Conference Agenda

In 1976 Siegmund introduced the notion of ``duality of Markov processes with respect to a function''. This concept has since then served as a powerful tool for analyzing Markov processes in fields like population genetics, risk theory, and queuing models, as it allows to connect long-term behavior and fluctuation theory.

In this talk, we will discuss Siegmund duality of Lévy-type processes, focusing on structural properties of (dual) generators and drawing relations to their adjoint operators. We will also investigate the connection of duality to the time reversal of soutions to Lévy-driven SDEs, with a focus on the example of generalized Ornstein-Uhlenbeck processes. Under certain conditions, we will show that the two concepts coincide.

Polynomial processes, which include affine processes as a subclass, are a class of Markov processes characterized by the property that their conditional polynomial moments can be computed in closed form. Due to their computational tractability, polynomial models are widely utilized in mathematical finance, with notable examples being the Heston model and Lévy-driven models. The aim of our research is to estimate the parameters of discretely observed polynomial models. In asymptotic statistics, the maximum likelihood estimator is highly desirable for its favorable properties, such as consistency, asymptotic normality, and minimal asymptotic error. However, in practice, the score function is often not available in closed form, motivating the use of alternative approaches.

We propose using martingale estimating functions in place of the score function, with a focus on the Heyde-optimal martingale estimating function, which minimizes the distance to the score function in an $L^2$ sense within a specified class of estimating functions. Our framework constructs a specialized class of polynomial martingale estimating functions for general polynomial processes, requiring only the calculation of polynomial conditional moments. Specifically, we consider: $$G_n(\vartheta) = \sum_{m=1}^n \sum_{|\pmb{\alpha}| \leq k} \lambda_{\vartheta, \pmb{\alpha}}(m) \left( \Delta X(m)^{\pmb{\alpha}} - \mathbb{E}_\vartheta[\Delta X(m)^{\pmb{\alpha}} | \mathcal{F}_{m-1}]\right)$$ where the maximum degree $k$ is fixed, and the integrand $\lambda_{\vartheta, \pmb{\alpha}}(m)$, measurable with respect to $\mathcal{F}_{m-1}$, can be freely chosen. By applying ergodic theory for Markov processes, we establish both the consistency and asymptotic normality of these estimating functions. Additionally, we demonstrate how to explicitly compute the Heyde-optimal estimating function within this class.