Viscosity methods for large deviations estimates of multiscale stochastic processes

Abstract

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.

Singular perturbations for a subelliptic operator

“…for some k 1 > 4, k 2 > 4, k 3 > 0 (see A2)). This choice of the drift is reminiscent of other similar conditions about recurrence of diffusion processes in the whole space (see for example [13] and references therein). By standard theory (see [10]), the value function V ǫ is the unique (viscosity) solution to the following Cauchy problem for an Hamilton-Jacobi-Bellman equation For the sake of completeness, in order to exhibit the degeneracy and the unboundedness of the operator, we write explicitly the second order term of L:…”

“…In this framework, the singular perturbation problems are strictly related to homogenization problems (see also [20]); Alvarez and Bardi [1,2] extended to singular perturbation problems with periodic fast variables the celebrated perturbed test function method by Evans (see also [3] for some cases in hypoelliptic periodic setting). Let us also recall that, the papers [5,6,13] studied singular perturbation problems of uniformly elliptic operators on the whole space.…”

Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations

“…We adapt the classical perturbed test function method (see [1,6,9]) to prove the convergence. To this end, we argue as in [13, Theorem 2.1] using the Liouville property for L, the regularity of the corrector and the existence of a Lyapunov function (W (y) = y 2 1 + y 2 2 ).…”

A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

“…This property is the crucial tool used in [18] to prove (7) and the existence and uniqueness of solutions for (9). Let us also mention the works of Bardi-Cesaroni-Ghilli [2] and Ghilli [20] for local equations, where (7) are obtained for constant nondegenerate diffusions and bounded solutions but for equations with possibly quadratic coercive Hamiltonians.…”

“…Then, there are three ingredients that we use to derive estimate (20): (i) the first one consists in using the supersolution φ µ to control the growth of different terms near infinity; (ii) for |x − y| small we use the ellipticity of the diffusion and we control the bad terms via Ishii-Lions' method (see [22,7] and Section 4); (iii) for |x − y| big we control those terms by the Ornstein-Uhlbenbeck drift (3).…”