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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688680
• 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归结为做出某种最佳选择。 例如，考虑一个储蓄退休的人。 基本的投资选择涉及确定投资股票指数的投资组合的一部分，其余的投资组合投资于债券。 这个比率应该如何改变，具体取决于总的累积财富和退休时间的时间？ 这里的问题是权益指数的随机（随机）行为。 这是最佳随机控制中的一个典型问题。******同样，拥有在线经纪人的任何人都可能遇到了使用最佳随机控制算法的使用。 例如，假设投资者提交命令以指定的限额价格购买1000股。 几分钟后，投资者可能会收到订单已填写的通知。 但是，如果检查了实际的贸易历史记录（通常可在线客户使用），则总买入订单将由许多小订单（100-200股）组成，所有股票都以略有不同的价格执行。 这里的想法是将大订单分为较小的单位，以避免过度的“价格影响”。当然，将大订单分解为许多较小的单位，可以使卖方在销售期间的价格下降。在这种情况下，我们有一个最佳贸易执行算法的示例，该算法基于最佳随机控制的解决方案。******该提案与为汉密尔顿 - 雅各布·贝尔曼（HJB）局部局部数量差异方程（PIDE）在金融应用中开发用于解决方案的数值算法有关。 最佳随机控制问题通常可以通过解决此类HJB方程来提出。 我们专注于数值算法，因为实际问题通常具有限制因素，如果寻求封闭式解决方案，则难以处理。 例如，任何投资实际退休帐户的人都将受到他们可以使用的杠杆率的限制。 但是，如果需要对HJB方程的封闭形式解决方案进行施加杠杆限制，同时在数值上下文中强加此类约束是相当简单的。********非线性HJB方程通常具有多个非平滑态度（即非不同的）解决方案。 这打开了许多问题，第一个问题是解决解决方案无法区分的微分方程意味着什么？ 此外，在许多可能的解决方案中，我们想要哪种解决方案？ 在这种情况下，我们必须在“粘度意义”中定义解决方案，这是对差分方程的解决方案的合适概括。 开发数值算法是一个非平凡的问题，这些算法可以保证会收敛到粘度解决方案。 该提案专门针对设计算法，这些算法可被证明与HJB Pides的粘度解决方案。*****