Gaussian Semimartingales and Moving Averages Andreas Basse Thiele - PowerPoint PPT Presentation

Stochastics in Turbulence and Finance Gaussian Semimartingales and Moving Averages Andreas Basse Thiele Centre, University of Aarhus, Denmark. Stochastics in Turbulence and Finance Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance The set-up where ( W t ) t ∈ R is a (two-sided) Brownian motion and K = K t ( s ) is a deterministic We are interested in the semimartingale property of processes ( X t ) t ≥ 0 on the form � t X t = K t ( s ) dW s , t ≥ 0 , (1) −∞ kernel such that the integral exists. Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance The set-up where ( W t ) t ∈ R is a (two-sided) Brownian motion and K = K t ( s ) is a deterministic We are interested in the semimartingale property of processes ( X t ) t ≥ 0 on the form � t X t = K t ( s ) dW s , t ≥ 0 , (1) −∞ kernel such that the integral exists. Two observations: If K t ( s ) does not depend on t , then ( X t ) t ≥ 0 is a martingale. If K t ( s ) = 1 [ 0 , 1 ] ( t − s ) , then X t = W t − W t − 1 , which is not a semimartingale. Andreas Basse Gaussian Semimartingales and Moving Averages

R , ( X t ) t ∈ R is called a moving average process. Stochastics in Turbulence and Finance Moving average processes In the case where K t ( s ) = ϕ ( t − s ) − ψ ( − s ) , that is � t X t = ϕ ( t − s ) − ψ ( − s ) dW s , t ∈ (2) −∞ Andreas Basse Gaussian Semimartingales and Moving Averages

R , ( X t ) t ∈ R is called a moving average process. Stochastics in Turbulence and Finance Moving average processes In the case where K t ( s ) = ϕ ( t − s ) − ψ ( − s ) , that is � t the special case of constant intermittency ( σ t ) t ∈ R reduces to a moving average X t = ϕ ( t − s ) − ψ ( − s ) dW s , t ∈ (2) −∞ Some examples: The OU process, in this case ψ = 0 and ϕ ( t ) = e − β t 1 [ 0 , ∞ ) ( t ) (this is a semimartingale). The fBm with Hurst parameter H ∈ ( 0 , 1 ) , in this case ψ ( t ) = ϕ ( t ) = ( t ∨ 0 ) H − 1 / 2 (this is not a semimartingale for H � = 1 / 2). The model for the turbulent velocity field by Barndorff-Nielsen and Schmiegel in process. Andreas Basse Gaussian Semimartingales and Moving Averages

We will use the following notation: For each process ( Y t ) t ∈ R , we let ( F Y Stochastics in Turbulence and Finance Definitions and notation t ) t ≥ 0 denote = σ ( Y r : r ∈ [ 0 , t ]) and let ( F Y , ∞ the filtration given by F Y ) t ≥ 0 denote the filtration t t given by F Y , ∞ = σ ( Y r : r ∈ ( −∞ , t ]) . t Let ( F t ) t ≥ 0 denote a filtration. Then ( Y t ) t ≥ 0 is said to be an ( F t ) t ≥ 0 -semimartingale if it can be written as Y t = Y 0 + M t + A t , t ≥ 0 , where ( M t ) t ≥ 0 is a càdlàg ( F t ) t ≥ 0 local martingale, ( A t ) t ≥ 0 is an ( F t ) t ≥ 0 -adapted càdlàg process of bounded variation and X 0 is F 0 -measurable. As seen from the definition, the semimartingale property is very filtration dependent. We have the following relation: Let ( G t ) t ≥ 0 and ( F t ) t ≥ 0 denote two filtrations satisfying G t ⊆ F t for all t ≥ 0 . Moreover, let ( Y t ) t ≥ 0 denote an ( F t ) t ≥ 0 -semimartingale which is ( G t ) t ≥ 0 -adapted then ( Y t ) t ≥ 0 is also a ( G t ) t ≥ 0 -semimartingale. Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance Overview over results Let ( X t ) t ≥ 0 be given by (1). In this talk we consider the semimartingale property of ( X t ) t ≥ 0 in the following three filtrations: ( F X , ∞ ( F W , ∞ ( F X t ) t ≥ 0 , ) t ≥ 0 and ) t ≥ 0 . t t Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance Overview over results In Basse(b) we let ( X t ) t ∈ R be a moving average process given by (2). We obtain Let ( X t ) t ≥ 0 be given by (1). In this talk we consider the semimartingale property of ( X t ) t ≥ 0 in the following three filtrations: general Gaussian process ( X t ) t ∈ R with stationary increments which is an ( F X , ∞ ( F W , ∞ ( F X t ) t ≥ 0 , ) t ≥ 0 and ) t ≥ 0 . t t t ) t ≥ 0 and ( F W , ∞ In Basse(a) we let ( X t ) t ≥ 0 given by (1). In the filtrations ( F X ) t ≥ 0 t we derive necessary and sufficient conditions on the kernel K for ( X t ) t ≥ 0 to be a semimartingale. necessary and sufficient conditions on ϕ and ψ for ( X t ) t ≥ 0 to be an ( F X , ∞ ) t ≥ 0 -semimartingale. We also characterize the spectral measure of a t ( F X , ∞ ) t ≥ 0 -semimartingale. t In Basse(c) we study general Gaussian semimartingale. We derive a representation result for them and use it to obtain necessary and sufficient conditions on the covariance function for a Gaussian process to be an ( F X t ) t ≥ 0 -semimartingale. Andreas Basse Gaussian Semimartingales and Moving Averages

R and h ∈ L 2 ( λ ) . Stochastics in Turbulence and Finance A generalisation of F. Knight’s result The following result is due to F. Knight: Let ( X t ) t ≥ 0 be a moving average process given by (2). Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale if and only if t � t ϕ ( t ) = α + h ( r ) dr , t ≥ 0 , 0 where α ∈ Andreas Basse Gaussian Semimartingales and Moving Averages

R and h ∈ L 2 ( λ ) . Let us rewrite this result: Stochastics in Turbulence and Finance A generalisation of F. Knight’s result The following result is due to F. Knight: Let ( X t ) t ≥ 0 be a moving average process given by (2). Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale if and only if R and h ∈ L 2 ( λ ) is 0 on ( −∞ , 0 ) . t � t ϕ ( t ) = α + h ( r ) dr , t ≥ 0 , 0 where α ∈ Let ( X t ) t ≥ 0 be given by (1) and assume K t ( s ) = ϕ ( t − s ) − ϕ ( − s ) . Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale if and only if t � t K t ( s ) = α 1 [ 0 , ∞ ) ( s ) + h ( r + s ) dr , s ≤ t , 0 where α ∈ Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance R and h ∈ L 2 ( λ ) is 0 on ( −∞ , 0 ) . A generalisation of F. Knight’s result Let ( X t ) t ≥ 0 be given by (1) and assume K t ( s ) = ϕ ( t − s ) − ϕ ( − s ) . Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale if and only if t � t K t ( s ) = α 1 [ 0 , ∞ ) ( s ) + h ( r + s ) dr , s ≤ t , 0 where α ∈ Andreas Basse Gaussian Semimartingales and Moving Averages

Stochastics in Turbulence and Finance R and h ∈ L 2 ( λ ) is 0 on ( −∞ , 0 ) . A generalisation of F. Knight’s result Let ( X t ) t ≥ 0 be given by (1) and assume K t ( s ) = ϕ ( t − s ) − ϕ ( − s ) . Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale if and only if R → R is square integrable on ( −∞ , t ] for all t ≥ 0 , µ is a Radon measure t R + and ( t , s ) �→ Ψ r ( s ) is a measurable mapping such that � Ψ r � L 2 ( µ ) = 1 for all � t K t ( s ) = α 1 [ 0 , ∞ ) ( s ) + h ( r + s ) dr , s ≤ t , 0 where α ∈ Theorem: Let ( X t ) t ≥ 0 be given by (1). Then ( X t ) t ≥ 0 is an ( F W , ∞ ) t ≥ 0 -semimartingale t if and only if � t K t ( s ) = g ( s ) + Ψ r ( s ) µ ( dr ) , s ≤ t , 0 where g : on r ≥ 0 and Ψ t ( s ) = 0 if t ≥ s . Andreas Basse Gaussian Semimartingales and Moving Averages

C : | z | = 1 } and for each measurable function f : R → S 1 satisfying R → R be given by R . Stochastics in Turbulence and Finance Theorem: Let ( X t ) t ∈ R denote a moving average process given by (2) with ϕ = ψ. Semimartingales w.r.t. ( F X , ∞ ) t ≥ 0 t R , Let S 1 := { z ∈ R , h ∈ L 2 ( λ ) and f : R → S 1 is measurable and satisfies f = f ( −· ) . If f = f ( −· ) , let ˜ f : � ∞ e its − 1 [ − 1 , 1 ] ( s ) ˜ f ( t ) = f ( s ) ds , t ∈ is −∞ Then ( X t ) t ≥ 0 is an ( F X , ∞ ) t ≥ 0 -semimartingale if and only if ϕ is on the form t � t � ϕ ( t ) = β + α ˜ f ˆ f ( t ) + h ( s ) ds , t ∈ 0 where α, β ∈ α � = 0 , h is 0 on ( 0 , ∞ ) . Moreover, ( X t ) t ≥ 0 is of bounded variation if and only if α = 0 and ( X t ) t ≥ 0 is an ( F X , ∞ ) t ≥ 0 -martingale if and only if h = 0 . t Andreas Basse Gaussian Semimartingales and Moving Averages

Let ( X t ) t ∈ R be a moving average process given by R . Then ( X t ) t ∈ R is a (two-sided) Brownian motion if and only if R → S 1 satisfying f = f ( −· ) . Stochastics in Turbulence and Finance Some applications � X t = ϕ ( t − s ) − ϕ ( − s ) dW s , t ∈ ϕ ( t ) = β + α ˜ f ( t ) for some f : Andreas Basse Gaussian Semimartingales and Moving Averages

Let ( X t ) t ∈ R be a moving average process given by R . Then ( X t ) t ∈ R is a (two-sided) Brownian motion if and only if R → S 1 satisfying f = f ( −· ) . R + ( t ) . Thus Stochastics in Turbulence and Finance Some applications � X t = ϕ ( t − s ) − ϕ ( − s ) dW s , t ∈ ϕ ( t ) = β + α ˜ f ( t ) for some f : Setting f ( t ) = ( t + i )( t − i ) − 1 we obtain ˜ f equals ϕ : t �→ ( e − t − 1 / 2 ) 1 � t X t = ϕ ( t − s ) − ϕ ( − s ) dW s , t ≥ 0 , −∞ is a Brownian motion. Andreas Basse Gaussian Semimartingales and Moving Averages