Fractional Risk Analysis of Stochastic Systems with Jumps and Memory
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arXiv
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| Format: | Preprint |
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2026
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| author | Sun, Yimeng Wang, Zhuoyuan Zhang, Xiaole Ping, Heng Xue, Jintang Bogdan, Paul Nakahira, Yorie |
| author_facet | Sun, Yimeng Wang, Zhuoyuan Zhang, Xiaole Ping, Heng Xue, Jintang Bogdan, Paul Nakahira, Yorie |
| contents | Accurate risk assessment is essential for safety-critical autonomous and control systems under uncertainty. In many real-world settings, stochastic dynamics exhibit asymmetric jumps and long-range memory, making long-term risk probabilities difficult to estimate across varying system dynamics, initial conditions, and time horizons. Existing sampling-based methods are computationally expensive due to repeated long-horizon simulations to capture rare events, while existing partial differential equation (PDE)-based formulations are largely limited to Gaussian or symmetric jump dynamics and typically treat memory effects in isolation. In this paper, we address these challenges by deriving a space- and time-fractional PDE that characterizes long-term safety and recovery probabilities for stochastic systems with both asymmetric Levy jumps and memory. This unified formulation captures nonlocal spatial effects and temporal memory within a single framework and enables the joint evaluation of risk across initial states and horizons. We show that the proposed PDE accurately characterizes long-term risk and reveals behaviors that differ fundamentally from systems without jumps or memory and from standard non-fractional PDEs. Building on this characterization, we further demonstrate how physics-informed learning can efficiently solve the fractional PDEs, enabling accurate risk prediction across diverse configurations and strong generalization to out-of-distribution dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional Risk Analysis of Stochastic Systems with Jumps and Memory Sun, Yimeng Wang, Zhuoyuan Zhang, Xiaole Ping, Heng Xue, Jintang Bogdan, Paul Nakahira, Yorie Systems and Control Optimization and Control Accurate risk assessment is essential for safety-critical autonomous and control systems under uncertainty. In many real-world settings, stochastic dynamics exhibit asymmetric jumps and long-range memory, making long-term risk probabilities difficult to estimate across varying system dynamics, initial conditions, and time horizons. Existing sampling-based methods are computationally expensive due to repeated long-horizon simulations to capture rare events, while existing partial differential equation (PDE)-based formulations are largely limited to Gaussian or symmetric jump dynamics and typically treat memory effects in isolation. In this paper, we address these challenges by deriving a space- and time-fractional PDE that characterizes long-term safety and recovery probabilities for stochastic systems with both asymmetric Levy jumps and memory. This unified formulation captures nonlocal spatial effects and temporal memory within a single framework and enables the joint evaluation of risk across initial states and horizons. We show that the proposed PDE accurately characterizes long-term risk and reveals behaviors that differ fundamentally from systems without jumps or memory and from standard non-fractional PDEs. Building on this characterization, we further demonstrate how physics-informed learning can efficiently solve the fractional PDEs, enabling accurate risk prediction across diverse configurations and strong generalization to out-of-distribution dynamics. |
| title | Fractional Risk Analysis of Stochastic Systems with Jumps and Memory |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2603.26009 |