BSS Processes and Intermittency/Volatility Realised Quadratic - - PowerPoint PPT Presentation

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

BSS Processes and Intermittency/Volatility

Turbulence Stochastics Ole E. Barndor¤-Nielsen and Jürgen Schmiegel

Thiele Centre Department of Mathematical Sciences University of Aarhus

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

I BSS models I Turbulence background I BSS models (cont.) I Inference on intermittency/volatility I Further ongoing work I Some open questions

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Brownian semistationary (BSS) processes: Yt =

Z t

∞ g(t s)σsdBs +

Z t

∞ q(t s)asds

(1) where B is Brownian motion, g and q are square integrable functions on R, with g (t) = q (t) = 0 for t < 0, and σ and a are cadlag processes. When σ and a are stationary, as will be assumed throughout this talk, then so is Y . It is sometimes convenient to indicate the formula for Y as Y = g σ B + q a Leb. (2)

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

We consider the BSS processes to be the natural analogue, in stationarity related settings, of the class BSM of Brownian semimartingales.

I The BSS processes are not in general semimartingales

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

The key object of interest is the integrated variance (IV) σ2+

t

=

Z t

0 σ2 s ds

We shall discuss to what extent realised quadratic variation

• f Y can be used to estimate σ2+

t . I Note that the relevant question here is whether a

suitably normalised version of the realised quadratic variation, and not necessarily the realised quadratic variation itself, converges in probability/law.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

In semimartingale theory the quadratic variation [Y ] of Y is de…ned in terms of the Ito integral Y Y , as [Y ] = Y 2 2Y Y . In that setting [Y ] equals the limit in probability as δ ! 0 of the realised quadratic variation [Yδ]

• f Y de…ned by

[Yδ]t =

bt/δc

∑

j=1

• Yjδ Y(j1)δ

2 (3) where bt/δc is the largest integer smaller than or equal to t/δ. It is this latter de…nition of quadratic variation that we will use here.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

White noise case: Yt (x) = µ +

Z

At(x) g (t s, jξ xj) σs (ξ) W (dξ, ds)

+

Z

Dt(x) q (t s, jξ xj) as (ξ) dξds.

Here At (σ) and Dt (σ) are termed ambit sets . Lévy case: σ2

t (x) =

Z

Ct(x) h (t s, jξ xj) L (dξ, ds)

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

(t(w), σ(w)) Xw At(w)(σ(w))

❅

• Figure: Ambit processes

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

In turbulence the basic notion of intermittency refers to the fact that the energy in a turbulent …eld is unevenly distributed in space and time. The present presentation is part of a project that aims to construct a stochastic process model of the …eld of velocity vectors representing the ‡uid motion, conceiving of the intermittency as a positive random …eld with random values σt (x) at positions (x, t) in space-time.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

However, most extensive data sets on turbulent velocities

• nly provide the time series of the main component (i.e. the

component in the main direction of the ‡uid ‡ow) of the velocity vector at a single location in space. In the present talk the focus is on this latter case, but in the concluding Section some discussion will be given on the further intriguing issues that arise when addressing tempo-spatial settings.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

For simplicity we now assume that σ? ?B and that q = 0, i.e. there is no drift term in Y and Yt =

Z t

0 g (t s) σsdBs.

However, at the end of the talk some discussion will be given

• n more general settings.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Let Z = fZtgt2R denote a second order stationary stochastic process, possibly complex valued, of mean 0 and continuous in quadratic mean. Recall that Z is said to be a moving average process if it is of the form Zt =

Z ∞

∞ φ (t s) dΞ s

(4) where φ is an, in general complex, deterministic and square integrable function and where the process Ξ has orthogonal increments with E n jdΞ

t j2o

= ̟dt for some constant ̟ > 0; …nally, the integral in (4) is de…ned in the quadratic mean sense.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

A second order stationary process (of mean 0 and continuous in quadratic mean) is called regular (or linearly regular) provided its future values cannot be predicted by linear

• perations on past values without error. Such processes can

be written in the continuous time Wold decomposition form Zt =

Z t

∞ φ (t s) dΞs + Vt

(5) where φ is an, in general complex, deterministic square integrable function satisfying φ (s) = 0 for s < 0 and where the process Ξ has orthogonal increments with E n jdΞtj2o = ̟dt for some constant ̟ > 0. Finally, the process V is nonregular (i.e. predictable by linear operations

• n past values without error).

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Given Z and neglecting sets of measure 0, both φ and Ξ are uniquely determined, φ up to a factor of modulus 1 and the driving process Ξ in the L2 sense and up to an additive constant.

I A slightly stronger condition than regularity is the

requirement that Z be completely nondeterministic, meaning that \t2Rsp fZs : s tg = f0g In this case Z = φ Ξ with φ real and uniquely determined up to a real constant of proportionality; and the same is therefore true of Ξ.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

We begin by recalling a classical necesssary and su¢cient condition, due to [Kni92], for the semimartingale property of a process X of the form Xt =

Z t

∞ g (t s) dBs.

(6) Knight’s Theorem says that (Xt)t0 is a semimartingale in the

• F B,∞

t

• t0 …ltration if and only if

g (t) = c +

Z t

0 b (s) ds

(7) for some c 2 R and a square integrable function b.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Example An example of some particular interest is where g (t) = tαeλt for t 2 (0, ∞) and some λ > 0. In order for the integral g σ B to exist α is required to be greater than 1

2, and for g to be of the

form (7) we must have α > 1

• 2. In other words, the

nonsemimartingale cases are α 2 1

2, 1 2

• .

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Recall that the integrated variance (IV) σ2+

t

=

Z t

0 σ2 s ds

(8) is the main object of interest and that we wish to discuss the extent to which realised quadratic variation of Y can be used to estimate σ2+

t .

Remark It is convenient to de…ne σ2+

t

also for t < 0 by the same expression.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Note that, as regards inference on σ2+, there may be notable di¤erences between cases where g is positive on all

• f (0, ∞) and those where g (t) = 0 for t > l for some

l 2 (0, ∞).

I Recall that σ?

?B.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Suppose g (t) = 0 for t > 1. For any t 2 R and u < t, Yt Ytu =

Z t

tu g (t s) σsdBs +

Z tu

∞ fg (t s) g (t s

It is illuminating to rewrite this as Yt Ytu =

Z 0

∞ φu (v) σv+tdBv+t

(9) de…ning φu by φu (v) = 8 < : g (v) for 0 v < u g (v) g (v u) for u v < ∞ .

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

This implies in particular that the conditional variance of Yt Ytu given the process σ takes the form E n (Yt Ytu)2 jσ

• =

Z ∞

ψu (v) σ2

tsds

(10) where ψu (v) = 8 < : g2 (v) for 0 v < u fg (v) g (v u)g2 for u v < ∞ . And unconditionally E n (Yt Ytu)2o = E

• σ2

Z ∞ ψu (v) dv. Hence

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

E n (Yt Ytu)2 jσ

• E

n (Yt Ytu)2o = R ∞

0 ψu (v) σ2 tvdv

E fσ2

0g

R ∞

0 ψu (v) dv =

Z ∞

¯ σ2

tv πu (d

where ¯ σ2 = σ2/E

• σ2
• .

and πu is the probability measure on [0, 1] having pdf ψu (v) /c (δ) with c (δ) = 2 kgk2 (1 r (u)) with r being the autocorrelation function of Y .

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Example g (t) = eλt1(0,1) (non-semimartingale case). Then ψδ (v) = e2λv 8 > > < > > : 1 for 0 v < δ

• eλδ 1

2 for δ v < 1 e2λδ for 1 v < 1 + δ for 1 + δ v . It follows that πδ ! π where π = 1 1 + e2λ δ0 + e2λ 1 + e2λ δ1 where δx denotes the delta measure at x.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Example g (t) = tα (1 t)β with 1

2 < α and β 1.

The …rst inequality ensures existence of the stochastic integral g σ B, and if α is less than 1

2 then we are in the

nonsemimartingale situation In this case π = δ0.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Now, de…ne the normed realised quadratic variation [Yδ] of Y as [Yδ] = δ c (δ) [Yδ] . Then E n [Yδ]tjσ

• =

Z ∞

( δ

bt/δc

∑

j=1

σ2

jδv

) πδ (dv) . Suppose that πδ converges weakly, as δ ! 0, to a probability measure π on [0, l], i.e. πδ

w

! π. (11)

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Then we have E n [Yδ]tjσ

• !

Z 1

• σ2+

tv σ2+ v

• π (dv) .

(12) In particular, if π = δ0, the delta measure at 0, then E n [Yδ]t

• ! σ2+

t .

We will refer to this case by saying that the model for Y is volatility memoryless. (Under a mild restriction the same holds for general g and σ provided the upper limit of the integral is taken as ∞.)

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

On the other hand, if π is a convex combination of the delta measure at 0 and l, i.e. π = θδ0 + (1 θ) δl for some θ 2 (0, 1), then E n [Yδ]

• ! θσ2+

t

+ (1 θ)

• σ2+

tl σ2+ l

• .

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Suppose that l = 1 and that g > 0 is a square integrable continuously di¤erentiable function on (0, 1). If, as δ ! 0, we have c (δ)1 δ2 = o (1) and if the probability measure πδ converges weakly to a probability measure π on [0, 1] then π is concentrated on the endpoints 0 and 1.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

To formulate conditions ensuring that π exists and equals δ0, let Ψδ (u) =

Z u

0 ψδ (v) dv

and ¯ Ψδ (u) =

Z 1+δ

1+δu ψδ (v) dv,

so that c (δ)1 Ψδ is the distribution function of πδ.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Proposition If (δ)1 δ2 = o (1) and if for some ε0 2 (0, 1) ¯ Ψδ (ε) Ψδ (ε) ! 0 for all ε 2 (0, ε0) then πδ

w

! δ0.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Remark In case c (δ) δ2 it may happen that π is absolutely continuous on [0, 1].

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Above it was assumed that l < ∞ (in fact, we took l = 1). The following Example has l = ∞. Example For g (t) = tαeλt (α > 1

2) it can be shown,

using detailed calculations for this special case given in [BNCP08], that π = δ0.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

We now seek conditions under which the conditional variance of the normalised realised quadratic variation tends to 0 as δ ! 0, i.e. Varf[Yδ]jσg ! 0. (13) In that case and provided E n [Yδ]jσ

• !

Z ∞

• σ2+

tv σ2+ v

• π (dv)

(14) holds we have that, conditionally on σ, [Yδ]t

p

!

Z ∞

• σ2+

tv σ2+ v

• π (dv) .

(15)

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Let n = bt/δc and ∆n

j Y = Yjδ Y(j1)δ. Via the

Cauchy-Schwarz inequality and using that for any pair X and Y of mean zero normal random variables we have CovfX 2, Y 2g = 2CovfX, Y g2. (16) it can be shown that

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Varf[Yδ]tjσg 2K (σ)2 δ + 2 δ

n1

∑

k=1

k ¯ ck (δ) +

∞

∑

k=n

¯ ck (δ) !! where K (σ) = sup0st σ2

s (as σ is assumed càdlàg,

K (σ) < ∞ a.s.) and ¯ ck (δ) = ck (δ) c (δ) with ck (δ) = δ

Z 1

0 ψδ ((k + u) δ) du.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Thus, for Varf[Yδ]jσg ! 0 to be valid it su¢ces to have δ

n1

∑

k=1

k ¯ ck (δ) ! 0 and

∞

∑

k=n

¯ ck (δ) ! 0. Example If g (t) = tα (1 t)β with 1

2 < α and β 1

then the above two conditions hold and [Yδ]

p

! σ2+.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

[Yδ]t bt/δc \ fVar fY0gg (1 ˆ r (δ))

p

!

Z ∞

• σ2+

tv σ2+ v

• π (dv) .

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

So far we have assumed that q = 0 and σ? ?B. In joint work (near completion) with José Manuel Corcuera and Mark Podolski these conditions have been substantially weakened. This more re…ned analysis has shown that [Yδ]

p

! σ2+ in wider generality and using the theory of multipower variation and recent powerful results of Malliavin calculus, due to Nualart, Peccati et al, a feasible CLT for [Yδ] has been established. The results are further extended to consistency and feasible CLTs for multipower variations, in particular for bipower variation (as is essential for inference on σ2+).

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Above only the case of time-wise behaviour at a single point in space was considered. In the general turbulence setting, space and the velocity vector are three dimensional. There the questions, analogous to those disussed above, are substantially more intricate, major di¤erences occurring

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

already for the case of a one-dimensional space component. The realised variation ratio (RVR) is de…ned by RVRt = π 2 [Yδ][1,1]

t

[Yδ]t where [Yδ][1,1] is the bipower variation, i.e. [Yδ][1,1]

t

=

bt/δc1

∑

j=1

• Yjδ Y(j1)δ
• Y(j+1)δ Yjδ
• .

The properties of RVR are presently under study in joint work with Neil Shephard.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

I r = g g

?

I In non-semimartingale cases, can one give a natural

meaning to dXt such that d [X]t = (dXt)2?

I Volatility modulated Volterra Processes (VMVP):

Yt =

Z ∞

∞ Kt (s) σsB (ds)

Yt (x) =

Z ∞

∞ Kt (ξ, s; x) σs (ξ) W (dξds) I Relevance for Finance?

Arbitrage?

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Barndor¤-Nielsen, O.E., Corcuera, J.M. and Podolskij,

• M. (2007): Power variation for Gaussian processes with

stationary increments. (Submitted.) Barndor¤-Nielsen, O.E., Corcuera, J.M. and Podolskij,

• M. (2008): Power variation for BSS processes. (In
• preparation. Title preliminary.)

Barndor¤-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M. and Shephard, N. (2006): A central limit theorem for realised power and bipower variations of continuous semimartingales. In Yu. Kabanov, R. Liptser and J. Stoyanov (Eds.): From Stochastic Calculus to Mathematical Finance. Festschrift in Honour of A.N.

• Shiryaev. Heidelberg: Springer. Pp. 33-68.

Barndor¤-Nielsen, O.E., Graversen, S.E., Jacod, J., and Shephard, N. (2006): Limit theorems for bipower variation in …nancial econometrics. Econometric Theory 22, 677-719.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Barndor¤-Nielsen, O.E., Blæsild, P. and Schmiegel, J. (2004): A parsimonious and universal description of turbulent velocity increments. Eur. Phys. J. B 41, 345-363. Barndor¤-Nielsen, O.E. and Schmiegel, J. (2004): Lévy-based tempo-spatial modelling; with applications to turbulence. Uspekhi Mat. NAUK 59, 65-91. Barndor¤-Nielsen, O.E. and Schmiegel, J. (2006b): Time change and universality in turbulence. Research Report 2007/8. Thiele Centre for Applied Mathematics in Natural Science. Barndor¤-Nielsen, O.E. and Schmiegel, J. (2007): Ambit processes; with applications to turbulence and cancer growth. In F.E. Benth, Nunno, G.D., Linstrøm, T., Øksendal, B. and Zhang, T. (Eds.): Stochastic Analysis and Applications: The Abel Symposium 2005. Heidelberg: Springer. Pp. 93-124.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Barndor¤-Nielsen, O.E. and Schmiegel, J. (2007): A stochastic di¤erential equation framework for the timewise dynamics of turbulent velocities. Theory Prob. Its Appl. (To appear.) Barndor¤-Nielsen, O.E., Schmiegel, J. and Shephard, N. (2008): Realised variation ratio. (In preparation.) Barndor¤-Nielsen, O.E. and Schmiegel, J. (2007): Time change, volatility and turbulence. To appear in Proceedings of the Workshop on Mathematical Control Theory and Finance. Lisbon 2007. Barndor¤-Nielsen, O.E. and Shephard, N. (2001): Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in …nancial economics (with Discussion). J. R. Statist. Soc. B 63, 167-241. Barndor¤-Nielsen, O.E. and Shephard, N. (2006): Multipower variation and stochastic volatility. In M. do Rosário Grossinho, A.N. Shiryaev, M.L. Esquível and

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

P.E. Oliveira: Stochastic Finance. New York: Springer.

• Pp. 73-82.

Barndor¤-Nielsen, O.E. and Shephard, N. (2008): Financial Volatility in Continous Time. (2007). Cambridge University Press. (To appear.) Barndor¤-Nielsen, O.E., Schmiegel, J. and Shephard, N. (2006): Time change and universality in turbulence and …nance. (Submitted.) Basse, A. (2007a): Spectral representation of Gaussian

• semimartingales. (Unpublished manuscript.)

Basse, A. (2007b): Gaussian moving averages and

• semimartingales. (Unpublished manuscript.)

Basse, A. (2007c): Representation of Gaussian semimartingales and applications to the covariance

• function. (Unpublished manuscript.)

Baudoin, F. and Nualart, D. (2003): Equivalence of Volterra processes. Stoch. Proc. Appl. 107, 327-350.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Bender, C. and Marquardt, T. (2007): Stochastic calculus for convoluted Lévy processes. (Unpublished manuscript.) Decreusefond, L. (2005): Stochastic integration with respect to Volterra processes. Ann. I. H. Poincaré PR41, 123-149. Decreusefond, L. and Savy, N. (2006): Anticipative calculus with respect to …ltered Poisson processes. Ann.

• I. H. Poincaré PR42, 343-372.

Guyon, X. (1987): Variations des champs gaussian stationaires: applications à l’identi…cation. Proba. Th.

• Rel. Fields 75, 179-193.

Guyon, X. and Leon, J. (1989): Convergence en loi des H-variation d’un processus gaussien stationaire. Ann.

• Inst. Henri Poincaré 25, 265-282.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Hult, H. (2003): Approximating some Volterra type stochastic intergrals with applications to parameter

• estimation. Stoch. Proc. Appl. 105, 1-32.

Istas, J. (2007): Quadratic variations of spherical fractional Brownian motions. Stoch. Proc. Appl. 117, 476-486. Knight, F. (1992): Foundations of the Prediction

• Process. Oxford: Clarendon Press.

Marquardt, T. (2006): Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099-1126. Shiryaev, A.N. (2006): Kolmogorov and the turbulence. Research Report 2006-4. Thiele Centre for Applied Mathematics in Natural Science. Shiryaev, A.N. (2007): On the classical, statistical and stochastic approaches to hydrodynamic turbulence.

Outline BSS models

De…nition Key object of interest Realised Quadratic Variation

Turbulence background

Ambit processes Intermittency

BSS models (cont.)

Canon Semi- and non-semimartingale questions

Inference on inter- mittency/volatility

Introduction Increment process Examples RQV and IV Conditions ensuring

πδ ! δ0

Consistency Feasible version

Further ongoing work

Relaxing assumptions Realised Variation Ratio

Some open

Research Report 2007-2. Thiele Centre for Applied Mathematics in Natural Science.