Stochastic Analysis and Statistics 2026

Nakahiro Yoshida (The University of Tokyo) - Deep learning of a diffusion process

We discuss an application of deep neural networks to estimation of diffusion coefficients from high frequency data of a mixing diffusion process. An oracle inequality and a rate of convergence are derived for the generalization error. This is a joint work with A. Gloter.

Michel Sørensen (University of Copenhagen) - Approximate likelihood inference for stochastic differential equations using splitting schemes

The complexity of likelihood inference for stochastic differential equations based on discrete time samples often necessitates the use of approximations. Approximate likelihood methods for high frequency data, such as Gaussian pseudo-likelihood functions, have been studied intensively and are popular in applications, for instance in financial econometrics. However, for strongly nonlinear models these methods usually do not perform well when the sampling frequency is not very high. New developments of approximate likelihood methods based on splitting schemes are presented. These methods perform well also for strongly nonlinear models and at moderate sampling frequencies. Splitting schemes were originally introduced to solve ODEs and SDEs numerically, but in Pilipovic, Samson and Ditlevsen (2024) it was proposed to use them for statistical inference. In the talk a more general approach is presented that is applicable to a broad class of diffusion models. The theory is developed in the framework of approximate martingale estimating functions, which provide approximations to the score function and estimators that are efficient for high frequency data. For Strang splitting an approximate martingale estimating function of order 3 is obtained. The lecture is based on joint work with Susanne Ditlevsen and Adeline Samson. Reference: Pilipovic, P., Samson, A. And Ditlevsen, S. (2024): Efficient estimation for ergodic diffusion processes sampled at high frequency. Ann. Statist., 52, 842 - 867.

Stefano Maria Iacus (European Commission, Harvard University) - Dynamic Attention (DynAttn): Interpretable High-Dimensional Spatio-Temporal Forecasting (with Application to Conflict Fatalities)

Forecasting conflict-related fatalities remains a central challenge in political science and policy analysis due to the sparse, bursty, and highly non-stationary nature of violence data. We introduce DynAttn, an interpretable dynamic-attention forecasting framework for high-dimensional spatio-temporal count processes. DynAttn combines rolling-window estimation, shared elastic-net feature gating, a compact weight-tied self-attention encoder, and a zero-inflated negative binomial (ZINB) likelihood. This architecture produces calibrated multi-horizon forecasts of expected casualties and exceedance probabilities, while retaining transparent diagnostics through feature gates, ablation analysis, and elasticity measures. We evaluate DynAttn using global country-level and high-resolution PRIO-grid-level conflict data from the VIEWS forecasting system, benchmarking it against established statistical and machine-learning approaches, including DynENet, LSTM, Prophet, PatchTST, and the official VIEWS baseline. Across forecast horizons from one to twelve months, DynAttn consistently achieves substantially higher predictive accuracy, with particularly large gains in sparse grid-level settings where competing models often become unstable or degrade sharply. Beyond predictive performance, DynAttn enables structured interpretation of regional conflict dynamics. In our application, cross-regional analyses show that short-run conflict persistence and spatial diffusion form the core predictive backbone, while climate stress acts either as a conditional amplifier or a primary driver depending on the conflict theater. joint work with Stefano M. Iacus, Haodong Qi, Marcello Carammia, Thomas Juneau

Hiroki Masuda (The University of Tokyo) - LAD estimation of locally stable SDE

We prove the asymptotic (mixed) normality of the least absolute deviation (LAD) estimator for a locally $\alpha$-stable stochastic differential equation (SDE) observed at high frequency. The objective function for the LAD estimator is expressed in a fully explicit form without necessitating numerical integration, offering a significant computational advantage over the existing non-Gaussian stable quasi-likelihood approach. The first-order results for both ergodic and non-ergodic cases are presented, where the terminal sampling time diverges or is fixed, respectively, under different sets of assumptions.

Mogens Bladt (University of Copenhagen) - Calibration of stochastic interest rate models using maximum likelihood estimation

We consider the so-called Markovian interest rate model, in which interest rates are assumed to follow an underlying Markov process. In each state of the Markov process, the spot interest rate is assumed either to be constant or a deterministic function. Either way, the Markovian interest rate models form a dense class that can approximate any model, e.g., SDE-driven models such as CIR or Vasicek. In this talk, we show how to achieve such approximations by linking Markovian interest rate models to the class of tractable, dense distributions known as phase-type distributions. Interest rate models are mostly applied for discounting purposes and are therefore modelled in terms of an implied zero-coupon bond. The link to phase-type theory arises from the fact that the zero-coupon bond price is identical to the survival function of a phase-type distribution. This simple but remarkable result has been unperceived in the applied probability community. Now with the phase-type toolbox in hand, we will show how to calibrate a Markovian interest model to observed bond prices and theoretical models, obtaining explicit formulae for related quantities such as yield curves and swap rates, and deal with negative interest rates. Some illustrative numerical examples will be provided.

Shota Yano (University of Tokyo) - Quasi-Likelihood Analysis for Adaptive Hybrid Estimation of Degenerate Diffusion Processes

We study adaptive estimation for discretely observed degenerate diffusion processes within the quasi-likelihood analysis framework. Using refined quasi-likelihoods derived from higher-order Itô–Taylor expansions, we construct a unified class of adaptive hybrid estimators that allows flexible combinations of quasi-maximum likelihood, quasi-Bayesian, and multi-step procedures (Newton method) at each stage. Under the relaxed balance condition nh^p→0, we establish asymptotic normality together with moment convergence in a unified manner. Numerical experiments for linear and FitzHugh–Nagumo models illustrate the robustness and practical effectiveness of the proposed hybrid strategy.

Yuta Koike (The University of Tokyo) - High-dimensional bootstrap and asymptotic expansion

The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this context, numerical experiments suggest that third-moment matching bootstrap approximations would outperform normal approximation even without studentization, but the existing theoretical results cannot explain this phenomenon. In this talk, we develop an asymptotic expansion formula for the bootstrap coverage probability and show that it can give an explanation for the above phenomenon. In particular, we find the following interesting blessing of dimensionality phenomenon: The third-moment matching wild bootstrap is second-order accurate in high dimensions even without studentization if the covariance matrix has identical diagonal entries and bounded eigenvalues. The validity of these results is established under the assumption that the underlying distributions admit Stein kernels.

Martin Bladt (University of Copenhagen) - Recent Nonparametric Advances in Jump Process Estimation

This talk considers recent developments in nonparametric inference for finite-state jump processes, covering both Markov and more general non-Markov settings. We begin with a review of the estimation of state occupation probabilities and transition probabilities in the Markov case, including handling of right-censoring. Next, we move beyond the Markov framework and adopt a conditional perspective, in which transition behaviour is described through joint and conditional occupation probabilities. This approach allows, through subsampling, for duration dependence and path effects, with semi-Markov models as a natural example, and provides a basis for incorporating both internal history and external covariates. We then address settings with possibly many covariates, and consider adaptive methods, including nearest-neighbour and forest-based estimators, which preserve the nonparametric character and allow for data-driven segmentation. Finally, we discuss estimation of transition rates via binning, also colloquially known as the Poisson regression method, applicable for both Markov and semi-Markov models.

Andrea Aveni (University of Copenhagen) - Smoothed Bayesian Inference for Time-Inhomogeneous Multi-State Markov Models

We study Bayesian inference for time-inhomogeneous multi-state Markov models. Our goal is to estimate transition dynamics in a form that is both flexible and smooth. A key difficulty is that convenient Bayesian nonparametric procedures often produce piecewise-constant cumulative transition estimates. Such estimators work well for cumulative hazards but are less suited to recovering smooth transition rates. We propose to smooth posterior draws by applying an explicit kernel regularization step. This preserves the computational advantages of the recently proposed Lévy-process-based construction. At the same time, it yields smooth trajectories that can be interpreted as time-varying transition rates. We show that the smoothed estimators remain asymptotically equivalent to the original cumulative estimators. In particular, they inherit consistency and asymptotic normality for cumulative transition functions. The added regularity also allows consistent estimation of the underlying transition rates. We further derive pointwise and finite-dimensional limit results that quantify the effect of smoothing. The talk highlights how a simple post-processing step can turn rough Bayesian output into statistically tractable smooth inference.

Masayuki Uchida (Osaka University) - Estimation for discretely observed linear parabolic SPDEs in two space dimensions with unknown damping parameters

We consider the parametric estimation of a class of second-order linear parabolic stochastic partial differential equations (SPDEs) in two spatial dimensions. These equations are driven by two types of Q-Wiener processes, and our analysis is based on high-frequency observations in both space and time. We construct estimators for the damping parameters of the Q-Wiener processes using realized quadratic variations derived from both temporal and spatial increments. In addition, we propose minimum contrast estimators for four coefficient parameters of the SPDEs and obtain adaptive estimators for the remaining unknown parameters through approximate coordinate processes. Finally, we examine the finite-sample properties of the proposed estimators using numerical simulations.