Conference Agenda

We introduce a novel mathematical framework for filtering problems in biophysical applications, focusing on data collected from confocal laser scanning microscopy that records the spatio-temporal evolution of intracellular wave dynamics. In these settings, the signals are modeled by stochastic partial differential equations (SPDEs), and the observations are represented as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems and demonstrate the asymptotic consistency of our approach, providing approximations for finite-dimensional observation schemes and partial observations. Our theoretical findings are validated through extensive simulations using both synthetic and real-world data. This work enhances the understanding of filtering with point process observations, state-space models and Bayesian estimation, establishing a robust framework for future research in this area.

Mean-field SDEs, also known as McKean-Vlasov SDEs, provide mathematical descriptions of random systems of interacting particles, whose time evolutions depend on the probability distribution of the entire system. They are frequently used in applied mathematics, particularly in mathematical finance. Stochastic Volterra equations on the other hand include memory effects and allow for generating non-Markovian stochastic processes.

In this talk, we unify these two concepts and consider the $d$-dimensional mean-field stochastic Volterra equation $$ X_t = X_0 +\int_0^t K_{b}(s,t) b(s,X_s,\mathcal{L}(X_s))\,\mathrm{d} s+\int_0^t K_{\sigma}(s,t)\sigma(s,X_s,\mathcal{L}(X_s))\,\mathrm{d} B_s,\quad t\in [0,T], $$ where $B$ is an $m$-dimensional Brownian motion and $\mathcal{L}(X_s)$ denotes the law of $X_s$. We establish a suitable local martingale problem and prove the existence of weak solutions to the above mean-field stochastic Volterra equation under mild assumptions on the kernels and non-Lipschitz coefficients following the idea of Skorokhod's existence theorem.

In this work, we investigate the long-time behaviour of Hilbert space-valued stochastic Volterra processes given as solutions of \[ u(t) = G(t) + \int_0^t E_b(t-s)b(u(s))\,ds + \int_0^t E_\sigma(t-s)\sigma(u(s))\,dW_s \] on $H$, where $E_b,E_\sigma$ form a family of bounded linear operators on $H$ with additional integrability conditions, $(W_t)_{t\geq0}$ is a cylindrical Wiener process on a Hilbert space $U$, $G\in L^2(\Omega;L^2_{loc}(\mathbb{R_+};H))$ is $\mathcal{F}_0$-measurable and $b\colon H\longrightarrow H$ and $\sigma\colon H\longrightarrow L_2(U,H)$ are Lipschitz. For Markovian systems, such problems are typically analyzed using various methods that leverage the Markov property. However, solutions to stochastic Volterra equations are generally neither Markov processes nor semimartingales, making their asymptotic analysis both intriguing and challenging. To remedy the lack of the Markov property, we consider a Hilbert space-valued Markovian lift $X$ for $u$ and study its asymptotic behaviour. Finally, a projection argument allows us to provide a full characterization of corresponding invariant measures, derive a law-of-large numbers, and show that in certain cases a central limit theorem with the usual Gaussian domain of attraction holds.

We present the asymptotic theory for integrated functions of increments in space of Brownian local times. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen, which delved into the special case of polynomials using the method of moments, Malliavin calculus and Ray-Knight theorems as well as Perkins’ semimartingale representation of local time and the Kailath-Segall formula. In contrast to those methodologies, our approach relies on infill limit theory for semimartingales, which allows us to establish a limit theorem for general functions that satisfy mild smoothness and growth conditions.