When approximating the expectation of a functional of a stochastic process, in particular for option pricing purposes, the performance of numerical integration methods based on deterministic quadra- ture, quasi-Monte Carlo (QMC), or multilevel Monte Carlo (MLMC) techniques may critically depend on the regularity of the integrated. To overcome this issue, we introduce in [1,2,3] different smoothing tech- niques. In the first part of the talk, we will discuss our novel numerical smoothing approach [1,2] in which we combine root-finding methods with one-dimensional integration with respect to a single well-selected variable, focusing on cases where the discretization of the asset price dynamics is necessary. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). Our analysis in [1] demonstrates the advantages of combining numerical smoothing with AS- GQ and QMC methods over ASGQ and QMC methods without smoothing, and the Monte Carlo approach. Moreover, our analysis in [2] shows that our numerical smoothing improves the robustness (the kurtosis at deep levels becomes bounded) and complexity of the MLMC method. In particular, we recover the optimal MLMC complexities obtained for Lipschitz functionals.In the second part of the talk, we will discuss our efficient Fourier-based method in [3] for pricing European multi-asset options under L ́evy models. Given that the integrand in the frequency space often has higher regularity than in the physical space, we extend the one-dimensional Fourier valuation formula to the multivariate case and employ two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed heuristic optimization rule. These parameters ensure integrability and control the regularity class of the integrand. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the numerical quadrature in high dimensions. We demonstrate the advantages of adaptivity and our damping parameter rule on the numerical complexity of the quadrature methods. Moreover, we reveal that our approach achieves substantial computational gains compared to the Monte Carlo method for different dimensions and parameter constellations.

We are organizing a workshop on Many Player Games and Applications in Berlin from August 29-31. This workshop brings together leadings experts from mathematics, economics, operations research and engineering departments to discuss recent developments in the theory of many player games and their applications to finance and engineering. The event follows up on a series of previous events, held at the Center for Interdisciplinary research (ZiF). It is sponsored by the CRC TRR 190 (Berlin-Munich), the IRTG 2544 (Berlin-Oxford) and the SFB 1238 (Bielefeld). Confirmed speakers include

• Peter Cains (McGill),
• Diogo Gomes (KAUST),  
• Johannes Muhle-Karbe (Emperial),
• Mathias Blonski (Frankfurt)
• Julio Backhoff-Veraguas (Vienna)
• Martin Herdegen (Warwick)
• Sujoy Mukerji (Queen Mary U)
• Chao Zhou (Singapore)

and many others. More information is available on the conference webpage. 

Meeting held in Berlin, August 22 - August 26.
Scientific Committee: P Bank, P Friz, U Horst, S Paycha, N Perkowski, W Stannat
Organizing Institutions: FU, HU and TU Berlin, U Potsdam and WAS Berlin
Support: DFG FOR 2402, ERC GPSART GPSART 683164, Berlin-Oxford IRTG 2544
This meeting is devoted to recent progress on stochastic & rough analysis and its applications. It is also a
welcome opportunity to congratulate our colleaque Michael Scheutzow (FOR 2402) on his retirement.

In one of its dynamic formulations, the optimal transport problem asks to determine the stochastic process that interpolates between given initial and terminal marginals and is as close as possible to the constant-speed particle. Typically, the answer to this question is a stochastic process with constant-speed trajectories. We explore the analogue problem in the setting of martingales, and ask: what is the martingale that interpolates between given initial and terminal marginals and is as close as possible to the constant-volatility particle? The answer this time is a process called ’stretched Brownian motion’, a generalization of the well-known Bass martingale. After introducing this process and discussing some of its properties, I will present current work in progress (with Mathias Beiglbo ̈ck, Walter Schachermayer and Bertram Tschiderer) concerning the fine structure of stretched Brownian motions.

We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The considered optimization problems concern the minimization of a discounted cost functional over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. In a multidimensional setting, we prove that the optimal control acts only when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. Our approach is based on the study of a suitable monotonicity property of the derivative of the value function through its interpretation as the value of an optimal stopping game. Such a monotonicity allows to construct nearly optimal policies which reflect the underlying diffusion at the boundary of approximating waiting regions. The limit of this approximation scheme then provides the desired characterization. Our result applies to a relevant class of linear-quadratic models, among others. Furthermore, it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature, where this important aspect was only partially addressed. This talk is based on a joint work with Giorgio Ferrari.

In this talk, we present recent results on the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via other approaches used in the previous literature. We then present a fictious play algorithm to approximate the mean-field game population dynamics in the context of the linear programming approach. Finally, we give an application of the theoretical and numerical contributions introduced in the first part of the talk to an entry-exit game in electricity markets. The talk is based on several works, joint with R. A ̈ıd, G. Bouveret, M. Leutscher and P. Tankov.

We present the multilevel Picard approximation method for high-dimensional semilinear parabolic PDEs which in particular appear in the pricing of financial derivatives. A key idea of our method is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of the proposed method grows polynomially both in the dimension and in the reciprocal of the required accuracy. Moreover, we present further applications of the multilevel Picard approximation method and illustrate its efficiency by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe Von Wurstemberger.