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，考虑一个人为退休储蓄。 基本的投资选择包括决定投资组合的哪一部分投资于股票指数，而投资组合的其余部分投资于债券。 根据累计财富总额和退休时间，这个比率应该如何变化？ 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典型问题。 同样，任何拥有在线经纪商的人都可能遇到过最优随机控制算法的使用。 例如，假设投资者提交了以指定限价购买 1000 股的订单。 几分钟后，投资者可能会收到订单已成交的通知。 然而，如果检查实际交易历史记录（通常可供在线客户使用），则总买入订单将由许多小订单（100-200 股）组成，所有订单的执行价格略有不同。 这里的想法是将大订单分解为较小的单位，以避免过度的“价格影响”。当然，将大订单分解为多个较小的单位会让卖家在销售期间面临价格下降的风险。在本例中，我们有一个最佳交易执行算法的示例，该算法基于最佳随机控制的解决方案。 该提案涉及开发数值算法来解决金融应用中的 Hamilton-Jacobi-Bellman (HJB) 偏积分微分方程 (PIDE)。 最优随机控制问题通常可以通过求解此类 HJB 方程来表述。 我们专注于数值算法，因为实际问题通常具有限制，如果寻求封闭形式的解决方案，这些限制很难处理。 例如，任何投资真实退休账户的人都会受到他们可以使用的杠杆数额的限制。 然而，如果需要 HJB 方程的封闭形式解，则施加杠杆约束似乎非常困难，而在数值环境中施加此类约束则相当简单。 非线性 HJB 方程通常有多个非光滑（即不可微）解。 这就提出了许多问题，第一个问题是求解解不可微的微分方程意味着什么？ 此外，在众多可能的解决方案中，哪一种是我们想要用于金融应用的解决方案？ 在这种情况下，我们必须定义“粘度意义上”的解，这是微分方程解的含义的适当概括。 开发保证收敛到粘度解的数值算法是一个重要的问题。 该提案专门针对设计可证明收敛于 HJB PIDE 粘度解的算法。