optimal control with random parameters: A multiscale approach, Eur. J. Control, 17 (2011), 30-45.
doi: 10.3166/ejc.17.30-45. |
[11] |
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, Mathématiques and Applications 17, ().
|
[12] |
G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Appl. Math. Optim., 21 (1990), 21-44.
doi: 10.1007/BF01445155. |
[13] |
F. Camilli, A. Cesaroni and C. Marchi, Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates, Adv. Nonlinear Stud., 11 (2011), 405-428. |
[14] |
F. Da Lio and O. Ley, Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications, SIAM J. Control Optim., 45 (2006), 74-106.
doi: 10.1137/S0363012904440897. |
[15] |
G. Dal Maso and H. Frankowska, Value functions for Boltza problems with discontinuous lagrangian and Hamilton-Jacobi inequality, ESAIM Control Optim. Calc. Var., 5 (2000), 369-393.
doi: 10.1051/cocv:2000114. |
[16] |
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998.
doi: 10.1007/978-1-4612-5320-4. |
[17] |
P. Dupuis and K. Spiliopoulos, Large deviations for multiscale problems via weak convergence methods, Stoch. Process. Appl., 122 (2012), 1947-1987.
doi: 10.1016/j.spa.2011.12.006. |
[18] |
L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.
doi: 10.1017/S0308210500018631. |
[19] |
L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 1-20. |
[20] |
J. Feng, M. Forde and J.-P. Fouque, Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, SIAM J. Financial Math., 1 (2010), 126-141.
doi: 10.1137/090745465. |
[21] |
J. Feng, J.-P. Fouque and R. Kumar, Small time asymptotic for fast mean-reverting stochstic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575.
doi: 10.1214/11-AAP801. |
[22] |
J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, American Mathematical Society, Providence, RI, 2006.
doi: 10.1090/surv/131. |
[23] |
W. H. Fleming and H. M. Soner, Controlled Markos Processes and Viscosity Solutions, Springer, New York, 2006. |
[24] |
J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge university press, Cambridge, 2000. |
[25] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), 1648-1665.
doi: 10.1137/S0036139902401550. |
[26] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics, Multiscale Model. Simul., 2 (2003), 22-42.
doi: 10.1137/030600291. |
[27] |
J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, 2011.
doi: 10.1017/CBO9781139020534. |
[28] |
D. Ghilli, Ph.D. thesis, University of Padova,, in preparation., ().
|
[29] |
Y. Kabanov and S. Pergamenshchikov, Two-scale Stochastic Systems. Asymptotic Analysis and Control, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-662-13242-5. |
[30] |
H. Kaise and S. Sheu, On the structure of solution of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2006), 284-320.
doi: 10.1214/009117905000000431. |
[31] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston 1990.
doi: 10.1007/978-1-4612-4482-0. |
[32] |
H.J. Kushner, Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.
doi: 10.1007/s00245-010-9104-y. |
[33] |
R. Lipster, Large deviations for two scaled diffusions, Probab. Theory Relat. Fields, 106 (1996), 71-104.
doi: 10.1007/s004400050058. |
[34] |
K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion, Appl Math Optim, 67 (2013), 123-161.
doi: 10.1007/s00245-012-9183-z. |
[35] |
A. Takahashi and K. Yamamoto, A remark on a singular perturbation method for option pricing under a stochastic volatility, Asia-Pacific Financial Markets, 16 (2009), 333-345. |
[36] |
A. Yu. Veretennikov, On large deviations for SDEs with small diffusion and averaging, Stochastic Process. Appl., 89 (2000), 69-79.
doi: 10.1016/S0304-4149(00)00013-2. |
[37] |
D. Williams, Probability with Martingales, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511813658. |
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