One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

翻译：在基于离散观测数据进行扩散模型参数贝叶斯推断时，核心挑战之一在于无法获得连续观测时刻间转移密度函数的解析表达式，而这正是推导似然函数所必需的。本文在先前利用归一化流求解Fokker-Planck型偏微分方程的研究基础上，提出一种新的归一化流架构，用于学习扩散过程在两个观测时刻间的转移密度函数。为此，我们采用神经Galerkin框架，以狄拉克δ质量作为初始条件，在初始数据与扩散系数的指定训练分布上求解相应的FP方程。我们特别关注扩散矩阵在某些不可达边界区域内为零的过程，例如满足Feller条件的随机波动率模型。沿观测轨迹计算的转移密度乘积可近似似然函数，从而通过马尔可夫链蒙特卡洛方法实现低成本后验抽样。在离线训练阶段完成后，推断效率显著提升，既无需为MCMC采样器提出的每个参数实时求解FP方程，也无需依赖需反复模拟扩散桥的其他无似然贝叶斯推断方法。