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We consider a class of discrete time stochastic control problems motivated by some financial applications. We use a pathwise stochastic control approach to provide a dual formulation of the problem. This enables us to develop a numerical…
This paper considers a class of stochastic control problems with implicitly defined objective functions, which are the sources of time-inconsistency. We study the closed-loop equilibrium solutions in a general controlled diffusion…
The time-fractional Black-Scholes equation (TFBSE) is intended to price the options for which the underlying price fluctuates within a correlated fractal transmission system. Although the TFBSE is an influential approach for grasping the…
This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$\sigma$-algebra of that of the leader. Necessary and sufficient…
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of…
We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its…
We present a new algorithms to discretize a decoupled forward backward stochastic differential equations driven by pure jump L\'evy process (FBSDEL in short). The method is built in two steps. Firstly, we approximate the FBSDEL by a forward…
Stochastic control problems with delay are challenging due to the path-dependent feature of the system and thus its intrinsic high dimensions. In this paper, we propose and systematically study deep neural networks-based algorithms to solve…
We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate…
We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the…
In this small note we are concerned with the solution of Forward-Backward Stochastic Differential Equations (FBSDE) with drivers that grow quadratically in the control component (quadratic growth FBSDE or qgFBSDE). The main theorem is a…
This paper presents a method to approximately solve stochastic optimal control problems in which the cost function and the system dynamics are polynomial. For stochastic systems with polynomial dynamics, the moments of the state can be…
In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $\Delta$ between the decision and execution times. The dynamics of the controlled system between two impulses is an arbitrary adapted…
We consider a system of Forward Backward Stochastic Differential Equations (FBSDEs), with time delayed generator and driven by L\`evy-type noise. We establish a non linear Feynman Kac representation formula associating the solution given by…
We explore the use of deep learning hierarchical models for problems in financial prediction and classification. Financial prediction problems -- such as those presented in designing and pricing securities, constructing portfolios, and risk…
This paper is concerned with a Stackelberg stochastic differential game, where the systems are driven by stochastic differential equation (SDE for short), in which the control enters the randomly disturbed coefficients (drift and…
In this paper we obtain results for the existence and uniqueness of solutions to coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with jumps defined on a random environment. This environment corresponds to a…
Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary nonlinear drift, and nonlinear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based…
We extend the notion of forward performance criteria to settings with random endowment in incomplete markets. Building on these results, we introduce and develop the novel concept of \textit{forward optimized certainty equivalent (forward…
Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number (less than…