Neural Jump ODEs as Generative Models
Robert A. Crowell, Florian Krach, Josef Teichmann
TL;DR
We address generative modeling for Itô processes that may be path‑dependent and irregularly observed by learning drift and diffusion coefficients with Neural Jump ODEs (NJODEs). The method first learns conditional expectations of the process and its moments to obtain estimators for $\mu$ and $\Sigma$, then generates samples by Euler–Maruyama using these learned coefficients; convergence guarantees tie the learned coefficients to the true law as the time step vanishes. A key advance is training in a predictive, non‑adversarial framework that naturally handles missing data and path‑dependent coefficients, with enhanced estimators for instantaneous drift and diffusion to reduce finite‑$Δ$ bias. Theoretical results establish convergence of estimators and the generated samples’ law to the true process under mild assumptions, and experiments on GBM and OU demonstrate accurate parameter recovery and realistic sample generation. Overall, the approach provides a scalable, prediction‑driven pathway to accurate generative modeling of complex Itô dynamics from irregular, partial observations.
Abstract
In this work, we explore how Neural Jump ODEs (NJODEs) can be used as generative models for Itô processes. Given (discrete observations of) samples of a fixed underlying Itô process, the NJODE framework can be used to approximate the drift and diffusion coefficients of the process. Under standard regularity assumptions on the Itô processes, we prove that, in the limit, we recover the true parameters with our approximation. Hence, using these learned coefficients to sample from the corresponding Itô process generates, in the limit, samples with the same law as the true underlying process. Compared to other generative machine learning models, our approach has the advantage that it does not need adversarial training and can be trained solely as a predictive model on the observed samples without the need to generate any samples during training to empirically approximate the distribution. Moreover, the NJODE framework naturally deals with irregularly sampled data with missing values as well as with path-dependent dynamics, allowing to apply this approach in real-world settings. In particular, in the case of path-dependent coefficients of the Itô processes, the NJODE learns their optimal approximation given the past observations and therefore allows generating new paths conditionally on discrete, irregular, and incomplete past observations in an optimal way.