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.
Mots clés : Viscosity solutions, Hamilton−Jacobi−Bellman equations, homogenization and singular perturbations, large deviations, stochastic volatility models
@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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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 | DOI | MR | Zbl
[2] and , Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61 | DOI | MR | Zbl
[3] and , Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. In vol. 960 of Memoirs of the American Mathematical Society (2010) vi+77 | MR | Zbl
[4] , and , Multiscale problems and homogenization for second-order Hamilton−Jacobi equations. J. Differ. Equ. 243 (2007) 349–387 | DOI | MR | Zbl
[5] and , On ergodic stochastic control. Commun. Partial Differ. Equ. 23 (1998) 2187–2217 | DOI | MR | Zbl
[6] , and , Convergence by viscosity methods in multiscale financial models with stochastic volatility. Siam J. Financial Math. 1 (2010) 230–265 | DOI | MR | Zbl
[7] , and , Large deviations for some fast stochastic volatility models by viscosity methods. Discrete Contin. Dyn. Syst. A. 35 (2015) 3965–3988 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[13] and , Comparison principle for Dirichlet-type Hamilton−Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21–44 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[17] , and , Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Amer. Math. Soc. 362 (2010) 4511–4536 | DOI | MR | Zbl
[18] , A note on the regularity of solutions of Hamilton−Jacobi equationswith superlinear growth in the gradient variable. ESAIM: COCV 15 (2009) 367–376 | Numdam | MR | Zbl
[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 | DOI | MR | Zbl
[20] , and , User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67 | DOI | MR | Zbl
[21] and , Uniqueness results for second order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74–106 | DOI | MR | Zbl
[22] and , Value functions for Boltza problems with discontinuous lagrangian and Hamilton−Jacobi inequality. ESAIM: COCV 5 (2000) 369–393 | Numdam | MR | Zbl
[23] , The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359–375 | DOI | MR | Zbl
[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 | DOI | Numdam | MR | Zbl
[25] , and , Small time asymptotic for fast mean-reverting stochastic volatility models. Ann. Appl. Probab. 22 (2012) 1541–1575 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[33] and , On the structure of solution of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34 (2006) 284–320 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[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 | DOI | Numdam | MR | Zbl
[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 | DOI | MR | Zbl
[40] and , On the Poisson equation and diffusion approximation II. Ann. Probab. 31 (2003) 1166–1192 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
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