Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy

This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solutions to a system of second order parabolic non-linear PDEs. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We provide an example of the methodology in the context of a swing contract (energy contract with flexibility in purchasing energy power), this allows us to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.