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 ambit set A t x � � � � Y t x t space x Figure: Ambit framework

X θ ( t ( θ ) , x ( θ )) ❅ A t ( 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 Z Y t ( x ) = µ + A t ( x ) g ( ξ , s ; t , x ) M ( d ξ , d s ) Z + D t ( x ) q ( ξ , s ; t , x ) a s ( ξ ) d ξ d s . Here A t ( x ) , and D t ( 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 R d � R . Typically, M will be constructed from a Lévy basis L . M ( d ξ , d s ) = σ s ( ξ ) L ( d ξ , d s ) where the random …eld Example : σ is referred to as the intermittency or volatility . This may itself be modeled as a positive ambit …eld. Ambit processes X θ = Y t ( 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 R d ) whose values are in…nitely divisible. L is said to be homogeneous if for any (bounded) Borel set the law of L ( B ) is the same as the law of L ( B + z ) for all z 2 R d . White noise is the special type of Lévy basis L for which L is Gaussian and homogeneous, with mean 0 and variance n L ( B ) 2 o E = 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. A t ( x ) is of the form A t ( x ) = A + ( x , t ) where A only involves negative time coordinates, and similarly for D t ( x ) . Furthermore, in case M ( d ξ , d s ) = σ s ( ξ ) L ( d ξ , d s ) , f σ s ( ξ ) g and f a s ( 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 ( d y ; d ξ , d s ) = ν ( d y ; d ξ ) d s .

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 Z t Z t Y t = � ∞ g ( t � s ) σ s W ( d s ) + � ∞ q ( t � s ) a s d s 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. Z t Z t Y t = 0 σ s d W s + 0 a s d s . Note The BSS processes Z t Z t Y t = � ∞ g ( t � s ) σ s W ( d s ) + � ∞ q ( t � s ) a s d s are, in general, not semimartingales

Important example Suppose Y = g � σ � W with g ( t ) = t ν � 1 e � λ t . 1 2 < ν < 1 nonSM ν = 1 SM 1 < ν < 3 nonSM 2

A key object of interest, whether for BSM or BSS processes, is the integrated squared volatility Z t σ 2 + 0 σ 2 = s d s t for any t 2 R . Realised multipower variations ( RMPV s) 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 σ 2 + or on the t small scale behaviour of the damping function g . However, because of the nonsemimartingale character of BSS processes the probabilistic limit theory of RMPV s for BSS processes is decisively di¤erent from that for BSM processes. Next : Brief review of the theory of RMPV s for BSS processes.

Multipower Variation In the following the process Y is assumed to be observed at time points t i = i ∆ n with i = 0 , . . . , [ t / ∆ n ] and ∆ n ! 0. A realised multipower variation of a stochastic process Y is de…ned as an object of the type [ t / ∆ n ] � k + 1 k j ∆ n i + j � 1 Y j p j ∑ ∏ i = 1 j = 1 where ∆ n i Y = Y i n � Y i � 1 and p 1 , . . . , p k � 0. n

For the case where Y 2 BSM , i.e. Y = σ � W + a � leb , it was established in [BNGJPS07] that Z t [ t / ∆ n ] � k + 1 k ucp n p + / 2 � 1 j ∆ n i + j � 1 Y j p j 0 j σ s j p + ds ∑ ∏ � ! µ p 1 � � � µ p k i = 1 j = 1 where p + = ∑ k j = 1 p j and µ p = E [ j u j p ] , u � N ( 0 , 1 ) .

Moreover, under a regularity condition on the volatility process σ , there is an associated stable central limit theorem: Z t � � [ t / ∆ n ] � k + 1 k p n n p + / 2 � 1 i + j � 1 Y j p j � µ p 1 � � � µ p k j ∆ n 0 j σ s j p + ds ∑ ∏ i = 1 j = 1 Z t p st 0 j σ s j p + dB s � ! C where B is another Brownian motion, de…ned on an extension of the probability space ( Ω , F , ( F t ) t � 0 , 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 of the processes, and especially the σ 2 + component, in the BSS setting the interest is not just in regard to σ 2 + but also concerns inference on the damping function g . For inference on g it is pertinent to study the behaviour of ratios of RMPV s and, in fact, not only of ’classical’ ( ∆ case) RMPV s but also RMPV s 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 RMPV s of types ∆ and � are de…ned as n ) � p + [ t / ∆ n ] � k + 1 k � 1 MPV ∆ ( Y , p 1 , . . . , p k ) n t = ∆ n ( τ ∆ j ∆ n i + l Y j p l + 1 ∑ ∏ i = 1 l = 0 (1) n ) � p + [ t / ∆ n ] � k + 1 k � 1 MPV � ( Y , p 1 , . . . , p k ) n t = ∆ n ( τ � j � n i + l Y j p l + 1 ∑ ∏ i = 2 l = 0 (2) where ∆ n i Y = Y i ∆ n � Y ( i � 1 ) ∆ n and � n i Y = X i ∆ n � 2 Y ( i � 1 ) ∆ n + Y ( i � 2 ) ∆ n , where p l � 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 σ 2 + . Except for the normalisation these results are similar in nature to those for BSM s 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 Ratio s ( RVR s), i.e. ratios of RMPV s. The relevant Realised Variation Ratio s ( RVR s), of types ∆ Note and � , do not involve normalisation of RMPV s. Thus, in this sense, the limit theory for RVR s is ’nonparametric’.

Inference on g . An example: To illustrate, consider the key case where g ( t ) = t ν � 1 e � λ t . � 1 � [ � � 1 , 3 The interesting situations are where ν 2 2 , 1 . Using � 1 � 2 ∆ - RMPV s it is only possible to establish a CLT for ν in 2 , 1 . Passing to � - RMPV s this can be strengthened to � 1 � [ � � 1 , 5 ν 2 2 , 1 . 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 p n 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