Ambit Processes: Aspects of Theory and Applications Ole E. - - PowerPoint PPT Presentation

Ambit Processes: Aspects of Theory and Applications

Ole E. Barndor¤-Nielsen

Aarhus University

July 21, 2010

Ambit …elds and processes

time space x t Y

t x

At x set ambit

Figure: Ambit framework

(t(θ), x(θ)) Xθ At(θ)(x(θ))

❅

• x

Figure: Ambit framework

This talk is based on collaborations with: Fred Espen Benth José Manuel Corcuera Svend Erik Graversen Mark Podolskij Jürgen Schmiegel Almut Veraart Supported by the Thiele Centre and by CREATES

Synopsis

I Ambit …elds and processes I Brownian semistationary processes (BSS processes) I Multipower variations I Turbulence I Forwards and spots in energy markets

Ambit …elds Yt (x) = µ +

Z

At(x) g (ξ, s; t, x) M (dξ, ds)

+

Z

Dt(x) q (ξ, s; t, x) as(ξ)dξds.

Here At (x), and Dt (x) are termed ambit sets, g and q are deterministic (matrix) functions, σ 0 and a are stochastic …elds, and M is a random measure on Rd R. Typically, M will be constructed from a Lévy basis L. Example: M (dξ, ds) = σs (ξ) L (dξ, ds) where the random …eld σ is referred to as the intermittency or volatility. This may itself be modeled as a positive ambit …eld. Ambit processes Xθ = Yt(θ) (x (θ)) where τ (θ) = (x (θ) , t (θ)) is a curve in space-time.

Lévy basis A Lévy basis L is an independently scattered random measure (on Rd) whose values are in…nitely divisible. L is said to be homogeneous if for any (bounded) Borel set the law

• f L (B) is the same as the law of L (B + z) for all z 2 Rd.

White noise is the special type of Lévy basis L for which L is Gaussian and homogeneous, with mean 0 and variance E n L (B)2o = leb (B).

Stationary regimes To model …elds and processes that are stationary in time and space, the damping functions g and q are chosen to have the form g (ξ, s; t, x) = g (x ξ, t s) q (ξ, s; t, x) = q (x ξ, t s) and the ambit sets are taken to be homogeneous and nonanticipative, i.e. At (x) is of the form At (x) = A + (x, t) where A only involves negative time coordinates, and similarly for Dt (x). Furthermore, in case M (dξ, ds) = σs (ξ) L (dξ, ds), fσs (ξ)g and fas (ξ)g are assumed to be stationary …elds, and the Lévy basis L is homogeneous..

If only stationarity in time is required then one may choose g (ξ, s; t, x) = g (ξ, t s; x) q (ξ, s; t, x) = q (ξ, t s; x) and the compensator n of the Lévy basis can be taken to satisfy n (dy; dξ, ds) = ν (dy; dξ) ds.

Turbulence Most extensive data sets on turbulent velocities only provide the time series of the main component of the velocity vector (i.e. the component in the main direction of the ‡uid ‡ow) at a single location in space. The turbulence modelling framework then particularises to the class of BSS models (Brownian semistationary processes). We discuss this class next, returning to turbulence settings later.

BSS processes The class of Brownian semistationary (BSS) processes is the subclass of the ambit processes corresponding to a degenerate space component (null-spatial case) and having the form Yt =

Z t

∞ g(t s)σsW (ds) +

Z t

∞ q(t s)asds

where W is Brownian motion on R, σ and a are cadlag processes and g and q are deterministic continuous memory function on R, with g (t) = q (t) = 0 for t 0. 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 W + q a leb.

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

Z t

0 σsdWs +

Z t

0 asds.

Note The BSS processes Yt =

Z t

∞ g(t s)σsW (ds) +

Z t

∞ q(t s)asds

are, in general, not semimartingales

Important example Suppose Y = g σ W with g (t) = tν1eλt.

1 2 < ν < 1

nonSM ν = 1 SM 1 < ν < 3

2

nonSM

A key object of interest, whether for BSM or BSS processes, is the integrated squared volatility σ2+

t

=

Z t

0 σ2 s ds

for any t 2 R. Realised multipower variations (RMPVs) of Y can be used to estimate elements of the main terms in Y , i.e. Y = g σ W . In particular they can be used to draw inference on σ2+

t

• r on the

small scale behaviour of the damping function g. However, because of the nonsemimartingale character of BSS processes the probabilistic limit theory of RMPVs for BSS processes is decisively di¤erent from that for BSM processes. Next: Brief review of the theory of RMPVs for BSS processes.

Multipower Variation In the following the process Y is assumed to be observed at time points ti = i∆n with i = 0, . . . , [t/∆n] and ∆n ! 0. A realised multipower variation of a stochastic process Y is de…ned as an

• bject of the type

[t/∆n]k+1

∑

i=1 k

∏

j=1

j∆n

i+j1Y jpj

where ∆n

i Y = Y i

n Y i1 n

and p1, . . . , pk 0.

For the case where Y 2 BSM, i.e. Y = σ W + a leb, it was established in [BNGJPS07] that np+/21

[t/∆n]k+1

∑

i=1 k

∏

j=1

j∆n

i+j1Y jpj ucp

• ! µp1 µpk

Z t

0 jσsjp+ds

where p+ = ∑k

j=1 pj and µp = E[jujp], u N(0, 1).

Moreover, under a regularity condition on the volatility process σ, there is an associated stable central limit theorem: pn

• np+/21

[t/∆n]k+1

∑

i=1 k

∏

j=1

j∆n

i+j1Y jpj µp1 µpk

Z t

0 jσsjp+ds

• st
• !

p C

Z t

0 jσsjp+dBs

where B is another Brownian motion, de…ned on an extension of the probability space (Ω, F, (Ft)t0, P) and independent of F, and C is a known constant.

Normalised RMPVs This was for BSM processes. When we pass on to BSS processes the situation changes signi…cantly (recall that BSS processes are generally not of BSM type). In order to obtain similar limit results one has to look at normalised RMPV. ([BNSch09], [BNCP08], [BNCP10]) The techniques for obtaining the theorems now, among other things, involves establishing a CLT for triangular arrays of Gaussian variables, using Malliavin calculus. (This CLT is of some independent interest.)

Furthermore, while the focus for BSM, and more generally for Ito processes, has been on inference concerning the quadratic variation

• f the processes, and especially the σ2+ component, in the BSS

setting the interest is not just in regard to σ2+ but also concerns inference on the damping function g. For inference on g it is pertinent to study the behaviour of ratios

• f RMPVs and, in fact, not only of ’classical’ (∆ case) RMPVs but

also RMPVs based on second order di¤erences ( case) instead of …rst order di¤erences (something that would make no di¤erence in the BSM case).

Normalised RMPVs The normalised RMPVs of types ∆ and are de…ned as MPV ∆(Y , p1, . . . , pk)n

t = ∆n(τ∆ n )p+ [t/∆n]k+1

∑

i=1 k1

∏

l=0

j∆n

i+lY jpl+1

(1) MPV (Y , p1, . . . , pk)n

t = ∆n(τ n )p+ [t/∆n]k+1

∑

i=2 k1

∏

l=0

jn

i+lY jpl+1

(2) where ∆n

i Y = Yi∆n Y(i1)∆n and

n

i Y = Xi∆n 2Y(i1)∆n + Y(i2)∆n, where pl 0 and

p+ = ∑k

l=1 p and where τ∆ n is a known function of g.

We have established LLNs and CLTs for inference on σ2+. Except for the normalisation these results are similar in nature to those for BSMs but the proofs are very di¤erent. (All speci…cs omitted here.)

Inference on g For inference on g it is (as already mentioned) pertinent to study Realised Variation Ratios (RVRs), i.e. ratios of RMPVs. Note The relevant Realised Variation Ratios (RVRs), of types ∆ and , do not involve normalisation of RMPVs. Thus, in this sense, the limit theory for RVRs is ’nonparametric’.

Inference on g. An example: To illustrate, consider the key case where g (t) = tν1eλt. The interesting situations are where ν 2 1

2, 1

[

• 1, 3

2

• . Using

∆-RMPVs it is only possible to establish a CLT for ν in 1

2, 1

• .

Passing to -RMPVs this can be strengthened to ν 2 1

2, 1

[

• 1, 5

4

• .

But the range v 2 [ 5

4, 1) is of particular interest in the context of

turbulence, and that can be covered only by modi…cations for which the convergence rate is somewhat slower than the pn rate that holds in the other intervals. (Details omitted.)

Turbulence Null-spatial case (spatial dimension 0). As mentioned earlier, most extensive data sets on turbulent velocities only provide the time series of the main component of the velocity vector (i.e. the component in the main direction of the ‡uid ‡ow) at a single location in space. The turbulence modelling framework then particularises to the class of BSS models

• 20

40 60 80 100 lag 5000 10000 15000 20000 RQV

Realised quadratic variation for the Brookhaven data set

Figure: S2 Brookhaven data

• 100

104 106 lag 0.01 0.1 1 10 S2

Loglogplot of the second order structure function for the Brookhaven data set. Red line has slope 23.

Figure: S2 Brookhaven data

Detailed analysis of the data and physical insights have made it possible to determine speci…cations of the ingredients of the BSS framework resulting in a closely realistic general model, describing the observations from a variety of extensive empirical and experimental recordings. The model does, in particular, reproduce the empirical traits just shown and also give a universal description

• f the distributions of the velocity di¤erences. ([BNBSch04],

[BNSch07a], [BNSch08a], [BNSch08b]) Here we omit the details, except for a brief discussion of the use of Realised Variation Ratios for the study of the small scale nature of g.

Small scale nature of g and Realised Variation Ratios Example Realised Bipower Variation/Realised Quadratic Variation: (∆ case) RVRn

t = MPV ∆(Y , 1, 1)n t

MPV ∆(Y , 2, 0)n

t

=

π 2 ∑ [t/∆n]k+1 i=1

j∆n

i Y jj∆n i+1Y j

∑

[t/∆n]k+1 i=1

j∆n

i Y j2

.

500 1000 1500 2000 0.8 1.0 1.2 1.4

RVR∆

t (δ)

t = 10000 δ = 1

Figure: RVR

Turbulence Tempo-spatial settings: d 1 case (spatial dimension d) We specify the d-dimensional velocity vector Y of a stationary turbulent ‡uid, at position x in Rd and at time time t, by Yt (x) = µ +

Z

A+(x,t) g (t s, x ξ) σs (ξ) W (dξ, ds)

+

Z

D+(x,t) q (t s, x ξ) σ2 s (ξ)dξds.

where W is Brownian white noise. Next Choice of ambit sets and volatility …elds from physical theory and stylised features.

✲ ✻

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

x t

Figure: Sound cone

Choice of volatity …eld ([BNSch04]) Arithmetic: σ2

t (x) =

Z

C +(x,t) h (t s, x ξ) L (dξ, ds)

Example: OU^ …elds Let L be a positive homogeneous Lévy basis and de…ne σ2

t (x) =

Z

^+(x,t) eλ(ts)L (dξ, ds) .

For …xed x this determines a stationary Markovian process.

Choice of volatity …eld Geometric: σ2

t (x) = exp

Z

C +(x,t) h (t s, x ξ) L (dξ, ds)

• Example: SI …elds

By suitable choice of the kernel h and the ambit set C we obtain a version of σ that constitutes a continuous analogue of multiplicative cascade processes in the description of turbulent energy-dissipation …elds.

The SI speci…cation allows explicit analytic calculations, in particular showing scaling relations of n-point correlators. The correlators are de…ned by c ((x1, t1) m1; , (xn, tn) mn) = m ((x1, t1) , m1; , (xn, tn) , mn) m ((x1, t1) , m1) m ((xn, tn) , mn) where m ((x1, t1) m1; , (xn, tn) mn) = E fYt1 (x1)m1 Ytn (xn)mng . Note Cancellation of factors

t t′

• ✲

✻

The probabilistic limit behaviour of normalised RQV in this tempo-spatial (1 + 1) setting is presently under study. In particular, the limit in probability of the nRQV is typically an integral of the squared volatility …eld over the boundary of the ambit set A, the measure on the boundary being determined by the nature of the damping function g.

Figure: Hornsrev

Forwards and Spots in Energy Markets joint work with Fred Espen Benth and Almut Veraart Using the ambit framework we aim to model both the forward and the spot price processes directly and coherently, encompassing main stylised features. Stylised features:

I Samuelson e¤ect: This e¤ect refers to the empirical trait,

• bserved on forward prices in power markets, that when the

time to maturity approaches 0 the volatility of the forward starts increasing and converges to the volatility of the spot eventually.

I High correlation between neighbouring contracts near

maturity.

Forward The forward price is modelled as an ambit …eld ft(u) that is stationary in time: ft(x) =

Z

At(x) k (ξ, t s; x) σs (ξ) L (dξ, ds)

where the ’space variable’ x is the time to maturity. Three ingredients for speci…cation:

I Deterministic damping function k I Volatility …eld σ (σs (ξ) stationary in s) I Family of ambit sets At (x) = A0 (x) + (0, t)

T=t+u t ξ s t u T=t+u t ξ s t u

Figure: Forwards

Under weak conditions, the forward ft(u) will converge, as time to maturity tends to zero, to a process st of the form st =

Z t

∞ k (ξ, t s) σs (ξ) L (dξ, ds) .

This process is then taken as the model for the spot price. Further, in this setup, we can indeed ‡exibly model

I The Samuelson e¤ect I High correlation between neighbouring contracts near maturity

Extension to include delivery periods Forward price ft(x, τ) =

Z

At(x,τ) k (ξ, χ, t s; x, τ) σs (ξ, χ) L (dξ, dχ, ds)

where τ denotes the period of delivery, starting at time t + x.

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