Hedging one's bets by high-dimensional stochastic control

The first half of this seminar will discuss the hedging problem faced by a large sports betting agent who has to risk-manage an unwanted position in a bet on the simultaneous outcome of multiple football matches, by trading in moderately liquid simple bets on individual results. The resulting mathematical framework is that of a coupled system of multi-dimensional HJB equations.

This leads to the wider question of the numerical approximation of such problems. Dynamic programming with PDEs, while very accurate in low dimensions, becomes practically intractable as the dimensionality increases. Monte Carlo methods, while robust for computing linear expectations in high dimensions, are not per se well suited to dynamic programming. This leaves high-dimensional stochastic control problems to be considered computationally infeasible in general.

In the second half of the seminar, we will outline ongoing work in this area by sparse grid techniques and asymptotic expansions, the former exploiting smoothness of the value function, the latter a fast decay in the importance of principal components. We hope to instigate a discussion of other possible approaches including e.g. BSDEs.