Risk Sensitive Path Integral Control

Recently path integral methods have been developed for stochastic optimal control for a wide class of models with non-linear dynamics in continuous space-time. Path integral methods find the control that minimizes the expected cost-to-go. In this paper we show that under the same assumptions, path integral methods generalize directly to risk sensitive stochastic optimal control. Here the method minimizes in expectation an exponentially weighted cost-to-go. Depending on the exponential weight, risk seeking or risk averse behaviour is obtained. We demonstrate the approach on risk sensitive stochastic optimal control problems beyond the linear-quadratic case, showing the intricate interaction of multi-modal control with risk sensitivity.

💡 Research Summary

The paper extends the recently developed path‑integral (PI) framework for stochastic optimal control to the risk‑sensitive setting. Classical PI methods solve a stochastic control problem by converting the Hamilton‑Jacobi‑Bellman (HJB) equation into a linear partial differential equation, whose solution can be expressed as a path‑integral over stochastic trajectories. The optimal control is then obtained as a gradient of the logarithm of a “partition function” that aggregates costs along sampled paths. This approach works for a broad class of nonlinear dynamics in continuous time and space, provided the control appears linearly in the dynamics and the cost is quadratic in the control.

Risk‑sensitive control replaces the usual expected cost objective with an exponential‑utility criterion:
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