Numerical methods for Hamilton Jacobi Bellman equations in computational finance

计算金融中 Hamilton Jacobi Bellman 方程的数值方法

• 批准号: RGPIN-2017-03760
• 负责人: Forsyth, Peter
• 金额: $ 3.13万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710520
• 关键词: Numerical; methods; Hamilton; Jacobi; Bellman

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.

金融中的大多数问题都归结为做出某种最优选择。 例如，考虑一个人为退休储蓄。 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，其余部分投资于债券。 这个比例应该如何变化，取决于总的积累财富和退休前的时间？ 这里的问题是股票指数的随机行为。 这是最优随机控制中的一个典型问题。 类似地，任何拥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，假设一个投资者提交了一个以指定限价购买1000股股票的订单。 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（通常在线客户可以获得），总的买入订单将包括一些小订单（100-200股），所有订单都以略有不同的价格执行。 这里的想法是将大订单分解成小单位，以避免过度的“价格影响”。当然，将大订单分解成一些小的单位，会使卖方在销售的过程中价格下降。在这种情况下，我们有一个最优交易执行算法的例子，它是基于最优随机控制的解决方案。 本文主要研究在金融领域中求解Hamilton-Jacobi-Bellman（HJB）偏积分微分方程（PIDE）的数值算法。 最优随机控制问题通常可以用求解这样的HJB方程来表示。 我们专注于数值算法，因为实际问题通常有限制，这是难以处理的，如果寻求封闭形式的解决方案。 例如，任何投资于真实的退休账户的人都将受到他们可以使用的杠杆金额的限制。 然而，施加杠杆约束似乎是非常困难的，如果一个封闭的形式解决方案的HJB方程是期望的，而施加这些种类的约束在数值上下文中是相当简单的。 非线性HJB方程通常具有多个非光滑（即不可微）解。 这就引出了许多问题，第一个问题是，解一个不可微的微分方程意味着什么？ 此外，在许多可能的解决方案中，哪一个是我们想要的金融应用程序？ 在这种情况下，我们必须在“粘性意义”上定义一个解，这是对微分方程解的适当概括。 这是一个不平凡的问题，发展的数值算法，保证收敛到粘性的解决方案。 这个建议是专门针对设计算法，可证明收敛到HJB PIDE的粘度解决方案。