Stochastics and Dynamics

In this paper, we consider a stochastic 2D liquid crystal system with a small multiplicative noise of Gaussian type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces. We derive a large deviation principle for the model. The proof relies on the weak convergence method that was introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat. 47(3) (2011) 725–747] and based on a variational representation on infinite-dimensional Brownian motion.

In this paper, using the mean-field game theory, we study a problem of equilibrium price formation among many investors with exponential utility in the presence of liabilities unspanned by the security prices. The investors are heterogeneous in their initial wealth, risk-averseness parameter, as well as stochastic liability at the terminal time. We characterize the equilibrium risk-premium process of the risky stocks in terms of the solution to a novel mean-field backward stochastic differential equation (BSDE), whose driver has quadratic growth both in the stochastic integrands and in their conditional expectations. We prove the existence of a solution to the mean-field BSDE under several conditions and show that the resultant risk-premium process actually clears the market in the large population limit.

We study large deviations in the context of stochastic gradient descent for one-hidden-layer neural networks with quadratic loss. We derive a quenched large deviation principle, where we condition on an initial weight measure, and an annealed large deviation principle for the empirical weight evolution during training when letting the number of neurons and the number of training iterations simultaneously tend to infinity. The weight evolution is treated as an interacting dynamic particle system. The distinctive aspect compared to prior work on interacting particle systems lies in the discrete particle updates, simultaneously with a growing number of particles.

In this paper, we provide necessary and sufficient conditions for invariance of finite dimensional submanifolds for rough differential equations (RDEs) with values in a Banach space. Furthermore, we apply our findings to the particular situation of random RDEs driven by $Q$-Wiener processes and random RDEs driven by $Q$-fractional Brownian motion.

In this paper, we show that stochastic differential equations (SDEs) with Markovian switching driven by Lévy noises generate random dynamical systems (RDSs), and study the existence and uniqueness of stationary solutions. Besides, we obtain sufficient conditions for the existence of invariant measures.