This paper studies continuous-time reinforcement learning for controlled jump-diffusion models by featuring the q-function (the continuous-time counterpart of Q-function) and the q-learning algorithms under the Tsallis entropy regularization. Contrary to the conventional Shannon entropy, the general form of Tsallis entropy renders the optimal policy not necessary a Gibbs measure, where some Lagrange multiplier and KKT multiplier naturally arise from certain constraints to ensure the learnt policy to be a probability distribution. As a consequence,the relationship between the optimal policy and the q-function also involves the Lagrange multiplier. In response, we establish the martingale characterization of the q-function under Tsallis entropy and devise two q-learning algorithms depending on whether the Lagrange multiplier can be derived explicitly or not. In the latter case, we need to consider different parameterizations of the q-function and the policy and update them alternatively. Finally, we examine two financial applications, namely an optimal portfolio liquidation problem and a non-LQ control problem. It is interesting to see therein that the optimal policies under the Tsallis entropy regularization can be characterized explicitly, which are distributions concentrate on some compact support. The satisfactory performance of our q-learning algorithm is illustrated in both examples.

The option pricing problem is formulated as a partial integro-differential equation, which is approximated via a new implicit-explicit minimizing movement time-stepping approach, involving approximation by deep, residual-type Artificial Neural Networks (ANNs) for each time step. The integral operator is discretized via two different approaches: a) a sparse-grid Gauss-Hermite approximation following localised coordinate axes arising from singular value decompositions, and b) an ANN-based high-dimensional special-purpose quadrature rule. Crucially, the proposed ANN is constructed to ensure the asymptotic behavior of the solution for large values of the underlyings and also leads to consistent outputs with respect to a priori known qualitative properties of the solution. The performance and robustness with respect to the dimension of the methods are assessed in a series of numerical experiments involving the Merton jump-diffusion model. A central problem in Mathematical Finance is the fast and accurate computation of arbitrage-free prices of financial derivatives, especially for advanced stochastic models and for multi-asset derivatives. A basket option is a contractual agreement between two parties, the buyer and the seller, to buy or sell a derivative whose value fluctuates over time based on the prices of a set of underlying assets (the "basket").

Trading in real market has lots of risks and limits such as transactions costs, discrete time hedging dates, illiquidity and non-tradable risk factors. These factors make results under the completeness assumption unreliable in most of time. This paper aims to solve hedging problems with three sources of incompleteness: volume risks, discrete tradable dates, and illiquidity constraints. Based on Merton's jump-diffusion model, many studies have been done over the simulation of extreme value movements. In [1],bilateral gamma distribution have excellent degree of fitting the German stock index(DAX).In this paper, a degraded bilateral gamma distribution: variance gamma distribution [2] is taken to simulate the jump size in the classic jump-diffusion model.