Numerical methods for Hamilton Jacobi Bellman equations in computational finance

金融领域的大多数问题都可以归结为做出某种最优选择。例如，考虑一个为退休储蓄的人。基本的投资选择包括决定投资组合中的哪一部分投资于股票指数，其余投资组合投资于债券。这一比率应该如何变化，取决于总积累的财富和退休前的时间？这里的问题是股票指数的随机(随机)行为。这是最优随机控制中的一个典型问题。类似地，任何有在线经纪人的人都可能遇到过最优随机控制算法的使用。例如，假设一位投资者提交了一份订单，要求以指定的限价购买1000股。几分钟后，投资者可能会收到订单已完成的通知。然而，如果检查实际的交易历史(通常是在线客户可以获得的)，总的买入订单将由一些小订单(100-200股)组成，所有订单都以略有不同的价格执行。这里的想法是将一个大订单分成几个小单元，以避免过度的“价格冲击”。当然，将大订单分成几个较小的单位，卖家可能会在整个销售过程中出现价格下跌。在这种情况下，我们有一个最优交易执行算法的例子，它基于最优随机控制的解。*这项建议涉及到在金融应用中发展求解哈密顿-雅各比-贝尔曼(HJB)偏积分微分方程组(PIDES)的数值算法。最优随机控制问题通常可以通过求解这样的HJB方程来表示。我们将重点放在数值算法上，因为实际问题通常有一些约束，如果寻求闭合形式的解，这些约束很难处理。例如，任何投资于真实退休账户的人都将受到他们可以使用的杠杆量的限制。然而，如果需要HJB方程的闭合形式解，施加杠杆约束似乎是非常困难的，而在数值环境中施加这些类型的约束是相当直接的。*非线性HJB方程通常有多个非光滑(即不可微)解。这就提出了一些问题，第一个问题是，在解不可微的情况下，解一个微分方程意味着什么？此外，在众多可能的解决方案中，我们希望为我们的金融应用程序提供哪种解决方案？在这种情况下，我们必须定义“粘性意义”的解，这是微分方程解的适当推广。开发保证收敛到粘性解的数值算法是一个不平凡的问题。该建议专门针对设计可证明收敛于HJB PIDS的粘性解的算法。

Most problems in finance boil down to making some sort of optimal choice. For example, consider a person saving for retirement. The basic investment choice involves deciding what fraction of the portfolio to invest in a stock index, with the remainder of the portfolio invested in bonds. How should this ratio change, depending on the total accumulated wealth and time until retirement? The problem here is the stochastic (random) behaviour of equity indices. This is a typical problem in optimal stochastic control.******Similarly, anyone who has an on-line broker has likely encountered the use of an optimal stochastic control algorithm. For example, suppose an investor submits an order to buy 1000 shares at a specified limit price. After a few minutes, the investor likely receives notification that the order was filled. However, if the actual trade history is examined (which is usually available to on-line clients), the total buy order will consist of a number of small orders (100-200 shares) all executed at slightly different prices. The idea here is to break up a large order into smaller units, to avoid excessive "price impact". Of course breaking up big orders into a number of smaller units opens up the seller to price drops over the length of the sale. In this case we have an example of an optimal trade execution algorithm, which is based on solution of an optimal stochastic control.******This proposal is concerned with developing numerical algorithms for solution of Hamilton-Jacobi-Bellman (HJB) Partial Integro Differential Equations (PIDEs) in financial applications. Optimal stochastic control problems can often be formulated in terms of solving such HJB equations. We focus on numerical algorithms, since practical problems usually have constraints which are difficult to handle if closed form solutions are sought. For example, anyone investing in a real retirement account will be constrained on the amount of leverage they can employ. However, imposing a leverage constraint seems to be very difficult if a closed form solution to the HJB equation is desired, while imposing these sorts of constraints in a numerical context is fairly straightforward.******Non-linear HJB equations typically have multiple non-smooth (i.e. non-differentiable) solutions. This opens up a number of questions, the first being what does it mean to solve a differential equation where the solution is not differentiable? In addition, of the many possible solutions, which is the one we want for our financial applications? In this case, we have to define a solution in the "viscosity sense", which is a suitable generalization of what is meant by a solution to a differential equation. It is a non-trivial issue to develop numerical algorithms which are guaranteed to converge to the viscosity solution. This proposal is directed specifically towards devising algorithms which are provably convergent to the viscosity solution of HJB PIDEs.*****