Conjugate Processes: Theory and Application to Risk Forecasting

TL;DR

This paper introduces conjugate processes, a new class of models for cyclic phenomena with stochastically evolving distributions, providing theoretical foundations and applications to financial risk forecasting.

Contribution

It develops the theory of conjugate processes, including laws of large numbers, and demonstrates their application to risk forecasting in finance.

Findings

Established laws of large numbers for conjugate processes

Provided a constructive example illustrating the theory

Applied the framework to financial risk forecasting

Abstract

Many dynamical phenomena display a cyclic behavior, in the sense that time can be partitioned into units within which distributional aspects of a process are homogeneous. In this paper, we introduce a class of models - called conjugate processes - allowing the sequence of marginal distributions of a cyclic, continuous-time process to evolve stochastically in time. The connection between the two processes is given by a fundamental compatibility equation. Key results include Laws of Large Numbers in the presented framework. We provide a constructive example which illustrates the theory, and give a statistical implementation to risk forecasting in financial data.