Student processes

Abstract

Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

A Gamma Activity Time Process with Noninteger Parameter and Self-Similar Limit

“…This construction validates the use of long-range dependent t and variance-gamma subordinator models for actual financial data as advocated in Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting. …”

“…Using Taqqu (1975), we can show that var(A k ) and var(B k ) are both O(k 2H ), and that (A kt − E A kt )/k H converges weakly as k tends to ∞ to a self-similar process with parameter H (Heyde and Leonenko (2005) and Finlay and Seneta (2006) showed this for ω = 1 and α = 2, but the proof can be extended to cover the more general case where ω > 0 and 0 < α ≤ 2). We give a proof in Theorem 2, below, that (1/k H )(B k − E B k ) converges in probability to 0, which is enough to demonstrate that our new discrete-time {T t } process (7) has asymptotically a self-similar limit structurally coincident with that of the original {T t } process used in Finlay and Seneta (2006).…”

“…To construct their LRD t process, Heyde and Leonenko (2005) started with distributed increments with integer ν. Their construction from then on does not require integer ν however; see Heyde and Leonenko (2005, Sections 3.3 and 5.1).…”

“…The processes constructed in Heyde and Leonenko (2005) and Finlay and Seneta (2006) are restricted to integer values of ν, however, a condition not consistent with estimation with actual data. The purpose of this note is therefore to extend the constructions to allow for noninteger ν.…”

Optimal Long-Term Investment Model with Memory

“…To name some related contributions, let us mention here Comte and Renault [7,8], Rogers [30], Heyde [16], Willinger et al [32], Barndorff-Nielsen and Shephard [5], BarndorffNielsen et al [4], Hu and Øksendal [18], Hu et al [19], Elliott and van der Hoek [9], and Heyde and Leonenko [17]. In most of these references, driving noise processes are assumed to have stationary increments since this is a natural requirement of simplicity.…”