On stochastic integration for volatility modulated Brownian-driven Volterra processes via white noise analysis

Abstract

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G * of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.(VMBV), and this is the class of processes that we will concentrate our attention on in this paper; so from now on we fix L = B in Equation (1.1). Barndorff-Nielsen et al. (2012) use methods of Malliavin calculus to validate the following definition of the integral: t 0 Y (s) dX(s) = t 0 K g (Y )(t, s)σ(s) δ M B(s) + t 0

Some Recent Developments in Ambit Stochastics

“…In particular, we have discussed the existence of the ambit fields driven by metatime changed Lévy bases, selfdecomposability of random fields [13], applications of BSS processes in the modelling of turbulent time series [35] and new results on the distributional collapse in financial data. Some of the topics not mentioned here but also under development are the integration theory with respect to time-changed volatility modulated Lévy bases [7]; integration with respect to volatility Gaussian processes in the White Noise Analysis setting in the spirit of [34] and extending [6]; modelling of multidimensional turbulence based on ambit fields; and in-depth study of parsimony and universality in BSS and LSS processes motivated by some of the discussions in the present paper.…”

Ambit Fields: Survey and New Challenges

“…for a stochastic integrand Z, strongly depends on whether the driving process L is a Brownian motion or a pure jump Lévy motion, since the main notions of Malliavin calculus differ in those two cases. An alternative way of defining the stochastic integral at (3.1) without imposing L 2 -structure of the integrand Z is proposed in [4]. The authors apply white noise analysis to construct the integral in the situation, where X is driven by a Brownian motion.…”

Selfdecomposable Fields

“…Finally, L denotes a Lévy basis (i.e., an independently scattered and infinitely divisible random measure). For aspects of the theory and applications of ambit processes and fields, see [8,10,12,14,15,23,34,39,52] and [55].…”

“…At this point, the concept of a measure in the cylindrical σ -field that does not charge zero plays an important role. However, the uniqueness can be obtained without condition (10).…”