Voir la notice de l'article provenant de la source Numdam
We study singular perturbation problems for second order HJB equations in an unbounded setting. The main applications are large deviations estimates for the short maturity asymptotics of stochastic systems affected by a stochastic volatility, where the volatility is modelled by a process evolving at a faster time scale and satisfying some condition implying ergodicity.
Ghilli, Daria 1
@article{COCV_2018__24_2_605_0,
author = {Ghilli, Daria},
title = {Viscosity methods for large deviations estimates of multiscale stochastic processes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {605--637},
publisher = {EDP-Sciences},
volume = {24},
number = {2},
year = {2018},
doi = {10.1051/cocv/2017051},
zbl = {1403.35334},
mrnumber = {3816407},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017051/}
}
TY - JOUR AU - Ghilli, Daria TI - Viscosity methods for large deviations estimates of multiscale stochastic processes JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 605 EP - 637 VL - 24 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017051/ DO - 10.1051/cocv/2017051 LA - en ID - COCV_2018__24_2_605_0 ER -
%0 Journal Article %A Ghilli, Daria %T Viscosity methods for large deviations estimates of multiscale stochastic processes %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 605-637 %V 24 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017051/ %R 10.1051/cocv/2017051 %G en %F COCV_2018__24_2_605_0
Ghilli, Daria. Viscosity methods for large deviations estimates of multiscale stochastic processes. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 605-637. doi : 10.1051/cocv/2017051. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017051/
[1] and , Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001/02) 1159–1188 | Zbl | MR | DOI
[2] and , Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61 | Zbl | MR | DOI
[3] and , Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. In vol. 960 of Memoirs of the American Mathematical Society (2010) vi+77 | Zbl | MR
[4] , and , Multiscale problems and homogenization for second-order Hamilton−Jacobi equations. J. Differ. Equ. 243 (2007) 349–387 | Zbl | MR | DOI
[5] and , On ergodic stochastic control. Commun. Partial Differ. Equ. 23 (1998) 2187–2217 | Zbl | MR | DOI
[6] , and , Convergence by viscosity methods in multiscale financial models with stochastic volatility. Siam J. Financial Math. 1 (2010) 230–265 | MR | Zbl | DOI
[7] , and , Large deviations for some fast stochastic volatility models by viscosity methods. Discrete Contin. Dyn. Syst. A. 35 (2015) 3965–3988 | MR | Zbl | DOI
[8] , A weak bernstein method for fully nonlinear elliptic equations. Differ. Integral Equ. 4 (1991) 241–262 | MR | Zbl
[9] , C0,α-regularity and estimates for solutions of elliptic and parabolic equations by the Ishii−Lions method. Gakuto International Series, Math. Sci. Appl. 30 (2008) 33–47
[10] , A short proof of the C0,α-regularity of viscosity subsolutions for superquadratic viscous Hamilton−Jacobi equations and applications. Nonlinear Anal. 73 (2010) 31–47 | MR | Zbl | DOI
[11] , Solutions de viscosité des équations de Hamilton−Jacobi. In Vol. 17 of Math. Appl. Springer Verlag (1994) | MR
[12] , and , A new PDE approach to the large time asymptotics of solutions of Hamilton−Jacobi equations. Bull. Math. Sci. 3 (2013) 363–388 | MR | Zbl | DOI
[13] and , Comparison principle for Dirichlet-type Hamilton−Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21–44 | MR | Zbl | DOI
[14] and , Space-time periodic solutions and long-time behavior of solutions of quasilinear parabolic equations. SIAM J. Math. Anal. (electronic) 32 (2001) 1311–1323 | MR | Zbl | DOI
[15] , Perturbation Methods in Optimal Control, John Wiley and Sons, Montrouge, France (1988) | MR | Zbl
[16] and , Singular perturbations in ergodic control of diffusions. SIAM J. Control Optim. 46 (2007) 1562–1577 | MR | Zbl | DOI
[17] , and , Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Amer. Math. Soc. 362 (2010) 4511–4536 | MR | Zbl | DOI
[18] , A note on the regularity of solutions of Hamilton−Jacobi equationswith superlinear growth in the gradient variable. ESAIM: COCV 15 (2009) 367–376 | MR | Zbl | mathdoc-id
[19] and , Hölder continuity to Hamilton−Jacobi equationswith superquadratic growth in the gradient and unbounded right-hand side. Comm. Partial Differ. Equ. 37 (2012) 1668–1688 | MR | Zbl | DOI
[20] , and , User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67 | MR | Zbl | DOI
[21] and , Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74–106 | MR | Zbl | DOI
[22] and , Value functions for Boltza problems with discontinuous lagrangian and Hamilton−Jacobi inequality. ESAIM: COCV 5 (2000) 369–393 | MR | Zbl | mathdoc-id
[23] , The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359–375 | MR | Zbl | DOI
[24] and , A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities. Ann. Inst. Henri Poincaré Anal. Non Linéaire 2 (1985) 1–20 | MR | Zbl | mathdoc-id | DOI
[25] , and , Small time asymptotic for fast mean-reverting stochastic volatility models. Ann. Appl. Probab. 22 (2012) 1541–1575 | MR | Zbl | DOI
[26] and , Large deviations for stochastic processes. Amer. Math. Soc. Providence, RI (2006) | MR | Zbl
[27] , and , Derivatives in financial markets with stochastic volatility. Cambridge university press, Cambridge (2000) | MR | Zbl
[28] , Some results in nonlinear PDEs: large deviations problems, nonlocal operators and stability for some isoperimetric problems. Ph.D. thesis, Dept. of Mathematics, Univ. of Padua (2016)
[29] , Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan denRijn, The Netherlands, Germantown, MD (1980) | MR
[30] , Recurrence and transience of optimal feedback processes associated with Bellman equations of ergodic type. SIAM J. Control Optim. 49 (2011) 1938–1960 | MR | Zbl | DOI
[31] and , Large time behavior of solutions of Hamilton−Jacobi−Bellman equations with quadratic nonlinearity in gradient. J. Math. Anal. 45 (2013) 279–306 | MR | Zbl
[32] and , Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83 (1990) 26–78 | MR | Zbl | DOI
[33] and , On the structure of solution of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34 (2006) 284–320 | MR | Zbl | DOI
[34] and Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems. Nonlinear Analysis: Theory, Methods and Applications 130 (2016) 76–101 | MR | Zbl | DOI
[35] , Generalized solutions of Hamilton−Jacobi equations. Pitman (Advanced Publishing Program), Boston, Mass. (1982) | MR | Zbl
[36] and , Ergodicity of Diffusion Processes. manuscript 2002
[37] and , Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. Henri Poincaré – Ann. Non Lin. 22 (2005) 667–677 | MR | Zbl | mathdoc-id | DOI
[38] and , Analytical Methods for Markov Semigroups, Pure and Applied Mathematics (Boca Raton) 283. Chapman and Hall/, Boca Raton, FL (2007) xxxii+526 | MR | Zbl
[39] and , On the Poisson equation and diffusion approximation, I. Ann. Probab. 29 (2001) 1061–1085 | MR | Zbl | DOI
[40] and , On the Poisson equation and diffusion approximation II. Ann. Probab. 31 (2003) 1166–1192 | MR | Zbl | DOI
[41] and , On the Poisson equation and diffusion approximation. III, Ann. Probab. 33 (2005) 1111–1133 | MR | Zbl
[42] , On the classical solution of nonlinear elliptic equations of second order. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988) 1272–1287 | MR | Zbl
[43] , On regularity and existence of viscoity solutions of nonlinear second order, elliptic equations. Partial differential equations and the Calculus of variations: Essays in Honor of Ennio de Giorgi, Progress in Nonlinear Differential Equations and theirApplications. Birkhaüser Boston Inc. (1989) | MR | Zbl
[44] , On large deviations for SDEs with small diffusion and averaging. Stochastic Process. Appl. 89 (2000) 69–79 | MR | Zbl | DOI
Cité par Sources :