THIELE CENTRE for applied mathematics in natural science Matrix Subordinators and Multivariate OU-based Volatility Models Ole E. Barndorff-Nielsen Thiele Centre Department of Mathematical Sciences University of Aarhus
THIELE CENTRE for applied mathematics in natural science . Based mainly on joint papers with: Victor Perez-Abreu CIMAT Robert Stelzer TUM Matrix Subordinators and Multivariate OU-based Volatility Models , page 2 of 40
THIELE CENTRE for applied mathematics in natural science Synopsis � Intro � Volatility and OU processes � Matrix subordinators � In�nite divisibility in cones � CLT for RMPV � Positive de�nite matrix processes of OU type � Roots of positive de�nite processes Matrix Subordinators and Multivariate OU-based Volatility Models , page 3 of 40
THIELE CENTRE for applied mathematics in natural science Intro Let Υ t denote a d -dimensional vector of log prices, modelled as a Brownian semimartingale Z t Z t Υ t = 0 a s d s + 0 σ s d W s ? OU modelling of Σ = σ > σ . One-dimensional case: realism and ana- lytical tractability = R t RMPV: Basis for inference on Σ + ? Multipower Variation 0 Σ s d s t = R t s σ s and more generally on Σ + r where Σ s = σ > 0 Σ r s d s . t ? The MPV theory uses SDE representations of d σ (not d Σ ). Need SDE representations of Σ r , in particular Σ 1/2 Matrix Subordinators and Multivariate OU-based Volatility Models , page 4 of 40
THIELE CENTRE for applied mathematics in natural science Volatility and OU processes Univariate OU volatility d σ 2 t = � λσ 2 t � dt + dL λ t where λ > 0 is a parameter and L is a subordinator , i.e. a Lévy process with nonnegative increments. Matrix Subordinators and Multivariate OU-based Volatility Models , page 5 of 40
THIELE CENTRE for applied mathematics in natural science Volatility and OU processes The solution can be shown to be Z t σ 2 t = e � λ t σ 2 0 e � λ ( t � s ) dL s λ 0 + Provided E ( log + ( L t )) < ∞ there is a unique stationary solution given by Z t σ 2 � ∞ e � λ ( t � s ) dL λ s t = Matrix Subordinators and Multivariate OU-based Volatility Models , page 6 of 40
THIELE CENTRE for applied mathematics in natural science Volatility and OU processes There is a vast literature concerning the extension of OU processes to R d -valued processes. By identifying M d , the class of d � d matrices, with R d 2 one imme- diately obtains matrix valued processes. So for a given Lévy process ( L t ) t 2 R with values in M d and a linear operator A : M d ! M d , a solution to the SDE dX t = A X t � dt + dL t is termed a matrix-valued process of Ornstein-Uhlenbeck type . Matrix Subordinators and Multivariate OU-based Volatility Models , page 7 of 40
THIELE CENTRE for applied mathematics in natural science Volatility and OU processes As in the univariate case one can show that for some given initial value X 0 the solution is unique and given by Z t X t = e A t X 0 + 0 e A ( t � s ) dL s . Provided E ( log + k L t k ) < ∞ and σ ( A ) 2 ( � ∞ , 0 ) + i R , there exists a unique stationary solution given by Z t � ∞ e A ( t � s ) dL s . X t = Matrix Subordinators and Multivariate OU-based Volatility Models , page 8 of 40
THIELE CENTRE for applied mathematics in natural science Matrix subordinators However, in order to obtain positive semide�nite Ornstein-Uhlenbeck processes we need to consider matrix subordinators as driving Lévy processes. Let ¯ S + d be the closure of the cone S + d of positive de�nite matrices in M d . S + A process L with values in ¯ De�nition d and having independent stationary increments is called a matrix subordinator Matrix Subordinators and Multivariate OU-based Volatility Models , page 9 of 40
THIELE CENTRE for applied mathematics in natural science In�nite divisibility in the cone ¯ S + d A random matrix M is in�nitely divisible in ¯ S + d if and only if for each integer p � 1 there exist p independent identically distributed ran- d such that M law dom matrices M 1 , ..., M p in ¯ S + = M 1 + ... + M p . Lévy-Khintchine representation (Skorohod (1991)) A random matrix M 2 ¯ S + d is in�nitely divisible in ¯ S + if and only if its d cumulant transform is of the form Z ( e i tr ( X Θ ) � 1 ) ρ ( d X ) , Θ 2 S + C ( Θ ; M ) = i tr ( γ Θ ) + d , S + ¯ d where γ 2 ¯ S + d is called the drift and the Lévy measure ρ is such that ρ ( S + d n ¯ S + d ) = 0 and ρ has order of singularity Z min ( 1, tr ( X )) ρ ( d X ) < ∞ . S + ¯ d Matrix Subordinators and Multivariate OU-based Volatility Models , page 10 of 40
THIELE CENTRE for applied mathematics in natural science In�nite divisibility in the cone ¯ S + d Lévy-Ito decomposition: If f L t g is a matrix subordinator with the above Lévy-Khintchine rep- resentation then it has a Lévy-Itô decomposition Z t Z L t = t γ + d nf 0 g x µ ( ds , dx ) S + 0 where γ 2 ¯ S + d is a deterministic drift and µ ( ds , dx ) a Poisson ran- dom measure on R + � ¯ S + d with E ( µ ( ds , dx )) = Leb ( ds ) ν ( dx ) , Leb denoting the Lebesgue measure and ν the Lévy measure of L t . Matrix Subordinators and Multivariate OU-based Volatility Models , page 11 of 40
THIELE CENTRE for applied mathematics in natural science Examples ? Quadratic Covariation of d -dimensional Lévy processes ? Gamma type matrix distribution Lévy density: j Σ j � < d > ( tr ( X Σ � 1 )) [ d ] e tr ( � X Σ � 1 ) where < d > = ( d + 1 ) /2 and [ d ] = ( d + 1 ) d /2 . Kumulant transform: Z log ( 1 + tr ( U Σ 1/2 ΘΣ 1/2 )) � 1 d U . K ( Θ , R ) = S + ¯ d Matrix Subordinators and Multivariate OU-based Volatility Models , page 12 of 40
THIELE CENTRE for applied mathematics in natural science Examples ? Bessel matrix distribution Lévy density: Z n o � � � [ d ] � β d Υ X Υ � 1 + Σ � 1 Υ ) j Σ j � < d > tr ( Υ Σ � 1 ) Υ > 0 etr ( � j Υ j < d > . where X and Υ are the anti-matrices of X and Υ . Matrix Subordinators and Multivariate OU-based Volatility Models , page 13 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV Central Limit Theory for Realised Multipower Variation (B-N, Jacod, Graversen, Podolskij and Shephard (2006)) Recall : For a wide class of real–valued processes Υ , including all semi- martingales, the realised quadratic variation process [ nt ] V ( Υ ; 2 ) n n ) 2 ∑ t = ( Υ i n � Υ i � 1 i = 1 converges in probability, as n ! ∞ and for all t � 0 , towards the quadratic variation process V ( Υ ; 2 ) t (usually denoted by [ Υ , Υ ] t ). Matrix Subordinators and Multivariate OU-based Volatility Models , page 14 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV Next, let r , s be nonnegative numbers. The realised bipower varia- tion process of order ( r , s ) is the increasing processes de�ned as: [ nt ] r + s n j r j Υ i + 1 V ( Υ ; r , s ) n 2 � 1 n j s . ∑ j Υ i n � Υ i � 1 n � Υ i t = n i = 1 Clearly V ( Υ ; 2 ) n = V ( Υ ; 2, 0 ) n . The bipower variation process of order ( r , s ) for Υ , denoted by V ( Υ ; r , s ) t , is the limit in probability, if it exists for all t � 0 , of V ( Υ ; r , s ) n t . Testing for jumps; Estimation of R t Uses : 0 σ 4 s d s in the presence of jumps; ... Matrix Subordinators and Multivariate OU-based Volatility Models , page 15 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV Extension to the multidimensional case. Now Υ = ( Υ j ) 1 � j � d is taken as d –dimensional. The realised cross–multipower variation processes are de�ned by V ( Υ j 1 , . . . , Υ j N ; r 1 , . . . , r N ) n t [ nt ] r 1 + ... + rN j Υ j 1 n � Υ j 1 n j r 1 . . . j Υ j N � Υ j N � 1 j r N . ∑ = n 2 i i � 1 i + N � 1 i + N � 2 n n i = 1 Matrix Subordinators and Multivariate OU-based Volatility Models , page 16 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV More generally still, let [ nt ] g ( p i Υ ) h ( p X n ( g , h ) t = 1 n ∆ n n ∆ n ∑ i + 1 Υ ) n i = 1 where ∆ n n , g and h are two maps on R d , taking vakues i Υ = Υ i n � Υ i � 1 in M d 1 , d 2 and M d 2 , d 3 respectively. So X n ( g , h ) t takes its values in M d 1 , d 3 . We refer to X n ( g , h ) as the realised multipower variation (RMPV) associated to g and h . Matrix Subordinators and Multivariate OU-based Volatility Models , page 17 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV To derive a CLT for RMPV we need the following structural assumptions: Hypothesis (H) : We have Z t Z t Υ t = Υ 0 + 0 a s ds + 0 σ s � dW s , where W is a standard d 0 –dimensional BM, a is predictable R d –valued locally bounded, and σ is M d , d 0 –valued càdlàg with Σ = σσ > invertible. Matrix Subordinators and Multivariate OU-based Volatility Models , page 18 of 40
THIELE CENTRE for applied mathematics in natural science Interlude: CLT for RMPV Hypothesis (H') : We have Z t Z t Z t 0 a 0 0 σ 0 σ t = σ 0 + s ds + s � dW s + 0 v s � dV s Z t Z + E ϕ � w ( s � , x )( µ � ν )( ds , dx ) 0 Z t Z + E ( w � ϕ � w )( s � , x ) µ ( ds , dx ) . 0 where **** Matrix Subordinators and Multivariate OU-based Volatility Models , page 19 of 40
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