We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.
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.
Nonsmooth nonconvex optimization problems involving the p quasi-norm, p ∈ (0, 1], of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developped and necessary optimality conditions for the original prolem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction.keywords: nonsmooth nonconvex optimization and active-set method and monotone algorithm and optimal control problems and image reconstruction and fracture mechanics.
An inverse breaking line identification problem formulated as an optimal control problem with a suitable PDE constraint is studied.
The constraint is a boundary value problem describing the anti-plane equilibrium of an elastic body with a stress-free breaking line under the action of a traction force at the boundary.
The behavior of the displacement is observed on a subset of the boundary, and the optimal breaking line is identified by minimizing the {L^{2}}-distance between the displacement and the observation.
Then the optimal control problem is solved by shape optimization techniques via a Lagrangian approach.
Several numerical experiments are carried out to show its performance in diverse situations.
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