In this talk I will describe how deep learning methods are being applied to forecast stock returns from high frequency order book states. I will review the literature in this area and describe a working paper where we evaluate return forecasts for several deep learning models for a large subset of symbols traded on the Nasdaq exchange. We investigate whether transformation of the order book states is necessary and we relate the performance of deep learning models for a symbol to its microstructural properties. This is joint work with Petter Kolm (NYU), Jeremy Turiel (UCL) and Antonio Briola (UCL).

We study finite-time horizon continuous-time linear-convex reinforcement learning problems in an episodic setting. In these problems, an unknown linear jump-diffusion process is controlled subject to nonsmooth convex costs. We start with the pure diffusion case with quadratic costs, and propose a least-squares algorithm which achieves a logarithmic regret bound of order O((lnM)(lnlnM)), with M being the number of learning episodes; the proof relies on the robustness of the associated Riccati differential equation and sub-exponential properties of the least-squares estimators. We then extend the least-squares algorithm to linear-convex learning problems with jumps, and establish a regret of the order O((MlnM)1/2); the analysis leverages the Lipschitz stability of the associated forward-backward stochastic differential equation and concentration properties of sub-Weibull random variables.

This is joint work with Matteo Basei, Xin Guo and Anran Hu.

Theoretical solutions to stochastic optimal control problems are well understood in many scenarios, however their practicability suffers from the assumption of known dynamics of the underlying stochastic process. This raises the challenge of developing purely data-driven strategies, which we explore for ergodic singular control problems associated to continuous diffusions and Lévy processes. In case of diffusion processes, the primary challenge consists of solving an exploration/exploitation tradeoff based on a minimax optimal estimation procedure of the optimal reflection boundaries with data collected in the exploration periods. Even though for Lévy processes such exploration/exploitation problem does not occur due to spatial homogeneity of the process, in this scenario we face the statistical challenge of estimating a generator functional of a subordinator associated to the Lévy process which a) cannot be observed directly from the data and b) is non-ergodic in time. We solve this problem by considering a space/time transformation of the process in form of its overshoots such that we can work with a spatially ergodic process that allows the construction of an unbiased estimator of the generator functional determining the optimal reflection boundary. We compare the results of our statistical procedure with those from deep learning approaches.

In this talk, we study a class of reflected backward stochastic differential equations (BSDEs) of mean-field type, where the mean-field interaction in terms of the expected value $\E[Y]$ of the $Y$-component of the solution enters both the driver and the lower obstacle. Under mild Lipschitz and integrability conditions on the coefficients, we obtain the well-posedness  of such a class of equations.  Under further monotonicity conditions we show convergence of the  standard penalization scheme to the  solution of the equation. This class of models is motivated by applications in pricing life insurance contracts with surrender options.

Fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equations will be discussed for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and the value function of the optimal stochastic control problem is the unique viscosity solution to the associated stochastic HJB equation. Applications in mathematical finance and some recent developments will be reported as well.