arXiv subjects

Georg Schlüchtermann

Publications and source records attributed to Georg Schlüchtermann.

3 recordsLinked to original sources

arXiv2026

Pontryagin Maximum Principle in Free Probability Theory

Motivated by the classical stochastic maximum principle, random matrices and free stochastic differential equations we, develop an analog maximum principle for control problems driven by non-commutative random variables, e.g. random matrices. We formulate an optimal control problem in the setting of free probability, consisting of the controlled forward equation, a free backward stochastic differential equation. For both, we give global existence theorems. Due to the non-commutative Itô-formula, the definition of the Hamiltonian differs to the commutative case. Our strategy is to stay as close as possible to the commutative case. Finally we formulate and proof the maximum principle in the context of free probability. Several examples show the application of the maximum principle, where explicit solutions can be found.

math.PR↗

arXiv2026

On Milstein-Type Methods for Free Stochastic Differential Equations

Previously, the authors derived an analog of the Euler-Maru\-yama method (fEMM) for free stochastic differential equations (fSDEs) and proved strong convergence of order $γ=0.5$ in $L_1(φ)$-norm under certain assumptions. In this paper, we study the development of numerical methods for fSDEs which show strong convergence of order $γ=1$ in $L_\infty(φ)$. As a side effect, strong convergence of order $γ=0.5$ of fEMM can be extended to $L_p(φ)$ for $p\in[1,\infty]$. Utilizing the framework of multiple operator integrals (MOI) we derive a stochastic Itô-Taylor expansion of the solution of the fSDE. It is then possible to identify those free stochastic iterated integrals, which must be discretized in order to obtain strong convergence of order $γ=1$. The non-commutativity imposes additional difficulties showing that the iterated free stochastic integrals can be simulated directly only under special situations, different from the commutative case. We will show, which diffusion terms lead to a Milstein-type method of order $γ=1$. For the cases, where a direct calculation is not possible, we approximate the iterated integrals based on a subdivision of the discretization intervals. As for fEMM, all proposed methods obey strong convergence of order $γ=1$ in $L_p(φ),\, 1\leq p\leq \infty$. For all methods developed, we show that the numerical solution is uniformly bounded on finite time intervals.

math.PR↗

arXiv2021

Free CIR Processes

For stochastic processes of non-commuting random variables we formulate a Cox-Ingersoll-Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by Voicelescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the existence of a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion.

math.PR↗