O. Alvarez and M. Bardi, Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control, SIAM Journal on Control and Optimization, vol.40, issue.4, pp.1159-1188, 2001.
DOI : 10.1137/S0363012900366741

O. Alvarez and M. Bardi, Singular Perturbations of Nonlinear Degenerate Parabolic PDEs: a General Convergence Result, Archive for Rational Mechanics and Analysis, vol.170, issue.1, pp.17-61, 2003.
DOI : 10.1007/s00205-003-0266-5

O. Alvarez and M. , Bardi: Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer, Math. Soc, issue.960, 2010.

O. Alvarez, M. Bardi, and C. Marchi, Multiscale problems and homogenization for second-order Hamilton???Jacobi equations, Journal of Differential Equations, vol.243, issue.2, pp.349-387, 2007.
DOI : 10.1016/j.jde.2007.05.027

M. Avellaneda, D. Boyer-olson, J. Busca, and P. Friz, Application of large deviation methods to the pricing of index options in finance, Comptes Rendus Mathematique, vol.336, issue.3, pp.263-266, 2003.
DOI : 10.1016/S1631-073X(03)00032-3

M. Arisawa and P. Lions, On ergodic stochastic control, Communications in Partial Differential Equations, vol.318, issue.11-12, pp.2187-2217, 1998.
DOI : 10.1137/0319020

S. Balbinot, Valore critico per Hamiltoniane non coercive e applicazioni a problemi di omogeneizzazione, 2012.

M. Bardi and I. , Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton- Jacobi-Bellman equations, 1997.

M. Bardi, A. Cesaroni, and L. Manca, Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility, SIAM Journal on Financial Mathematics, vol.1, issue.1, pp.230-265, 2010.
DOI : 10.1137/090748147

M. Bardi and A. Cesaroni, Optimal Control with Random Parameters: A Multiscale Approach, European Journal of Control, vol.17, issue.1, pp.30-45, 2011.
DOI : 10.3166/ejc.17.30-45

URL : https://hal.archives-ouvertes.fr/hal-00664449

G. Barles, Solutions de viscosité deséquationsdeséquations de Hamilton-Jacobi, Mathématiques and Applications, vol.17

G. Barles and B. Perthame, Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Applied Mathematics & Optimization, vol.25, issue.1, pp.21-44, 1990.
DOI : 10.1007/BF01445155

F. Camilli, A. Cesaroni, and C. Marchi, Abstract, Advanced Nonlinear Studies, vol.11, issue.2, pp.405-428, 2011.
DOI : 10.1515/ans-2011-0210

A. Cutr-`-cutr-`-i and F. Da-lio, Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations, ESAIM: Control, Optimisation and Calculus of Variations, vol.13, issue.3, pp.484-502, 2007.
DOI : 10.1051/cocv:2007021

F. Da-lio and O. Ley, Uniqueness Results for Second-Order Bellman--Isaacs Equations under Quadratic Growth Assumptions and Applications, SIAM Journal on Control and Optimization, vol.45, issue.1, pp.74-106, 2006.
DOI : 10.1137/S0363012904440897

URL : https://hal.archives-ouvertes.fr/hal-00455993

G. Dal-maso and H. Frankowska, Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities, ESAIM: Control, Optimisation and Calculus of Variations, vol.5, pp.369-393, 2000.
DOI : 10.1051/cocv:2000114

P. Dupuis and K. Spiliopoulos, Large deviations for multiscale diffusion via weak convergence methods, Stochastic Processes and their Applications, vol.122, issue.4, pp.1947-1987, 2012.
DOI : 10.1016/j.spa.2011.12.006

L. C. Evans, Synopsis, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.1248, issue.3-4, pp.359-375, 1989.
DOI : 10.1137/0326063

L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.2, issue.1, pp.1-20, 1985.
DOI : 10.1016/S0294-1449(16)30409-7

J. Feng, M. Forde, and J. Fouque, Short-Maturity Asymptotics for a Fast Mean-Reverting Heston Stochastic Volatility Model, SIAM Journal on Financial Mathematics, vol.1, issue.1, pp.126-141, 2010.
DOI : 10.1137/090745465

J. Feng, J. Fouque, and R. Kumar, Small-time asymptotics for fast mean-reverting stochastic volatility models, The Annals of Applied Probability, vol.22, issue.4, pp.1541-1575, 2012.
DOI : 10.1214/11-AAP801

J. Feng and T. G. Kurtz, Large deviations for stochastic processes, 2006.

W. H. Fleming and H. M. Soner, Controlled Markos processes and viscosity solutions, 2006.

J. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in financial markets with stochastic volatility, 2000.

J. Fouque, G. Papanicolaou, R. Sircar, and K. , Solna: Singular perturbations in option pricing, SIAM J. Appl. Math, vol.63, issue.5, pp.1648-1665, 2003.

J. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, Multiscale Stochastic Volatility Asymptotics, Multiscale stochastic volatility asymptotics, pp.22-42, 2003.
DOI : 10.1137/030600291

J. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, Multiscale stochastic volatility for equity, interest rate, and credit derivatives, 2011.
DOI : 10.1017/CBO9781139020534

Y. Kabanov and S. Pergamenshchikov, Two-scale stochastic systems Asymptotic analysis and control, 2003.

H. Kaise and S. Sheu, On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control, The Annals of Probability, vol.34, issue.1, pp.284-320, 2006.
DOI : 10.1214/009117905000000431

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, 1990.
DOI : 10.1007/978-1-4612-4482-0

H. J. Kushner, Large Deviations for Two-Time-Scale Diffusions, with??Delays, Applied Mathematics & Optimization, vol.244, issue.1, pp.295-322, 2010.
DOI : 10.1007/s00245-010-9104-y

R. Lipster, Large deviations for two scaled diffusions, Probab. Theory Relat, pp.71-104, 1996.

K. Spiliopoulos, Large Deviations and Importance Sampling for Systems of Slow-Fast Motion, Applied Mathematics & Optimization, vol.85, issue.1, pp.123-161, 2013.
DOI : 10.1007/s00245-012-9183-z

A. Takahashi and K. Yamamoto, A Remark on a Singular Perturbation Method for Option Pricing under a Stochastic Volatility, Asia-Pacific Financial Markets, vol.16, pp.333-345, 2009.

A. Yu and . Veretennikov, On large deviations for SDEs with small diffusion and averaging. Stochastic Process, Appl, vol.89, issue.1, pp.69-79, 2000.

D. Williams, Probability with Martingales, 1991.
DOI : 10.1017/CBO9780511813658

M. Dipartimento-di, Università di Padova, via Trieste 63, 35121 Padova, Italy E-mail address: bardi@math