A free boundary problem arising from a stochastic optimal control model under controllable risk

Optimal dividend payout problem under both diffusion risk and Poisson risk in finite horizon

“…So there exists a u ∈ W 2,1 p, loc (Q L T )∩C α, α 2 (Q L T ) and a subsequence of u ε converging to u uniformly in C(Q L T ) and weekly in W 2,1 p (Q L T \ B r ) for any r > 0. It is not hard to check that u is a solution of the problem (16). Moreover, Lemma 5.2 implies the uniqueness.…”

“…Solvability of the equivalent obstacle problem. In this section, we will prove the existence of a solution to the problem (16). As usual, we adopt the standard penalty approximation method, see [10,17,35].…”

“…When studying a dividend payout problem in a Markovian model, the dynamic programming principle is a powerful tool. It leads to the study of the so-called Hamilton-Jacobi-Bellman (HJB for short) equation, which turns out to be a variational inequality with gradient constraint, see [18,33] for infinite time models and [16,17] for finite time models.…”

A free boundary problem for a class of nonlinear nonautonomous size-structured population model

“…The theory of free boundaries has seen great progress in the last century [7,11,15]. Recent decades have witnessed a rapid widening of the subject area by incorporation of important free boundary topics coming from different areas: In Finance, free boundary problems appear to determine the optimal exercise value in Black-Scholes models [12,13]; In Mathematical Biology, they indicate the moving fronts of populations or tumors [8,9,10].…”

A fully nonlinear free boundary problem arising from optimal dividend and risk control model

“…An insurance company is a typical example of a financial corporation in the problem of optimal dividend distribution and risk control, [15,16,9] considered some kinds of optimal risk and dividend control problems under some conditions. For a finite time model, we solved the problems of dividend control model and risk control model in [7] and [8], respectively. In this paper, we deal with a stochastic model of risk control and dividend optimization, which is an evolutionary problem of one of the models in [15].…”