Chapter 7: Hybrid scheme for Brownian semistationary processes⁴³

We study simulation methods for Brownian semistationary (BSS) processes, first introduced by Barndorff-Nielsen and Schmiegel [35, 36], which form a flexible class of stochastic processes that are able to capture some common features of empirical time series, such as stochastic volatility (intermittency), roughness, stationarity, and strong dependence. These processes have been applied in various contexts, most notably in the study of turbulence in physics [34, 103] and in finance as models of energy prices [31, 48]. A BSS process X is defined via the integral representationwhere W is a two-sided Brownian motion providing the fundamental noise innovations, the amplitude of which is modulated by a stochastic volatility (intermittency) process σ that may depend on W. This driving noise is then convolved with a deterministic kernel function g that specifies the dependence structure of X. The process X can also be viewed as a moving average of volatility-modulated Brownian noise; setting σ_s= 1, we see that stationary Brownian moving averages are nested in this class of processes.

⁴³ This chapter is derived from “Hybrid scheme for Brownian semistationary processes,” Finance and Stochastics, 21 (2017), pp. 931-965, https://dx.doi.org/10.1007/s00780-017-0335-5.