When is a Moving Average a Semimartingale?

Transcription

1 29 Barrett Lectures Ph.D.-student under supervision of Jan Pedersen, Thiele Centre, University of Aarhus, Denmark. 29 Barrett Lectures at The University of Tennessee: Stochastic Analysis and its Applications

2 Semimartingales 29 Barrett Lectures Let F = (F t ) t denote a filtration. Then X = (X t ) t is said to be an F-semimartingale if it can be written as X t = X + M t + A t, t, where M is a càdlàg F-local martingale, A is an F-adapted càdlàg process of bounded variation and X is F -measurable. A càdlàg process X induces a reasonable stochastic integral Ys dxs if and only if it is a semimartingale.

3 Moving averages 29 Barrett Lectures The focus of this talk is on the semimartingale property of moving averages X = (X t ) t, given by X t = ϕ(t s) dz s, t, where Z = (Z t ) t is a Lévy process and ϕ is a deterministic function for which the integral exists. Some examples: If ϕ is constant then X is a semimartingale. If Z Brownian motion, ϕ = 1 [,1] then X t = Z t Z (t 1) (is not a semimartingale). The Ornstein-Uhlenbeck type process; here ϕ(t) = e βt (is a semimartingale). The Riemann-Liouville fractional integral; here ϕ(t) = t β 1/2 (in most cases not a semimartingale).

4 F. Knight 29 Barrett Lectures Let X = (X t ) t be given by where Z is a Brownian motion. Theorem (F. Knight (1992)) X t = ϕ(t s) dz s, t, X is an F Z -semimartingale if and only if ϕ W 1,2 loc (Ê+), i.e. ϕ(t) = α + h(s) ds, t, where α Ê and h L 2 loc (λ).

5 29 Barrett Lectures Lévy driving moving averages Let X = (X t ) t be given by X t = ϕ(t s) dz s, t, where Z = (Z t ) t is a Lévy process with characteristic triplet (γ, σ 2, ν). Theorem X is an F Z -semimartingale if and only if 1 Z is of bounded variation: ϕ is of bounded variation, 2 σ 2 > : ϕ W 1,2 loc (Ê+), 3 Z is of unbounded variation and σ 2 = : ϕ is absolutely continuous on Ê + with a density ϕ satisfying 1 ( xϕ (s) 2 xϕ (s) ) ν(dx) ds <, for all t >. 1

6 A corollary 29 Barrett Lectures Let X be given by X t = (t s) β 1 2 dz s, ( β 1 ). 2 Corollary (The Riemann-Liouville fractional integral) σ 2 > : X is an F Z -semimartingale if and only if β > 1. σ 2 = : X is an F Z -semimartingale if and only if β > 1, β = 1 and 1 1 x2 log( x ) ν(dx) <, 1 2 < β < 1 and 1 1 x 2/(3 2β) ν(dx) <. Thus if Z is α-stable then X is an F Z -semimartingale if and only if β > when α [1, 2], α β > 1 when α (, 1). 2

7 29 Barrett Lectures Gaussian chaos processes Let (Z t ) t denote a Brownian motion. (Y t ) t is said to be a Gaussian chaos process if there exists n 1 such that Y t n k= H k for all t. Let us now consider X of the form X t = ϕ(t s)σ s dz s, t, where σ is an F Z -adapted Gaussian chaos process which is continuous in probability and ϕ L 2 loc is a deterministic function. Theorem X is an F Z -semimartingale if and only if ϕ W 1,2 loc (Ê+).

8 29 Barrett Lectures Chaos semimartingales Let S p denote the space of special semimartingales X = (X t ) t [,T] with canonical decomposition X = X + M + A satisfying [M] p/2 T, V [,T] (A) L 1. To show the previous theorem we use and prove the following result: Theorem Let X denote an F-semimartingale satisfying E[X t F s] n k= H k, s t T. Then X S p for all p 1, and in particular a quasimartingale. Futhermore, if X = X + A + M denotes the F-canonical decomposition of X then A t, M t n k= H k for all t [, T]. When X is a Gaussian processes this result was shown by Stricker (1983).

9 Key references 29 Barrett Lectures Knight, F. B. (1992). Foundations of the prediction process, Volume 1 of Oxford Studies in Probability. New York: The Clarendon Press Oxford University Press. Oxford Science Publications. Basse, A. and J. Pedersen (29). Lévy driving moving averages and semimartingales. Stochastic Process. Appl. In Press. Basse, A. and S.-E. Graversen (29). Chaos processes and semimartingales. Work in progress Basse, A. (28a). Gaussian moving averages and semimartingales. Electron. J. Probab. 13, no. 39,