Martingale problems for controlled processes
Martingale problems for controlled processes
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受控过程的鞅问题
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发表时间：
1987
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通讯作者：
T. Kurtz
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作者：
T. Kurtz
鞅问题为刻画马尔可夫过程提供了一种强有力的方法。随机控制问题也可以自然地表示为鞅问题(见Fleming(1983))，我们的目标是利用这个公式给出随机控制问题在有限时间域和折扣情况下最优解的一般存在定理(第1，3节)，构造Nisio半群(第2节)，并根据Nisio半群的生成元给出最优解可以被控制是分段恒定的解逼近的条件(第2节，第3节)。最后一个结果的证明包括证明适当定义的折扣控制问题的值函数在Nisio半群的生成元的域中。
Martingale problems provide a powerful method for characterizing Markov processes. Stochastic control problems can also be formulated naturally as martingale problems (see for example Fleming (1983)), and our goal here is to exploit this formulation to give a general existence theorem for optimal solutions of a stochastic control problem in the finite time horizon and discounted cases (Sections 1,3), to construct the Nisio semigroup (Section 2), and to give conditions in terms of the generator of the Nisio semigroup under which the optimal solution can be approximated by solutions in which the control is piecewise constant (Sections 2,3). The proof of this last result involves showing that the value function of an appropriately defined discounted control problem is in the domain of the generator of the Nisio semigroup.