Numerical methods for Hamilton Jacobi Bellman equations in computational finance

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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.