Learning to reflect: A unifying approach for data-driven stochastic control strategies

TY - JOUR

T1 - Learning to reflect

T2 - A unifying approach for data-driven stochastic control strategies

AU - Christensen, Sören

AU - Strauch, Claudia

AU - Trottner, Lukas

PY - 2024/8

Y1 - 2024/8

N2 - Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the statistical challenge of developing purely data-driven strategies. For the mathematically separated classes of continuous diffusion processes and Lévy processes, we show that developing efficient strategies for related singular stochastic control problems can essentially be reduced to finding rate-optimal estimators with respect to the sup-norm risk of objects associated to the invariant distribution of ergodic processes which determine the theoretical solution of the control problem. From a statistical perspective, we exploit the exponential β-mixing property as the common factor of both scenarios to drive the convergence analysis, indicating that relying on general stability properties of Markov processes is a sufficiently powerful and flexible approach to treat complex applications re-quiring statistical methods. We show moreover that in the Lévy case—even though per se jump processes are more difficult to handle both in statistics and control theory—a fully data-driven strategy with regret of significantly better order than in the diffusion case can be constructed utilizing spatial ergodicity of a path-time transformation of the Lévy process in form of its overshoots.

AB - Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the statistical challenge of developing purely data-driven strategies. For the mathematically separated classes of continuous diffusion processes and Lévy processes, we show that developing efficient strategies for related singular stochastic control problems can essentially be reduced to finding rate-optimal estimators with respect to the sup-norm risk of objects associated to the invariant distribution of ergodic processes which determine the theoretical solution of the control problem. From a statistical perspective, we exploit the exponential β-mixing property as the common factor of both scenarios to drive the convergence analysis, indicating that relying on general stability properties of Markov processes is a sufficiently powerful and flexible approach to treat complex applications re-quiring statistical methods. We show moreover that in the Lévy case—even though per se jump processes are more difficult to handle both in statistics and control theory—a fully data-driven strategy with regret of significantly better order than in the diffusion case can be constructed utilizing spatial ergodicity of a path-time transformation of the Lévy process in form of its overshoots.

KW - Data-driven singular control

KW - Lévy processes

KW - diffusion processes

KW - exploration vs. exploitation

KW - nonparametric statistics

KW - overshoots

KW - reinforcement learning

KW - sup-norm risk

UR - http://www.scopus.com/inward/record.url?scp=85194375318&partnerID=8YFLogxK

U2 - 10.3150/23-BEJ1665

DO - 10.3150/23-BEJ1665

M3 - Journal article

SN - 1350-7265

VL - 30

SP - 2074

EP - 2101

JO - Bernoulli

JF - Bernoulli

IS - 3

ER -