Pathwise Stochastic Control Problems and Stochastic HJB Equations
Pathwise Stochastic Control Problems and Stochastic HJB Equations
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DOI：
10.1137/s036301290444335x
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发表时间：
2007
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影响因子：
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通讯作者：
R. Buckdahn;Jin Ma
中科院分区：
文献类型：
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作者：
R. Buckdahn;Jin Ma
本文研究了一类允许最优性依赖于外生噪声（或信息）路径的路径随机控制问题。这种现象可以通过考虑一个特定的投资者来说明，他想利用某些额外的信息，但以完全合法的方式。我们证明这样的控制问题甚至可能没有“最小化序列”，但（Bellman）动态规划原理仍然成立。然后我们证明了相应的Hamilton-Jacobi-Bellman方程是一个随机偏微分方程，正如Lion和Souganidis [C.]所预测的那样。R.学术科学。巴黎Se´r。我的数学。， 327(1998)，第735-741页。我们的主要设备是在我们之前的工作[随机过程]中引入的doss - sussmann型转换。达成。， 93 (2001), pp. 181-204]和[随机过程]。达成。， 93 (2001), pp. 205-228]。借助这种变换，我们将路径控制问题简化为一个更标准的松弛控制问题，由此我们可以验证路径随机控制问题的值函数是该随机偏微分方程在随机过程意义上的唯一随机粘度解。达成。， 93 (2001), pp. 181-204]和[随机过程]。达成。， 93 (2001), pp. 205-228]。
In this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information). Such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner. We show that such a control problem may not even have a “minimizing sequence,” but nevertheless the (Bellman) dynamical programming principle still holds. We then show that the corresponding Hamilton-Jacobi-Bellman equation is a stochastic partial differential equation, as was predicted by Lion and Souganidis [C. R. Acad. Sci. Paris Se´r. I Math., 327 (1998), pp. 735-741]. Our main device is a Doss-Sussmann-type transformation introduced in our previous work [Stochastic Process. Appl., 93 (2001), pp. 181-204] and [Stochastic Process. Appl., 93 (2001), pp. 205-228]. With the help of such a transformation we reduce the pathwise control problem to a more standard relaxed control problem, from which we are able to verify that the value function of the pathwise stochastic control problem is the unique stochastic viscosity solution to this stochastic partial differential equation, in the sense of [Stochastic Process. Appl., 93 (2001), pp. 181-204] and [Stochastic Process. Appl., 93 (2001), pp. 205-228].