Weak dependence of mixed moving average processes and applications - PowerPoint PPT Presentation

Page 1 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Weak dependence of mixed moving average processes and applications Robert Stelzer Institute of Mathematical Finance Ulm University Risk and Statistics 2nd ISM-UUlm Joint Workshop Villa Eberhardt, Ulm October 9th, 2019 Based on joint work with Imma Curato and Bennet Ströh

Page 2 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Continuous time stochastic volatility models Consider the log price process of a financial asset p t = a t + M t , where M is a local martingale and a is a cádlág adapted process of locally bounded variation. We take M to have a stochastic volatility: � t � M t = X s dW s 0 where the non-negative spot volatility X is assumed to have cádlág sample paths (which implies it can posses jumps!)

Page 3 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Continuous time stochastic volatility models ◮ X s independent of W ◮ X s can be Itô diffusion as in the Heston model (1993) � dX s = α ( β − X s ) ds + ν X s dZ s where Z is a Brownian motion correlated with W , and 2 αβ > ν 2 . ◮ X s can be a Lévy driven Ornstein Uhlenbeck process dX s = − a X s ds + dL s , a > 0 where L is a subordinator independent of W , (Lévy process with positive increments and no drift), Barndorff-Nielsen and Shepard (2001). For E [log( | L 1 | ∨ 1)] < ∞ and a > 0, a unique stationary solution exists: � t e − a ( t − s ) dL s X t = −∞

Page 4 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Lévy driven Ornstein Uhlenbeck Process For L a Lévy process with characteristic triplet ( γ, Σ , ν ), � t e − a ( t − s ) dL s X t = −∞ ◮ The parameter a is called mean reversion parameter ◮ Corr ( X 0 , X r ) = e − ar with r > 0.

Page 5 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Superposition of Ornstein-Uhlenbeck process ◮ Typically, the autocovariance function of the squared returns of financial prices decays much faster at the beginning than at higher lags. Hence: ◮ Add up countably many independent OU-type processes � t ∞ � e − a i ( t − s ) dL i , s X t = w i −∞ k =1 with independent identically distributed Lévy processes ( L i ) i ∈ N and appropriate a i > 0, w i > 0 with � ∞ i =1 w i = 1.

Page 6 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Superposition of Ornstein-Uhlenbeck process More generally we can “integrate” over all possible mean reversion parameters. � t � e A ( t − s ) Λ( dA , ds ) X t = R − −∞ where Λ is called a Lévy basis and the mean reversion parameter A becomes a random variable. The supOU process was first introduced by Barndorff-Nielsen (2001) and further investigated in Barndorff-Nielsen and St.(2011) and Fuchs and St. (2013).

Page 7 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Lévy basis I Let B ( S ) the Borel σ -field on S and π some probability measure on ( S , B ( S )). Definition A family Λ = { Λ( B ) : B ∈ B b ( S × R ) } of real-valued random variables is called a d -dimensional Lévy basis on S × R if: ◮ the distribution of Λ( B ) is infinitely divisible for all B ∈ B b ( S × R ), ◮ for arbitrary n ∈ N and pairwise disjoint sets B 1 , . . . , B n ∈ B b ( S × R ) the random variables Λ( B 1 ) , . . . , Λ( B n ) are independent and ◮ for any pairwise disjoint sets B 1 , B 2 , . . . ∈ B b ( S × R ) with � n ∈ N B n ∈ B b ( S × R ) we have, almost surely, Λ( � n ∈ N B n ) = � n ∈ N Λ( B n ).

Page 8 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Lévy basis II We restrict ourselves to time-homogeneous and factorisable Lévy bases, i.e. Lévy bases with characteristic function E [ e i � u , Λ( B ) � ] = e Φ( u )Π( B ) (1) for all u ∈ R d and B ∈ B b ( S × R ), where Π = π × λ is the product of the probability measure π on S and the Lebesgue measure λ on R and Φ( u ) = i � γ, u � − 1 � R d e i � u , x � − 1 − i � u , x � 1 [0 , 1] ( � x � ) ν ( dx ) , 2 � u , Σ u � + where γ ∈ R d , Σ ∈ S + d - i.e. the space of the positive semi-definite matrix- and ν is a Lévy measure. By L we denote the underlying Lévy process with characteristic triplet ( γ, Σ , ν ). The quadruple ( γ, Σ , ν, π ) determines the distribution of the Lévy bases completely and therefore it is called the generating quadruple.

Page 9 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Mixed Moving Average Processes The process � � f ( A , t − s ) Λ( dA , ds ) , X t = S R is infinitely divisible and strictly stationary and called a MMA process. f is a deterministic kernel function and integrable in the sense of Rajput and Rosiński (1989). ◮ The class of mixed moving average processes allows to obtain models with flexible autocorrelation structure and that at the same time can generate many kinds of marginal distribution by choosing an appropriate Lévy basis. ◮ In Fuchs and St. (2013), it is shown that a MMA process is mixing and consequently ergodic.

Page 10 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Example Let us assume that π is a probability distribution with support in R − defined as B ξ where B ∈ R − and ξ is Γ( α, 1) with α > 1 (distribution function of the random mean reverting parameter A ). The autocovariance of the supOU process � t � e A ( t − s ) Λ( dA , ds ) X t = R − −∞ is Cov ( X 0 , X k ) = − σ 2 (1 − Bk ) 1 − α , 2 B ( α − 1) where σ 2 = Var [ L 1 ].

Page 11 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Weak dependence Let � � F = F u and G = G v u ∈ N ∗ v ∈ N ∗ where F u and G v are respectively two classes of measurable functions from ( R d ) u to R and ( R d ) v to R .

Page 12 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Ψ -weak dependence A process X = ( X t ) t ∈ R with values in R d is called a Ψ-weakly dependent process if there exists a sequence ( ǫ ( r )) r ∈ R + converging to 0, satisfying | Cov ( F ( X i 1 , . . . , X i u ) , G ( X j 1 , . . . , X j v )) | ≤ c Ψ( F , G , u , v ) ǫ ( r ) for all ( u , v ) ∈ N ∗ × N ∗ ;   r ∈ R + ;     ( i 1 , . . . , i u ) ∈ R u and ( j 1 , . . . , j v ) ∈ R v ,   with i 1 ≤ . . . ≤ i u ≤ i u + r ≤ j 1 ≤ . . . ≤ j v ;  functions F : ( R d ) u → R and G : ( R d ) v → R      belonging respectively to F and G  and where c is a constant independent of r . The sequence ( ǫ ( r )) r ∈ R + corresponds to different sequences of weak dependence coefficients

Page 13 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer η -weak dependence Let F = G and F u be the class of bounded Lipschitz functions. We consider R d equipped with the Euclidean norm and | F ( x ) − F ( y ) | Lip ( F ) = sup x � = y � x 1 − y 1 � + � x 2 − y 2 � + ... + � x d − y d � . The η -coefficients correspond to Ψ( F , G , u , v ) = u � G � ∞ Lip ( F ) + v � F � ∞ Lip ( G ) and have been introduced in Doukhan and Louhichi (1999).

Page 14 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer θ -weak dependence Let F u the class of bounded measurable functions, G v be the Lipschitz functions. The θ -coefficients correspond to Ψ( F , G , u , v ) = v � F � ∞ Lip ( G ) . and have been introduced in Dedecker and Doukhan (2003).

Page 15 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer Remarks ◮ Let ( A t ) t ∈ R be the filtration generated by Λ defined as the σ -algebras A t generated by the set of random variables { Λ( B ) : B ∈ B ( S × ( −∞ , t ]) } for t ∈ R . If an MMA process is adapted to ( A t ) t ∈ R , we call it causal . Otherwise it is referred to as being non-causal . ◮ An MMA process is (under moment assumptions) always η -weakly dependent and in the causal case also θ -weakly dependent. ◮ Different versions and proofs of the above statement can be found in Curato and St. (2019) in function of different moment conditions on the underlying Lévy process.

Page 16 Weak dependence of mixed moving average processes | October 9th, 2019 | Robert Stelzer MMA: θ -weak dependence conditions (Curato and St., 2019) Let Λ be an R d -valued Lévy basis with characteristic quadruple � � x � > 1 � x � 2 ν ( dx ) < ∞ , ( γ, Σ , ν, π ) such that E [ L 1 ] = 0 and f : S × R + → M n × d ( R ) a B ( S × R + )-measurable function and f ∈ L 2 ( S × R + , π ⊗ λ ). Then, the resulting causal MMA process X is a θ -weakly dependent process with coefficients � − r � 1 � � 2 tr ( f ( A , − s )Σ L f ( A , − s ) ′ ) ds π ( dA ) θ X ( r ) = S −∞ for all r ≥ 0, where E [ L 1 L ′ � R d xx ′ ν ( dx ). 1 ] = Σ L = Σ +