Università di Milano – Bicocca Quaderni di Matematica Stochastic equations with delay : optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations
Università di Milano – Bicocca Quaderni di Matematica Stochastic equations with delay : optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations
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发表时间：
2008
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通讯作者：
M. Fuhrman;F. Masiero;G. Tessitore;Quaderno
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作者：
M. Fuhrman;F. Masiero;G. Tessitore;Quaderno
考虑一类由布朗运动驱动的具有时滞的伊藤随机微分方程，其解通过适当的变换定义了一个马尔可夫过程X，其值在连续函数空间C中，生成元为L.然后，我们考虑一个依赖于X的倒向随机微分方程，未知的过程（Y，Z），我们研究所得到的系统的性质，特别是我们确定的过程Z作为X的确定性功能。接下来，我们证明了向前向后系统提供了一个合适的解决方案，一类抛物型偏微分方程的空间C上的L驱动，我们应用这个结果证明了一个表征的公平价格和套期保值策略的金融市场的记忆效应。我们还包括应用程序的最优随机控制微分方程的延迟：特别是我们的最优控制的特征反馈法律的过程X。
We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions C, with generator L. We then consider a backward stochastic differential equation depending on X, with unknown processes (Y, Z), and we study properties of the resulting system, in particular we identify the process Z as a deterministic functional of X. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space C driven by L, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms of the process X.