Events
DMS Applied and Computational Mathematics Seminar |
| Time: Sep 30, 2022 (02:00 PM) |
| Location: 328 Parker Hall |
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Details: Speaker: Yuming Paul Zhang Title: McKean-Vlasov equations involving hitting times: blow-ups and global solvability Abstract: We study two McKean-Vlasov equations involving hitting times. The equations are used in financial networks, computational neuroscience and some fluid models. The first equation is \(X(t) = X(0) + B(t) - \alpha \mathbb{P}(\tau \le t)\), where \((B(t); \, t \ge 0)\) denotes the Brownian motion, and \(\tau:= \inf\{t \ge 0: X(t) \le 0\}\) is the hitting time to zero of the process \(X\). We provide a simple condition on \(\alpha\) and the distribution of \(X(0)\) such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time \(t \ge 0\). We take the PDE approach and develop a new comparison principle.
The second equation is \(X(t) = X(0) + \beta t + B(t) + \alpha \log \mathbb{P}(\tau \le t)\), \(t \ge 0\), whose Fokker-Planck equation is non-local. We prove that if \(\beta,1/\alpha > 0\) are sufficiently large, the McKean-Vlasov dynamics is well-defined for all time \(t \ge 0$\) The argument is based on a relative entropy analysis.
This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.
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