Stability and instability of a planar random dynamical system
We study a planar stochastic differential equation with additive noise for which the rotational speed is of the form ρ(R) where R is the radial part.
We investigate how phenomena like strong or weak synchronization, existence of a pullback or a point attractor and strong completeness of the associated random dynamical system depend on the function ρ. This is joint work (in progress) with Maximilian Engel and Dennis Chemnitz (FU Berlin).
Pathwise convergence of the Euler scheme for rough and stochastic differential equations
First and higher order Euler schemes play a central role in the numerical ap- proximations of stochastic differential equations. While the pathwise convergence of higher order Euler schemes can be adequately explained by rough path theory, the first order Euler scheme seems to be outside its scope, at least at first glance.
In this talk, we show the convergence of the first order Euler scheme for differen- tial equations driven by càdlàg rough paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with mesh size going to zero. This property is verified for almost all sample paths of various stochastic processes and time discretizations. Consequently, we obtain the pathwise conver- gence of the first order Euler scheme for rough stochastic differential equations driven by these stochastic processes.
The talk is based on joint work with A. L. Allan, A. P. Kwossek, and C. Liu.
Junior Researchers in Stochastic Optimal Control
The event “Workshop Junior Researchers in Stochastic Optimal Control”, is taking place on August 31 and September 01 in Berlin.
This workshop is funded by the IRTG and organized by IRTG-members from Berlin and Oxford. We are pleased to announce our impressive lineup of keynote speakers
- Roxana Dumitrescu (King’s College London),
- Johannes Muhle-Karbe (Imperial College London),
- David Siska (University of Edinburgh),
- Peter Tankov (ENSAE Paris),
- Gianmario Tessitore (University Milano-Bicocca).
Solving Discrete-Time Graphon Mean Field Games
Mean-field games (MFGs) facilitate otherwise intractable learning in game-theoretical equi- librium problems with many agents. The general approach is to analyze agents via their distribution, which allows to abstract multi-agent stochastic dynamical systems into a single representative agent and the mass of all other agents. We begin by focusing on discrete-time models. We show that fixed point iteration is insufficient for solving MFGs in general. We then present some algorithms based on entropy regularization, dynamic programming and reinforcement learning. In the second half of the talk, we in- corporate graph structure into the model via graphon limits. A numerical reduction of graphon MFGs to standard MFGs allows application of algorithms for general MFGs. Accordingly, we demonstrate intuitive numerical results for exemplary investment and epidemics control problems. Lastly, we briefly touch upon some extensions to other types of graphs. Overall, we obtain a scalable framework for solving large-scale dynamic game-theoretic problems.
A Principal-Agent Framework for Optimal Incentives in Renewable Investments
We investigate the optimal regulation of energy production reflecting the long-term goals of the Paris climate agreement.
We analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of a competitive market
is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal’s value function.
This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario.
In addition, we expand the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. We follow the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. We provide closed-form expression of the second-best contracts. In particular, we show that they are in rebate form involving time-dependent prices of each state variable.
(In collaboration with René Aïd and Nizar Touzi.)
High order splitting methods for stochastic differential equations
In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain “iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology. (Joint work with James Foster and Calum Strange)