A simulation based method for the numerical solution of PDEs with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behavior.2010 Mathematics Subject Classification. 35R60, 47B80, 60H35, 65C20, 65N12, 65N22, 65J10,65C05 . Key words and phrases. partial differential equations with random coefficients, random PDE, uncertainty quantification, Feynman-Kac, stochastic differential equations, stochastic simulation, stochastic regression, Monte-Carlo, Euler-Maruyama.We would like to thank the anonymous referees for their helpful comments. We are also grateful to Anders Szepessy for clarifying discussions.

This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a numerically implementable method to solve this problem in a general setting, allowing for general time-consistent ambiguity-averse preferences and general payoff processes driven by jump diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We also provide asymptotically optimal exercise rules. We analyze the limiting behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.

We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier-based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modelling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed.Reference to this paper should be made as follows: Ladkau, M., Schoenmakers, J. and Zhang, J. (2013) 'Libor model with expiry-wise stochastic volatility and displacement', Int.

This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for solving optimal stopping problems. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm. In this respect our new approach uses the multilevel idea in the context of the nested simulations, where each level corresponds to a specific number of inner simulations. A thorough analysis of the convergence rates in the multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to the standard Monte Carlo based policy iteration. The performance of the multilevel method is illustrated in the case of pricing a multidimensional American derivative.

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