Equilibrium Stochastic Delay Processes

Stochastic processes with temporal delay play an important role in science and engineering whenever finite speeds of signal transmission and processing occur. However, an exact mathematical analysis of their dynamics and thermodynamics is available for linear models only. We introduce a class of stochastic delay processes with nonlinear time-local forces and linear time-delayed forces that obey fluctuation theorems and converge to a Boltzmann equilibrium at long times. From the point of view of control theory, such ``equilibrium stochastic delay processes'' are stable and energetically passive, by construction. Computationally, they provide diverse exact constraints on general nonlinear stochastic delay problems and can, in various situations, serve as a starting point for their perturbative analysis. Physically, they admit an interpretation in terms of an underdamped Brownian particle that is either subjected to a time-local force in a non-Markovian thermal bath or to a delayed feedback force in a Markovian thermal bath. We illustrate these properties numerically for a setup familiar from feedback cooling and point out experimental implications.

Correlating Lévy processes with self-decomposability: applications to energy markets

Based on the concept of self-decomposability, we extend some recent multidimensional Lévy models built using multivariate subordination. Our aim is to construct multivariate Lévy processes that can model the propagation of the systematic risk in dependent markets with some stochastic delay instead of affecting all the markets at the same time. To this end, we extend some known approaches keeping their mathematical tractability, study the properties of the new processes, derive closed-form expressions for their characteristic functions and detail how Monte Carlo schemes can be implemented. We illustrate the applicability of our approach in the context of gas, power and emission markets focusing on the calibration and on the pricing of spread options written on different underlying commodities.