Dissertation Defense: Chanuka Dissanayake

The title to Mathematics student Chanuka Dissanayake's dissertation is "Principal-Agent Problems: From Merton’s Portfolio Optimization to Volatility Control and Numerical Solutions". This thesis studies the continuous-time principal-agent problem, where an investor (principal) contracts a portfolio manager (agent) to allocate wealth in a Merton-type geometric Brownian motion model, and the principal designs the compensation contract. In 2025, Chen, Qian, and Qiao introduced a partial differential equation (PDE) approach to the drift-control problem, where the agent’s allocation enters only the drift of the output process. The central contribution here is extending the PDE approach to volatility control, where the agent’s allocation enters both the drift and the volatility linearly. The main challenge is that the agent’s optimal effort depends on the second spatial derivative of his value function v. We resolve this by reparameterizing in terms of the processes vx and vxx. Further we prove that wy, the derivative of the principal’s value function with respect to the agent’s value function, is conserved along sample paths. Moreover, we show that the Hamilton-Jacobi-Bellman equations satisfied by the principal’s and the agent’s value functions reduce to transport-type PDEs, and we obtain explicit solutions in some simplified cases as benchmarks. This transport structure is available only in the volatility-control case, and it is what we exploit numerically. We develop a combined scheme that couples the PDE methods with stochastic differential equation–based Monte Carlo simulations, recovering the known contract structure in solved examples. Advisor: Dr. John Chadam