论文标题
随机的时间范围
The American Put Option with A Random Time Horizon
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
This paper is intended to provide a unique valuation formula for the American put option in a random time horizon; more precisely, such option restricts any stopping rules to be made before the last exit time of the price of the underlying assets at any fixed level. By exploiting the theory of enlargement of filtrations associated with random times, this pricing problem can be transformed into an equivalent optimal stopping problem with a semi-continuous, time-dependent gain function whose partial derivative is singular at certain point. The difficulties in establishing the monotonicity of the optimal stopping boundary and the regularity of the value function lie essentially in these somewhat unpleasant features of the gain function. However, it turns out that a successful analysis of the continuation and stopping sets with proper assumption can help us overcome these difficulties and obtain the important properties that possessed by the free-boundary, which eventually leads us to the desired free-boundary problem. After this, the nonlinear integral equations that characterise the free-boundary and the value function can be derived. The solution to these equations are examined in details.
