arXiv subjects

Amin Wu

Publications and source records attributed to Amin Wu.

4 recordsLinked to original sources

arXiv2026

Bayesian Inference for Partially Observed McKean-Vlasov SDEs with Full Distribution Dependence

McKean-Vlasov stochastic differential equations (MVSDEs) describe systems whose dynamics depend on both individual states and the population distribution, and they arise widely in neuroscience, finance, and epidemiology. In many applications the system is only partially observed, making inference very challenging when both drift and diffusion coefficients depend on the evolving empirical law. This paper develops a Bayesian framework for latent state inference and parameter estimation in such partially observed MVSDEs. We combine time-discretization with particle-based approximations to construct tractable likelihood estimators, and we design two particle Markov chain Monte Carlo (PMCMC) algorithms: a single-level PMCMC method and a multilevel PMCMC (MLPMCMC) method that couples particle systems across discretization levels. The multilevel construction yields correlated likelihood estimates and achieves mean square error $(O(\varepsilon^2))$ at computational cost $(O(\varepsilon^{-6}))$, improving on the $(O(\varepsilon^{-7}))$ complexity of single-level schemes. We address the fully law-dependent diffusion setting which is the most general formulation of MVSDEs, and provide theoretical guarantees under standard regularity assumptions. Numerical experiments confirm the efficiency and accuracy of the proposed methodology.

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arXiv2025

Bayesian Parameter Estimation for Partially Observed McKean-Vlasov Diffusions Using Multilevel Markov chain Monte Carlo

In this article we consider Bayesian estimation of static parameters for a class of partially observed McKean-Vlasov diffusion processes with discrete-time observations over a fixed time interval. This problem features several obstacles to its solution, which include that the posterior density is numerically intractable in continuous-time, even if the transition probabilities are available and even when one uses a time-discretization, the posterior still cannot be used by adopting well-known computational methods such as Markov chain Monte Carlo (MCMC). In this paper we provide a solution to this problem by using new MCMC algorithms which can solve the afore-mentioned issues. This MCMC algorithm is extended to use multilevel Monte Carlo (MLMC) methods. We prove convergence bounds on our parameter estimators and show that the MLMC-based MCMC algorithm reduces the computational cost to achieve a mean square error versus ordinary MCMC by an order of magnitude. We numerically illustrate our results on two models.

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arXiv2025

Bayesian Inference for Non-Synchronously Observed Diffusions

We consider the problem of Bayesian inference for bi-variate data observed in time but with observation times which occur non-synchronously. In particular, this occurs in a wide variety of applications in finance, such as high-frequency trading or crude oil futures trading. We adopt a diffusion model for the data and formulate a Bayesian model with priors on unknown parameters along with a latent representation for the the so-called missing data. We then consider computational methodology to fit the model using Markov chain Monte Carlo (MCMC). We have to resort to time-discretization methods as the complete data likelihood is intractable and this can cause considerable issues for MCMC when the data are observed in low frequencies. In a high frequency observation frequencies we present a simple particle MCMC method based on an Euler--Maruyama time discretization, which can be enhanced using multilevel Monte Carlo (MLMC). In the low frequency observation regime we introduce a novel bridging representation of the posterior in continuous time to deal with the issues of MCMC in this case. This representation is discretized and fitted using MCMC and MLMC. We apply our methodology to real and simulated data to establish the efficacy of our methodology.

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arXiv2024

Bayesian Parameter Inference for Partially Observed Stochastic Volterra Equations

In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest, SVEs must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. Based upon recent studies on time-discretization of SVEs (e.g. Richard et al. 2021), we use Euler-Maruyama methods for the afore-mentioned discretization. We then show how multilevel Markov chain Monte Carlo (MCMC) methods (Jasra et al. 2018) can be applied in this context. In the examples we study, we give a proof that shows that the cost to achieve a mean square error (MSE) of $\mathcal{O}(ε^2)$, $ε>0$, is {$\mathcal{O}(ε^{-\tfrac{4}{2H+1}})$, where $H$ is the Hurst parameter. If one uses a single level MCMC method then the cost is $\mathcal{O}(ε^{-\tfrac{2(2H+3)}{2H+1}})$} to achieve the same MSE. We illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real data.

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