Probabilistic numerical methods for high-dimensional stochastic control and valuation problems on electricity markets

Abstract : This thesis deals with the numerical solution of general stochastic control problems, with notable applications for electricity markets. We first propose a structural model for the price of electricity, allowing for price spikes well above the marginal fuel price under strained market conditions. This model allows to price and partially hedge electricity derivatives, using fuel forwards as hedging instruments. Then, we propose an algorithm, which combines Monte-Carlo simulations with local basis regressions, to solve general optimal switching problems. A comprehensive rate of convergence of the method is provided. Moreover, we manage to make the algorithm parcimonious in memory (and hence suitable for high dimensional problems) by generalizing to this framework a memory reduction method that avoids the storage of the sample paths. We illustrate this on the problem of investments in new power plants (our structural power price model allowing the new plants to impact the price of electricity). Finally, we study more general stochastic control problems (the control can be continuous and impact the drift and volatility of the state process), the solutions of which belong to the class of fully nonlinear Hamilton-Jacobi-Bellman equations, and can be handled via constrained Backward Stochastic Differential Equations, for which we develop a backward algorithm based on control randomization and parametric optimizations. A rate of convergence between the constrained BSDE and its discrete version is provided, as well as an estimate of the optimal control. This algorithm is then applied to the problem of superreplication of options under uncertain volatilities (and correlations).
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Nicolas Langrené. Probabilistic numerical methods for high-dimensional stochastic control and valuation problems on electricity markets. Computational Finance [q-fin.CP]. Université Paris-Diderot - Paris VII, 2014. English. ⟨tel-00957948v2⟩

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