Developments in Computational Finance and Stochastic Numerics - Honoring John Schoenmakers on the occasion of his retirement

Explicit and implicit Euler schemes are formulated for Caputo fractional differential equations (for short Caputo FDEs) of order α ∈ (0,1) whose vector fields satisfy a standard Lipschitz continuity condition in the state variable and a Hölder continuity condition in the time variable. The convergence rates of these schemes are given investigated. Stochastic counterparts are also considered. The corresponding equations do not satisfy the Itô formula.

Particle models are often used to simulate pollution transport. More than 15 years ago, John Schoenmakers, Grigori Milstein, Arnold Heemink and Daria Spyvakovskaya, a PhD student at Delft University of Technology, used the concept of reverse time diffusion to obtain an efficient forward-reverse estimator for solving specific pollution problems (Advances in Water Research, 2005). More than 10 years later, the methodology, surprisingly, turned out to be very useful for ocean dynamics problems too: Tracing the ventilation pathways of oceans (Journal of Physical Oceanography, 2017). In the presentation, the important contributions of John Schoenmakers to the methodology will be addressed, and a number of practical applications in pollution transport and ocean dynamics will be used to illustrate the methodology.

Fifteen years ago, John, Denis Belomestny, and I published a non-nested Monte Carlo algorithm for computing dual upper bounds for Bermudan option prices. The key idea of this method is to come up with a regression estimator for the Doob martingale part of the approximative Snell envelope, which is based on the martingale representation theorem and, thus, preserves the martingale property. In this talk, we discuss an algorithm which combines the advantages of regression-now algorithms (generic applicability) and regression-later algorithms (vanishing variance) for approximating the integrand in the martingale representation theorem via simulation-based least-squares regression. More generally, the algorithm can be applied to a class of regression problems with Malliavin weights. Our error analysis shows that the convergence in the number of Monte Carlo samples can take place at an arbitrarily fast polynomial rate, if the problem is sufficiently smooth. The talk is based on joint work with Nikolaus Schweizer (Tilburg).

We propose a statistical model based on Hawkes processes in which large financial losses can arise in close succession serially as well as cross-sectionally. We derive in closed form saddlepoint approximations to the tails of profit and loss distributions, both marginal and joint, and use them to construct explicit risk measure formulae that account for the fact that a given financial institution's losses make it more likely that that institution will experience further losses, and that other financial institutions will experience losses as well. These closed-form risk measures can be used for comparative statics, parameter calibration, and setting capital requirements and potential systemic risk charges. (Based on joint work with Yacine Aït-Sahalia.)

Reinforcement learning (RL) algorithms are designed to maximize the cumulative reward according to a carefully chosen reward function. However, creating such a reward function often requires task-specific prior knowledge, which may not be available in a quantifiable form. To address these challenges, preference-based reinforcement learning algorithms have been developed. These algorithms can learn directly from an experts preferences rather than relying on a manually designed numerical reward function. In this presentation, I will provide an overview of preference-based reinforcement learning and demonstrate its core principles using examples. Specifically, I will discuss the types of human feedback that can be assumed and how preferences can be incorporated into the optimization problem through penalization.

We consider high-dimensional asset price models that are reduced in their dimension in order to lower the computational complexity or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction to obtain a reduced system and use concepts known for deterministic control systems. We sketch different schemes leading to a good pathwise approximation of the underlying. The error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments.

We study statistical properties of the optimal value of the Sample Average Approximation. The focus is on the tail function of the absolute error induced by the Sample Average Approximation, deriving upper estimates of its outcomes dependent on the sample size. The estimates allow to conclude immediately convergence rates for the optimal value of the Sample Average Approximation. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity in the parameter is not imposed in advance as often in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the sample average approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of mean upper semideviations and divergence risk measures.

In this presentation, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by numerical examples that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works.Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for different problems, one being the one that failed for a direct extension of the deep BSDE Method.