We consider the single-leg airline revenue management problem in continuous time with Poisson arrivals. Earlier work on this problem generally uses the Hamilton-Jacobi-Bellman equation to find an optimal policy whenever the value function is differentiable and is a solution to this equation. In this paper, we employ a different probabilistic approach, which do not rely on the smoothness of the ...

A canonical problem in real option pricing, as described in the classic text of Dixit and Pindyck [2], is to determine the optimal time to invest at a fixed cost, to receive in return a stochastic cashflow. In this paper we are interested in this problem in an incomplete market where the cashflow is not spanned by the traded assets. We follow the formulation in Miao and Wang [21]; our contribut...

This paper treats the problem of consumption and portfolio choice in continuous time, with stochastic income that cannot be replicated by trading the available securities. The optimal controls and value functions are characterized in terms of the viscosity solution of the associated Hamilton-JacobiBellman equation, which is shown to exist and is characterized. The optimal policy is then given f...

We develop a computationally efficient learning-based forward–backward stochastic differential equations (FBSDE) controller for both continuous and hybrid dynamical (HD) systems subject to noise state constraints. Solutions optimal control (SOC) problems satisfy the Hamilton–Jacobi–Bellman (HJB) equation. Using current FBSDE-based solutions, can be obtained from HJB using deep neural networks (...

In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze via value function. Due to non-smoothness $L^0$ cost functional, in general, function is not differentiable domain. Then, characterize as viscosity solution associated Hamilton-Jacobi-Bellman (HJB) equation. Based on result, derive necessary suffici...

We obtain error bounds for monotone approximation schemes of Hamilton-Jacobi-Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton-Jacobi-Bellman equation.

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on bounded domain $${\mathcal {O}}\subset {\mathbb {R}}^N$$ ( $$N=1,2,3$$ ) with oblique derivatives boundary conditions. These appear naturally the study optimal control diffusion processes reflection at domain. The proposed scheme is shown to s...

This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by difference between nonnegative rate and fraction of historical maximum. running maximum process chosen as an auxiliary state process, hence value function depends on two variables. Hamilton–Jacobi–Bellman (HJB) equation can be heuristicall...

We consider the numerical solution of discretised Hamilton-Jacobi-Bellman (HJB) equations with applications in finance. For the discrete linear complementarity problem arising in American option pricing, we study a policy iteration method. We show, analytically and numerically, that, in standard situations, the computational cost of this approach is comparable to that of European option pricing...

A bstract . We study the ergodic control problem of switching diffusions representing a typical hybrid system that arises in numerous applications such as fault-tolerant control systems, flexible manufacturing systems, etc. Under fairly general conditions, we establish the existence of a stable, nonrandomized Markov policy which almost surely minimizes the pathwise long-run average cost. We the...