Weak Solutions and Optimal Control for Multivalued Stochastic Differential Equations
Weak Solutions and Optimal Control for Multivalued Stochastic Differential Equations
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DOI：
10.1007/s00030-008-7037-9
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发表时间：
2008-12
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通讯作者：
Adrian Zălinescu
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文献类型：
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作者：
Adrian Zălinescu
在本文中，我们首先证明了$$dX_{t} +A(X_{t}) dt \ni b (t, X) dt + \sigma (t, X) dB_{t}, t \ni [0, T]$$形式的有限维多值随机微分方程的弱解的存在性，其中A是最大单调算子，系数带σ是状态变量的连续泛函。使用的主要工具是鞅问题方法。其次，我们关注控制问题，其中系统由类似的方程驱动，并且控制策略在紧凑的空间中获取其值。使用鞅问题公式，我们证明了最优松弛控制的存在。在系数凸性的一些补充假设下，我们证明了初始问题的最优控制的存在。
In this paper we first prove the existence of a weak solution to a finite dimensional multivalued stochastic differential equation of the form $$dX_{t} +A(X_{t}) dt \ni b (t, X) dt + \sigma (t, X) dB_{t}, t \ni [0, T]$$, whereAis a maximal monotone operator, and the coefficientsband σ are continuous functionals of the state variable. The main tool used is the martingale problem approach.Secondly we are concerned with a control problem where the system is driven by a similar equation and the control policy takes its values in a compact space. Using the martingale problem formulation, we show the existence of an optimal relaxed control. Under some supplementary hypotheses of convexity on the coefficients, we prove the existence of an optimal control for the initial problem.