Abstract. Hilbert space-valued jump-diffusion models are employed for various markets and derivatives. Examples include swaptions, which depend on continuous forward curves, and basket options on stocks. Usually, no analytical pricing formulas are available for such products. Numerical methods, on the other hand, suffer from exponentially increasing computational effort with increasing dimension of the problem, the "curse of dimension." In this paper, we present an efficient approach using partial integro-differential equations. The key to this method is a dimension reduction technique based on a Karhunen-Loève expansion, which is also known as proper orthogonal decomposition. Using the eigenvectors of a covariance operator, the differential equation is projected to a low-dimensional problem. Convergence results for the projection are given, and the numerical aspects of the implementation are discussed. An approximate solution is computed using a sparse grid combination technique and discontinuous Galerkin discretization. The main goal of this article is to combine the different analytical and numerical techniques needed, presenting a computationally feasible method for pricing European options. Numerical experiments show the effectiveness of the algorithm.

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We present a numerical method for pricing Bermudan options depending on a large number of underlyings. The asset prices are modeled with exponential time-inhomogeneous jump-diffusion processes. We improve the least-squares Monte Carlo method proposed by Longstaff and Schwartz introducing an efficient variance reduction scheme. A control variable is obtained from a low-dimensional approximation of the multivariate Bermudan option. To this end, we adapt a model reduction method called proper orthogonal decomposition (POD), which is closely related to principal component analysis, to the case of Bermudan options. Our goal is to make use of the correlation structure of the assets in an optimal way. We compute the expectation of the control variable by either solving a low-dimensional partial integro-differential equation or by applying Fourier methods. The POD approximation can also be used as a candidate for the minimizing martingale in the dual pricing approach suggested by Rogers. We evaluate both approaches in numerical experiments.

The basic contracts traded on energy exchanges involve fixed-rate payments for the delivery of electricity over a certain period of time. It has been shown that options on these electricity swaps can be priced efficiently using a Hilbert space-valued time-inhomogeneous jump-diffusion model for the forward curve. We consider the mean-variance hedging problem for European options under this model. The computation of hedging strategies leads to quadratic optimization problems whose parameters depend on the solution of an infinite-dimensional partial integro-differential equation. The main objective of this article is to find an efficient numerical algorithm for this task. Using proper orthogonal decomposition (a dimension reduction method), approximately optimal strategies are computed. We prove convergence of the corresponding hedging error to the minimal achievable error in the electricity market. Numerical experiments are performed to analyze the resulting hedging strategies.

Variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena, most notably in weather prediction, and are usually solved via an ill-posed optimal control problem for the initial state at the time of the first available measurements. An alternative "forward" approach focuses on computation of the final state after this interval-which is just as suitable for prediction purposes-and is well-posed without additional regularization. Specifically, it is possible to compute projections of the unknown final state on all elements of an orthonormal basis, which theoretically allows for the complete reconstruction of the final state. In this paper, an efficient numerical method for linear evolution equations of diffusive type is presented, and convergence of the numerical approximation based on a discontinuous Galerkin discretization is proved. The key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition. Numerical examples for a scalar convectiondiffusion equation in two and three dimensions show the effectiveness of the method.