Adam Andersson

Adam Andersson, Stig Larsson

We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.

Adam Andersson, Mihály Kovács, Stig Larsson

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.

Adam Andersson, Raphael Kruse

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for the two-step BDF-Maruyama scheme and for the backward Euler-Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $\tfrac32$-volatility model from finance, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.

Kasper Bågmark, Adam Andersson, Stig Larsson et al.

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in numerical examples. In a prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, followed by an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem. The convergence analysis is complemented by a nonlinear $10$-dimensional numerical example demonstrating the robustness of the method.

Kasper Bågmark, Adam Andersson, Stig Larsson

A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using the well-known deep BSDE method and neural networks. The method is trained offline, which means that it can be applied online with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic Hörmander condition. The theoretical convergence rate is confirmed in two numerical examples.

Mika Persson, Jonas Lidman, Jacob Ljungberg et al.

This work presents a conceptual study on the application of Multi-Agent Reinforcement Learning (MARL) for decentralized control of unmanned aerial vehicles to relay a critical data package to a known position. For this purpose, a family of deterministic games is introduced, designed for scaling studies for MARL. A robust baseline policy is proposed, which is based on restricting agent motion envelopes and applying Dijkstra's algorithm. Experimental results show that two off-the-shelf MARL algorithms perform competitively with the baseline for a small number of agents, but scalability issues arise as the number of agents increase.

Kristoffer Andersson, Adam Andersson, Cornelis W. Oosterlee

In this paper, 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 a numerical example 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 three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Adam Andersson, Raphael Kruse, Stig Larsson

We introduce a new family of refined Sobolev-Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.