Stochastical control problems with limitations on the set of allowable controls.

摘要

The optimum dividend in the Radner-Shepp model for a firm was seen to involve the mathematically difficult concept of payouts in local time . In chapter 1, we make the model more realistic by requiring the payouts to occur only at certain designated times, say periodically or the random times which are the points of an independent Poisson process. We obtain the full and rigorous solution, which is not more complicated, and show that the more discrete solution converges to the continuous solution when the payout points become dense.; In Chapter 2, consider a (geometric) Brownian motion X_t (ω) with drift and its running maximum S _t = max_{0 ≤u ≤ t} X_u. Suppose there is a source that sends signals at random times τ₁ τ ₂ ···. Our objective is to choose an optimal stopping time τ* to maximize E_x,s[ g(S_τ*)] = max_τ∈{lcub}τ _i{rcub} E[g(S_τi )|X₀ = x, S₀ = s]. In this chapter we derive explicit solutions for a discounted linear payoff function g(·), assuming that the signal arrival is a Poisson process with frequency λ.; In Chapter 3, this chapter investigates a mixed regular-singular stochastic control problem where the drift of the dynamics is quadratic in the regular control variable. Moreover, the regular control variable is constrained. The value function of the problem is derived in closed form via solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, and optimal controls are obtained explicitly.; In Chapter 4, A. N. Shiryaev and M. Jeanblanc-Picque [20] (their case B), considered a stochastic dividend problem, in the spirit of [24], but where transaction cost is included. They correctly settled their problem in all cases except when the transaction cost is very high. In this chapter, we correct this case. We also obtain the correct asymptotic behavior when the transaction cost is near zero.