2009 Barrett Lectures

When is a Moving Average a Semimartingale?

Andreas Basse

Ph.D.-student under supervision of Jan Pedersen, Thiele Centre, University of Aarhus, Denmark.

2009 Barrett Lectures at The University of Tennessee: Stochastic Analysis and its Applications

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Semimartingales

Let F = (Ft )t≥0 denote a filtration. Then X = (Xt )t≥0 is said to be an F-semimartingale if it can be written as

Xt = X0 + Mt + At , t ≥ 0,

where M is a càdlàg F-local martingale, A is an F-adapted càdlàg process of bounded variation and X0 is F0-measurable. t A càdlàg process X induces a “reasonable” stochastic integral 0 Ys dXs if and only if it is a semimartingale. R

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Moving averages

The focus of this talk is on the semimartingale property of moving averages X = (Xt )t≥0, given by

t Xt = ϕ(t − s) dZs, t ≥ 0, Z0

where Z = (Zt )t≥0 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[0,1] then Xt = Zt − Z(t−1)∨0 (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) = tβ−1/2 (in most cases not a semimartingale).

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures F. Knight

Let X = (Xt )t≥0 be given by

t Xt = ϕ(t − s) dZs, t ≥ 0, Z0 where Z is a Brownian motion.

Theorem (F. Knight (1992))

Z 1,2 X isan F -semimartingale if and only if ϕ ∈ Wloc (Ê+), i.e.

t ϕ(t) = α + h(s) ds, t ≥ 0, Z0

2 where α ∈ Ê and h ∈ Lloc(λ).

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Lévy driving moving averages

Let X = (Xt )t≥0 be given by

t Xt = ϕ(t − s) dZs, t ≥ 0, Z0

2 where Z = (Zt )t≥0 is a Lévy process with characteristic triplet (γ, σ ,ν).

Theorem X isan F Z -semimartingale if and only if 1 Z is of bounded variation: ϕ is of bounded variation,

2 2 1,2 σ > 0: ϕ ∈ Wloc (Ê+), 3 2 Z is of unbounded variation and σ = 0: ϕ is absolutely continuous on Ê+ with a density ϕ0 satisfying

t 1 |xϕ0(s)|2 ∧|xϕ0(s)| ν(dx) ds < ∞, for all t > 0. Z Z 0 −1

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures A corollary

Let X be given by t β− 1 1 Xt = (t − s) 2 dZs, β 6= . Z0  2 

Corollary (The Riemann-Liouville fractional integral) σ2 > 0:X isan F Z -semimartingale if and only if β > 1. σ2 = 0:X isan F Z -semimartingale if and only if β > 1, 1 2 β = 1 and −1 x |log(|x|)| ν(dx) < ∞, 1 R 1 2/(3−2β) 2 <β< 1 and −1|x| ν(dx) < ∞. R

Thus if Z is α-stable then X is an F Z -semimartingale if and only if 3 1 β > 2 − α when α ∈ [1, 2], 1 β > 2 when α ∈ (0, 1).

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Gaussian chaos processes

Let (Zt )t≥0 denote a Brownian motion.

(Yt )t≥0 is said to be a Gaussian chaos process if there exists n ≥ 1 such that n Yt ∈⊕k=0Hk for all t ≥ 0. Let us now consider X of the form

t Xt = ϕ(t − s)σs dZs, t ≥ 0, Z0

where σ is an F Z -adapted Gaussian chaos process which is continuous in 2 probability and ϕ ∈ Lloc is a deterministic function.

Theorem

Z 1,2 X isan F -semimartingale if and only if ϕ ∈ Wloc (Ê+).

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Chaos semimartingales

p Let S denote the space of special semimartingales X = (Xt )t∈[0,T ] with p/2 1 canonical decomposition X = X0 + M + A satisfying [M]T , V[0,T ](A) ∈ L . To show the previous theorem we use and prove the following result:

Theorem Let X denote an F-semimartingale satisfying

n E[Xt |Fs] ∈⊕k=0Hk , ∀ 0 ≤ s ≤ t ≤ T . Then X ∈Sp for all p ≥ 1, and in particular a quasimartingale. Futhermore, if n X = X0 + A + M denotes the F-canonical decomposition of X then At , Mt ∈⊕k=0Hk for all t ∈ [0, T ].

When X is a Gaussian processes this result was shown by Stricker (1983).

Andreas Basse When is a Moving Average a Semimartingale? 2009 Barrett Lectures Key references

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 (2009). Lévy driving moving averages and semimartingales. Stochastic Process. Appl. In Press. Basse, A. and S.-E. Graversen (2009). Chaos processes and semimartingales. Work in progress Basse, A. (2008a). Gaussian moving averages and semimartingales. Electron. J. Probab. 13, no. 39, 1140–1165.

Andreas Basse When is a Moving Average a Semimartingale?