Controlled supermartingale functions for stochastic differential equations: inference and applications
(2025) 2025 IEEE 64th Conference on Decision and Control (CDC) — Location: Rio de Janeiro, Brazil (9.December.2025)
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- Abstract
- We study the problem of constructing controlled supermartingale functions to synthesize feedback laws that guarantee safety properties of stochastic differential equations (SDE) with control inputs. SDEs are widely used to model continuous time stochastic processes with applications ranging from financial markets to biology. In this paper, we extend classic notions from martingale theory for stochastic processes to prove that a given SDE will not exit a safe region over some finite time horizon with high probability. Our notion considers time-varying supermartingale functions that provide sharper probability bounds when compared to those that are time-independent. Furthermore, we study the controlled version of these supermartingales and the problem of synthesizing feedback control law that will maintain the state within a safe set with high probability over a given finite time horizon. We provide a projection-based algorithm for synthesizing polynomial, time-varying controlled supermartingales and corresponding feedback laws using sum-of-square (SOS) programming techniques. We implement our approach on some challenging numerical examples to demonstrate how it can synthesize control feedback laws that provide upper bounds on the probability of safety violations over a given time horizon.
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Ghanbarpour, M., Berger, G., & Sankaranarayanan, S. (2025). Controlled supermartingale functions for stochastic differential equations: inference and applications. Published. 2025 IEEE 64th Conference on Decision and Control (CDC), Rio de Janeiro, Brazil.