Multiple G-It\^{o} integral in the G-expectation space

In this paper, motivated by mathematic finance we introduce the multiple G-It\^{o} integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-It\^{o} integrals which is a natural extension of the classical result obtained by It\^{o} in 1951.Comment: 9 page

An overview of Viscosity Solutions of Path-Dependent PDEs

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12]

Stochastic control for a class of random evolution models

Hongler MO, Soner HM, Streit L. Stochastic control for a class of random evolution models. APPLIED MATHEMATICS AND OPTIMIZATION. 2004;49(2):113-121.We construct the explicit connection existing between a solvable model of the discrete velocities non-linear Boltzmann equation and the Hamilton-Bellman-Jacobi equation associated with a simple optimal control of a piecewise deterministic process. This study extends the known relation that exists between the Burgers equation and a simple controlled diffusion problem. In both cases the resulting partial differential equations can be linearized via a logarithmic transformation and hence offer the possibility to solve physically relevant non-linear field models in full generality

Leveraged exchange-traded funds with market closure and frictions

202208 bckwNot applicableOthersNational Natural Science Foundation of China; Hong Kong Polytechnic University start-up grant; Singapore MoE research grant; CUHK grant

Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints

International audienceThis paper investigates sufficient conditions for Lipschitz regularity of the value function for an infinite horizon optimal control problem subject to state constraints. We focus on problems with a cost functional that includes a discount rate factor and allow time dependent dynamics and Lagrangian. Furthermore, our state constraints may be unbounded and with nonsmooth boundary. The key technical result used in our proof is an estimate on the distance of a given trajectory from the set of all its viable (feasible) trajectories (provided the discount rate is sufficiently large). These distance estimates are derived under a uniform inward pointing condition on the state constraint and imply, in particular, that feasible trajectories dependon initial states in a Lipschitz way with an exponentially increasing in time Lipschitz constant. As a corollary, we show that the value function of the original problem coincides with the value function of the relaxed infinite horizon problem