Joint ORFE/PACM Talk
The theory of continuousstochastic processes has provided a wealth of modeling tools for random systems with complex interactions, via the building blocks of Brownian motion, martingales, and Markov processes, for instance via stochastic differential equations. These tools rely heavily on the independence of increments for Brownian motion, which translates into short‐range dependence for the resulting stochastic differential models. However, applications ranging from communications networks, to polymerstudies,to financial and geophysicaltime series, are frequently and increasingly calling for models with longer‐range interactions. For time‐evolution systems, this can be referred to as stochastic long memory, and requires a change in framework. The basic building block in continuoustime then becomesthe so‐called fractional Brownian motion, which creates technical challenges in the interpretation of stochastic differential systems. We will present a broad overview of some of the mathematics developed recently to overcome several of these challenges, including elements of analysis on Wiener space, the Malliavin calculus, and other tools in stochastic analysis. We will also discuss questions ofstatistical estimation and calibration, and applications of long‐range dependence to stochastic volatility tracking via options markets, and to proxy‐based paleo‐temperature Bayesian reconstruction.