Ambit Processes; with Applications to Turbulence and Tumour Growth

Abstract

Summary. The concept of ambit processes is outlined. Such stochastic processes are of interest in spatio-temporal modelling, and they play a central role in recent studies of velocity fields in turbulence and of the growth of cancer tumours. These studies are reviewed, and some open problems are outlined.

Hybrid scheme for Brownian semistationary processes

“…To make numerical implementation feasible, we truncate the second sum on the right-hand side of (2.6) so that both sums have N n ≥ κ + 1 terms in total. Thus, we arrive at a discretization scheme for X t , which we call a hybrid scheme, given by X n t :=X n t +X n t , 8) and…”

Time Change, Volatility, and Turbulence

“…It is important to note that in spite of the long range dependence the autocorrelation of the log price process is essentially zero whereas the velocity field shows algebraic decay of the autocorrelation function. Other important differences are the skewness of the densities of velocity increments in contrast to the symmetry of the distribution of log returns in FX markets 1 and the different behaviour of bipower variation [BNS04;BNSchS07]. Table 1 gives an overview of the differences and similarities between turbulence and finance.…”

“…In that more general context, the role of the Brownian motion is taken over by a Gaussian white noise field (or Brownian sheet) and the volatility is expressed as a random field, which may, for instance, be generated from a Lévy basis as in [BNSch04]. The paper [BNSch07a] gives a first discussion of the theoretical properties of such processes and describes some applications to turbulence and cancer growth.…”

“…The model type (20) is a one-dimensional limit of a spatio-temporal modelling framework introduced under the name of Ambit processes in [BNSch07a]. In that more general context, the role of the Brownian motion is taken over by a Gaussian white noise field (or Brownian sheet) and the volatility is expressed as a random field, which may, for instance, be generated from a Lévy basis as in [BNSch04].…”

Stochastic integration on the real line

“…The setting (α) is in [1]- [3] used for modeling of an interesting new class of moving averages; here we recall, e.g., from [13] that moving averages provide a large class of stationary processes. Note that when in (α) σ and Z are independent it is easy to define the integral by conditioning on σ and using integration of a deterministic integrand with respect to an independently scattered random measure, cf., e.g., [23].…”