Outline BSS models De…nition Key object of interest BSS Processes and Intermittency/Volatility Realised Quadratic Variation Turbulence Turbulence Stochastics background Ambit processes Intermittency BSS models Ole E. Barndor¤-Nielsen and Jürgen Schmiegel (cont.) Canon Semi- and non-semimartingale Thiele Centre questions Department of Mathematical Sciences Inference on inter- mittency/volatility University of Aarhus Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline BSS models De…nition Key object of interest Realised Quadratic I BSS models Variation Turbulence I Turbulence background background Ambit processes I BSS models (cont.) Intermittency BSS models I Inference on intermittency/volatility (cont.) Canon I Further ongoing work Semi- and non-semimartingale questions I Some open questions Inference on inter- mittency/volatility Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Brownian semistationary ( BSS ) processes: Outline Z t Z t BSS models Y t = � ∞ g ( t � s ) σ s d B s + � ∞ q ( t � s ) a s d s (1) De…nition Key object of interest Realised Quadratic Variation where B is Brownian motion, g and q are square integrable Turbulence background functions on R , with g ( t ) = q ( t ) = 0 for t < 0, and σ and Ambit processes Intermittency a are cadlag processes. BSS models (cont.) Canon When σ and a are stationary, as will be assumed throughout Semi- and non-semimartingale questions this talk, then so is Y . Inference on inter- mittency/volatility Introduction It is sometimes convenient to indicate the formula for Y as Increment process Examples RQV and IV Conditions ensuring Y = g � σ � B + q � a � Leb . (2) π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline BSS models De…nition Key object of interest Realised Quadratic Variation We consider the BSS processes to be the natural analogue, Turbulence in stationarity related settings, of the class BSM of background Ambit processes Brownian semimartingales. Intermittency BSS models (cont.) Canon I The BSS processes are not in general semimartingales Semi- and non-semimartingale questions Inference on inter- mittency/volatility Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline The key object of interest is the integrated variance (IV) BSS models De…nition Key object of interest Z t Realised Quadratic σ 2 + 0 σ 2 Variation = s d s t Turbulence background Ambit processes Intermittency We shall discuss to what extent realised quadratic variation BSS models of Y can be used to estimate σ 2 + t . (cont.) Canon Semi- and I Note that the relevant question here is whether a non-semimartingale questions suitably normalised version of the realised quadratic Inference on inter- mittency/volatility variation, and not necessarily the realised quadratic Introduction Increment process variation itself, converges in probability/law. Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline In semimartingale theory the quadratic variation [ Y ] of Y is BSS models De…nition de…ned in terms of the Ito integral Y � Y , as Key object of interest [ Y ] = Y 2 � 2 Y � Y . In that setting [ Y ] equals the limit in Realised Quadratic Variation Turbulence probability as δ ! 0 of the realised quadratic variation [ Y δ ] background of Y de…ned by Ambit processes Intermittency BSS models � � 2 b t / δ c (cont.) ∑ [ Y δ ] t = Y j δ � Y ( j � 1 ) δ (3) Canon Semi- and non-semimartingale j = 1 questions Inference on inter- mittency/volatility where b t / δ c is the largest integer smaller than or equal to Introduction t / δ . It is this latter de…nition of quadratic variation that we Increment process Examples will use here. RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
White noise case: Outline BSS models Z De…nition Key object of interest Y t ( x ) = µ + A t ( x ) g ( t � s , j ξ � x j ) σ s ( ξ ) W ( d ξ , d s ) Realised Quadratic Variation Z Turbulence background + D t ( x ) q ( t � s , j ξ � x j ) a s ( ξ ) d ξ d s . Ambit processes Intermittency BSS models (cont.) Canon Here A t ( σ ) and D t ( σ ) are termed ambit sets . Semi- and non-semimartingale questions Inference on inter- Lévy case: mittency/volatility Introduction Z Increment process σ 2 Examples t ( x ) = C t ( x ) h ( t � s , j ξ � x j ) L ( d ξ , d s ) RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
( t ( w ) , σ ( w )) X w ❅ Outline BSS models De…nition Key object of interest Realised Quadratic Variation A t ( w ) ( σ ( w )) � 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 π δ ! δ 0 Consistency Feasible version Further ongoing work Figure: Ambit processes Relaxing assumptions Realised Variation Ratio Some open
Outline BSS models De…nition In turbulence the basic notion of intermittency refers to the Key object of interest Realised Quadratic fact that the energy in a turbulent …eld is unevenly Variation Turbulence distributed in space and time. background Ambit processes Intermittency The present presentation is part of a project that aims to BSS models (cont.) construct a stochastic process model of the …eld of velocity Canon Semi- and vectors representing the ‡uid motion, conceiving of the non-semimartingale questions intermittency as a positive random …eld with random values Inference on inter- mittency/volatility σ t ( x ) at positions ( x , t ) in space-time. Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline BSS models De…nition However, most extensive data sets on turbulent velocities Key object of interest Realised Quadratic only provide the time series of the main component (i.e. the Variation Turbulence component in the main direction of the ‡uid ‡ow) of the background Ambit processes velocity vector at a single location in space. Intermittency BSS models (cont.) In the present talk the focus is on this latter case, but in the Canon Semi- and concluding Section some discussion will be given on the non-semimartingale questions further intriguing issues that arise when addressing Inference on inter- mittency/volatility tempo-spatial settings. Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline BSS models De…nition For simplicity we now assume that σ ? ? B and that q = 0, Key object of interest Realised Quadratic Variation i.e. there is no drift term in Y and Turbulence Z t background Ambit processes Y t = 0 g ( t � s ) σ s d B s . Intermittency BSS models (cont.) Canon Semi- and non-semimartingale questions However, at the end of the talk some discussion will be given Inference on inter- mittency/volatility on more general settings. Introduction Increment process Examples RQV and IV Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
Outline Let Z = f Z t g t 2 R denote a second order stationary BSS models stochastic process, possibly complex valued, of mean 0 and De…nition Key object of interest continuous in quadratic mean. Recall that Z is said to be a Realised Quadratic Variation moving average process if it is of the form Turbulence background Z ∞ Ambit processes � ∞ φ � ( t � s ) d Ξ � Intermittency Z t = (4) s BSS models (cont.) Canon where φ � is an, in general complex, deterministic and square Semi- and non-semimartingale integrable function and where the process Ξ � has orthogonal questions n t j 2 o Inference on inter- mittency/volatility j d Ξ � = ̟ � d t for some constant increments with E Introduction ̟ � > 0; …nally, the integral in (4) is de…ned in the quadratic Increment process Examples RQV and IV mean sense. Conditions ensuring π δ ! δ 0 Consistency Feasible version Further ongoing work Relaxing assumptions Realised Variation Ratio Some open
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