THIELE CENTRE for applied mathematics in natural science Volatility Modulated Volterra Processes Ole E. Barndorff-Nielsen Thiele Centre Department of Mathematical Sciences University of Aarhus
THIELE CENTRE for applied mathematics in natural science Synopsis . � Intro: Turbulence and Finance; MultipowerVariation � Volterra processes � Volatility modulated Volterra Processes ( VMVP ) � Ambit processes � 1 -dim MA BM setting: Y = g � σ � B � Concrete model type � Realised Variation Ratio Volatility Modulated Volterra Processes , page 2 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Modelling framework: in Finance The basic framework for stochastic volatility modeling in �nance is that of Brownian semimartingales Z t Z t Y t = Y 0 + 0 σ s d B s + 0 a s d s where σ and a are cadlag processes and B is Brownian motion, with σ expressing the volatility. In general, Y , σ , B and a will be multidimensional. Volatility Modulated Volterra Processes , page 3 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Modelling framework: Turbulence (Phenomenological approach) Whereas Brownian semimartingales are 'cumulative' in nature, for free turbulence it is physically natural to model timewise velocity dy- namics by stationary processes: At time t and at a �xed position x in the turbulent �eld, the velocity vector is speci�ed as V t = µ + Y t with Volatility Modulated Volterra Processes , page 4 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Z t Z Y t ( x ) = R 3 g ( t � s , x � ξ ) σ s ( ξ ) W ( d ξ d s ) � ∞ Z t Z + R 3 q ( t � s , x � ξ ) a s ( ξ ) d ξ d s . � ∞ where W is white noise, with σ expressing the intermittency (= volatility). In general, Y , g , σ , W , q , and a will be multidimensional. Volatility Modulated Volterra Processes , page 5 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Multipower Variations For any stochastic process Y = f Y t g t � 0 (or Y = f Y t g t 2 R ) the quadratic variation (QV) process [ Y ] and the bipower variation (BV) process f Y g are, respectively, the lim- its in probability, when they exist, of the realised quadratic variation (RQV) [ Y δ ] and the realised bipower variation (RBV) f Y δ g . To de�ne RVR and RBP , for any δ > 0 let Y δ denote the δ -discretisation of Y , i.e. ( Y δ ) t = Y b t / δ c δ , and recall that for a standard normal vari- able u we have p µ 1 = E fj u jg = 2/ π . Furthermore, for positive integers n and δ = n � 1 , let ∆ n j Y = Y j δ � Y ( j � 1 ) δ . Volatility Modulated Volterra Processes , page 6 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Then RVR and RBP are given, respectively, by � � 2 b nt c ∆ n ∑ [ Y δ ] t = j Y j = 1 and f Y δ g t = π 2 [ Y δ ] [ 1,1 ] t with � � � � b t / n c � � � � [ Y δ ] [ 1,1 ] � ∆ n � ∆ n ∑ = j � 1 Y � j Y � . t j = 2 Volatility Modulated Volterra Processes , page 7 of 72
THIELE CENTRE for applied mathematics in natural science Introduction General multipower: n o Y [ r ] t = c r [ Y δ ] [ r ] t δ where � � � � b nt c r k � � � r 0 . � � � � [ Y δ ] [ r ] � ∆ n � ∆ n ∑ = j � k Y � j Y � t j = k + 1 More generally, � � � � b nt c ∆ n ∆ n ∑ f 1 j � k Y � � � f k j Y j = k + 1 Volatility Modulated Volterra Processes , page 8 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Applications In Finance � � � � � � L � stably 1 1 δ � 1 [ Y δ ] t � σ 2 + t , f Y δ g t � σ 2 + σ 4 + ! N 2 ( 0, 0 ) , 2 2 t 1 1 + ϑ t where ϑ = π 2 /4 + π � 5 ( . = 0.609 ) . Feasible results . Volatility Modulated Volterra Processes , page 9 of 72
THIELE CENTRE for applied mathematics in natural science Introduction Applications in Turbulence Volatility Modulated Volterra Processes , page 10 of 72
THIELE CENTRE for applied mathematics in natural science Volterra processes Brownian Volterra processes ( BVP ): Z ∞ Z ∞ Y t = � ∞ K t ( s ) d B s + � ∞ Q t ( s ) d s , Here K and Q are deterministic functions, suf�ciently regular to give suitable meaning to the integrals. Backward type: Z t Z t Y t = � ∞ K t ( s ) d B s + � ∞ Q t ( s ) d s . Volatility Modulated Volterra Processes , page 11 of 72
THIELE CENTRE for applied mathematics in natural science Volterra processes Lévy Volterra processes ( LVP ): Z ∞ Z ∞ Y t = � ∞ K t ( s ) d L s + � ∞ Q t ( s ) d s Here L denotes a Lévy process on R and K and Q are deterministic kernels, satisfying certain regularity conditions. Backward type: Z t Z t Y t = � ∞ K t ( s ) d L s + � ∞ Q t ( s ) d s . Volatility Modulated Volterra Processes , page 12 of 72
THIELE CENTRE for applied mathematics in natural science Volterra processes Stochastic integration in this kind of setting is discussed for BVP in [Hu03], [Dec05], [DecSa06] and for LVP in [BeMar07]. When is Y a semimartingale? In that case what is the character of its spectral representation? Andreas Basse [Bas07a], [Bas07b], [Bas07c], for Brownian case. Volatility Modulated Volterra Processes , page 13 of 72
THIELE CENTRE for applied mathematics in natural science Volterra processes Tempo-spatial Volterra processes : Z ∞ Z Z ∞ Z Ξ K t ( ξ , s ; x ) L # ( d ξ d s ) + Y t ( x ) = Ξ Q t ( ξ , s ; x ) d ξ d s � ∞ � ∞ Here K and Q are deterministic functions, Ξ is a region in R d and L # is a homogeneous Lévy basis on Ξ � R . Backward type: Z t Z Z t Z Ξ K t ( ξ , s ; x ) L # ( d ξ d s ) + Y t ( x ) = Ξ Q t ( ξ , s ; x ) d ξ d s � ∞ � ∞ Volatility Modulated Volterra Processes , page 14 of 72
THIELE CENTRE for applied mathematics in natural science Volatility modulated Volterra processes Volatility modulated Volterra Processes ( VMVP ): Z ∞ Z Z ∞ Z Ξ K t ( ξ , s ; x ) σ s ( ξ ) L # ( d ξ d s ) + Y t ( x ) = Ξ Q t ( ξ , s ; x ) a s ( ξ ) d ξ d s � ∞ � ∞ where σ is a positive stochastic process, embodying the volatility or intermittency. ( K and Q deterministic, σ and a stochastic.) Backwards moving average type : Z t Z Ξ g ( ξ � x , t � s ) σ s ( ξ ) L # ( d ξ d s ) Y t ( x ) = � ∞ Z t Z + Ξ q ( ξ � x , t � s ) a s ( ξ ) d ξ d s � ∞ Volatility Modulated Volterra Processes , page 15 of 72
THIELE CENTRE for applied mathematics in natural science Inference on the volatility A central issue in these settings is how to draw inference on the volatility process σ . In cases where the processes are semimartingales, the theory of multipower variations provides effective tools for this. ([BNGJPS07], [BNGJS06] and references given there) However, VMVP processes are generally not of semimartingale type and the question of how to proceed then is largely unsolved and poses mathematically challenging problems. Volatility Modulated Volterra Processes , page 16 of 72
THIELE CENTRE for applied mathematics in natural science Inference on the volatility It is further of interest to consider cases where processes express- ing possible jumps or noise in the dynamics are added. Some of these problems are presently under study in joint work with Jose-Manuel Corcuera, Neil Shephard, Jürgen Schmiegel and Mark Podolski. Volatility Modulated Volterra Processes , page 17 of 72
THIELE CENTRE for applied mathematics in natural science Ambit processes Ambit processes : ([BNSch07a]) Z Y t ( x ) = µ + A t ( σ ) g ( t � s , j ξ � x j ) σ s ( ξ ) W ( d ξ , d s ) Z + D t ( σ ) q ( t � s , j ξ � x j ) a s ( ξ ) d ξ d s Here A t ( σ ) and D t ( σ ) are termed ambit sets . Volatility Modulated Volterra Processes , page 18 of 72
THIELE CENTRE for applied mathematics in natural science Ambit processes X w ( t ( w ) , σ ( w )) ❅ A t ( w ) ( σ ( w )) � � � Ambit processes Volatility Modulated Volterra Processes , page 19 of 72
THIELE CENTRE for applied mathematics in natural science Ambit processes ✻ t ′ • t • ✲ σ ′ σ Two overlappng ambit sets Volatility Modulated Volterra Processes , page 20 of 72
THIELE CENTRE for applied mathematics in natural science 1-dim. BM MA setting Recall: Modelling time series by stochastic processes of the form V = µ + Y with Z t Z t Y t = � ∞ g ( t � u ) σ u d B u + � ∞ h ( t � u ) a s d u . (1) Here B is Brownian motion, the kernels h and g are determinis- tic, positive and square integrable functions on ( 0, ∞ ) , presumed known, and σ is a stationary process which expresses the time- dependent variation or volatility of the process Y . Moreover, a and σ are stochastic processes satisfying the same as- sumptions as are usual for Brownian semimartingales; in particular, σ is square integrable. Volatility Modulated Volterra Processes , page 21 of 72
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