BSS Processes and Intermittency/Volatility

Transcription

1 BSS Processes and /Volatility Stochastics Ole E. Barndor -Nielsen and Jürgen Schmiegel Thiele Centre Department of Mathematical Sciences University of Aarhus

2 I I I I I I Some open

3 Brownian semistationary (BSS) processes: Y t = Z t Z t g(t s)σ s db s + q(t s)a s ds (1) where B is Brownian motion, g and q are square integrable functions on R, with g (t) = q (t) = 0 for t < 0, and σ and a are cadlag processes. When σ and a are stationary, as will be assumed throughout this talk, then so is Y. It is sometimes convenient to indicate the formula for Y as Y = g σ B + q a Leb. (2)

4 We consider the BSS processes to be the natural analogue, in stationarity related settings, of the class BSM of Brownian semimartingales. I The BSS processes are not in general semimartingales

5 The key object of interest is the integrated variance (IV) Z t σ 2+ t = σ 2 s ds 0 We shall discuss to what extent realised quadratic variation of Y can be used to estimate σ 2+ t. I Note that the relevant question here is whether a suitably normalised version of the realised quadratic variation, and not necessarily the realised quadratic variation itself, converges in probability/law.

6 In semimartingale theory the quadratic variation [Y ] of Y is de ned in terms of the Ito integral Y Y, as [Y ] = Y 2 2Y Y. In that setting [Y ] equals the limit in probability as δ! 0 of the realised quadratic variation [Y δ ] of Y de ned by [Y δ ] t = bt/δc Y jδ Y (j 2 1)δ (3) j=1 where bt/δc is the largest integer smaller than or equal to t/δ. It is this latter de nition of quadratic variation that we will use here.

7 White noise case: Z Y t (x) = µ + A t (x ) g (t s, jξ xj) σ s (ξ) W (dξ, ds) Z + q (t s, jξ xj) a s (ξ) dξds. D t (x ) Here A t (σ) and D t (σ) are termed ambit sets. Lévy case: Z σ 2 t (x) = C t (x ) h (t s, jξ xj) L (dξ, ds)

8 X w Figure: (t(w), σ(w)) A t(w) (σ(w))

9 In turbulence the basic notion of intermittency refers to the fact that the energy in a turbulent eld is unevenly distributed in space and time. The present presentation is part of a project that aims to construct a stochastic process model of the eld of velocity vectors representing the uid motion, conceiving of the intermittency as a positive random eld with random values σ t (x) at positions (x, t) in space-time.

10 However, most extensive data sets on turbulent velocities only provide the time series of the main component (i.e. the component in the main direction of the uid ow) of the velocity vector at a single location in space. In the present talk the focus is on this latter case, but in the concluding Section some discussion will be given on the further intriguing issues that arise when addressing tempo-spatial settings.

11 For simplicity we now assume that σ??b and that q = 0, i.e. there is no drift term in Y and Y t = Z t 0 g (t s) σ s db s. However, at the end of the talk some discussion will be given on more general settings.

12 Let Z = fz t g t2r denote a second order stationary stochastic process, possibly complex valued, of mean 0 and continuous in quadratic mean. Recall that Z is said to be a moving average process if it is of the form Z t = Z φ (t s) dξ s (4) where φ is an, in general complex, deterministic and square integrable functionnand where the process Ξ has orthogonal increments with E jdξt j 2o = ϖ dt for some constant ϖ > 0; nally, the integral in (4) is de ned in the quadratic mean sense.

13 A second order stationary process (of mean 0 and continuous in quadratic mean) is called regular (or linearly regular) provided its future values cannot be predicted by linear operations on past values without error. Such processes can be written in the continuous time Wold decomposition form Z t = Z t φ (t s) dξ s + V t (5) where φ is an, in general complex, deterministic square integrable function satisfying φ (s) = 0 for s < 0 and where the n process Ξ has orthogonal increments with E jdξ t j 2o = ϖdt for some constant ϖ > 0. Finally, the process V is nonregular (i.e. predictable by linear operations on past values without error).

14 Given Z and neglecting sets of measure 0, both φ and Ξ are uniquely determined, φ up to a factor of modulus 1 and the driving process Ξ in the L 2 sense and up to an additive constant. I A slightly stronger condition than regularity is the requirement that Z be completely nondeterministic, meaning that \ t2r sp fz s : s tg = f0g In this case Z = φ Ξ with φ real and uniquely determined up to a real constant of proportionality; and the same is therefore true of Ξ.

15 We begin by recalling a classical necesssary and su cient condition, due to [Kni92], for the semimartingale property of a process X of the form X t = Z t g (t s) db s. (6) Knight s Theorem says that (X t ) t0 is a semimartingale in the ltration if and only if F B, t t0 Z t g (t) = c + b (s) ds (7) 0 for some c 2 R and a square integrable function b.

16 Example An example of some particular interest is where g (t) = t α e λt for t 2 (0, ) and some λ > 0. In order for the integral g σ B to exist 1 α is required to be greater than 2, and for g to be of the form (7) we must have α > 1 2. In other words, the 1 nonsemimartingale cases are α 2 2, 1 2.

17 Recall that the integrated variance (IV) Z t σ 2+ t = 0 σ 2 s ds (8) is the main object of interest and that we wish to discuss the extent to which realised quadratic variation of Y can be used to estimate σ 2+ t. Remark It is convenient to de ne σ 2+ t the same expression. also for t < 0 by

18 Note that, as regards inference on σ 2+, there may be notable di erences between cases where g is positive on all of (0, ) and those where g (t) = 0 for t > l for some l 2 (0, ). I Recall that σ??b.

19 Suppose g (t) = 0 for t > 1. For any t 2 R and u < t, Y t Y t u = Z t t u Z t g (t s) σ s db s + It is illuminating to rewrite this as de ning φ u by Y t Y t u = Z 0 u fg (t s) g (t s φ u ( v) σ v +t db v +t (9) 8 < g (v) for 0 v < u φ u (v) =. : g (v) g (v u) for u v <

20 This implies in particular that the conditional variance of Y t Y t u given the process σ takes the form n o Z E (Y t Y t u ) 2 jσ = ψ u (v) σ 2 t sds (10) 0 where 8 < g 2 (v) for 0 v < u ψ u (v) = : fg (v) g (v u)g 2 for u v < And unconditionally Hence n E (Y t Y t u ) 2o = E σ 2 0 Z 0 ψ u (v) dv..

21 n o E (Y t Y t u ) 2 jσ E n(y t Y t u ) 2o = where R 0 ψ u (v) σ2 t v dv E fσ 2 0 g R 0 ψ = u (v) dv σ 2 = σ 2 /E σ 2 0. Z and π u is the probability measure on [0, 1] having pdf ψ u (v) /c (δ) with c (δ) = 2 kgk 2 (1 r (u)) with r being the autocorrelation function of Y. 0 σ 2 t v π u (d

22 Example g (t) = e λt 1 (0,1) ( case). Then 8 1 for 0 v < δ >< ψ δ (v) = e 2λv e λδ 1 2 for δ v < 1 e >: 2λδ. for 1 v < 1 + δ 0 for 1 + δ v It follows that π δ! π where π = e 2λ δ 0 + e 2λ 1 + e 2λ δ 1 where δ x denotes the delta measure at x.

23 Example g (t) = t α (1 t) β 1 with 2 < α and β 1. The rst inequality ensures existence of the stochastic integral g σ B, and if α is less than 1 2 then we are in the nonsemimartingale situation In this case π = δ 0.

24 Now, de ne the normed realised quadratic variation [Y δ ] of Y as [Y δ ] = δ c (δ) [Y δ]. Then ( n o Z bt/δc E [Y δ ] t jσ = δ 0 j=1 σ 2 jδ v ) π δ (dv). Suppose that π δ converges weakly, as δ! 0, to a probability measure π on [0, l], i.e. π δ w! π. (11)

25 Then we have n o Z 1 E [Y δ ] t jσ! 0 σ 2+ t v σ 2+ v π (dv). (12) In particular, if π = δ 0, the delta measure at 0, then o E n[y δ ] t! σ 2+ t. We will refer to this case by saying that the model for Y is volatility memoryless. (Under a mild restriction the same holds for general g and σ provided the upper limit of the integral is taken as.)

26 On the other hand, if π is a convex combination of the delta measure at 0 and l, i.e. π = θδ 0 + (1 θ) δ l for some θ 2 (0, 1), then n o E [Y δ ]! θσ 2+ t + (1 θ) σ 2+ t l σ 2+ l.

27 Suppose that l = 1 and that g > 0 is a square integrable continuously di erentiable function on (0, 1). If, as δ! 0, we have c (δ) 1 δ 2 = o (1) and if the probability measure π δ converges weakly to a probability measure π on [0, 1] then π is concentrated on the endpoints 0 and 1.

28 To formulate conditions ensuring that π exists and equals δ 0, let Ψ δ (u) = Z u 0 ψ δ (v) dv and Ψ δ (u) = Z 1+δ 1+δ u so that c (δ) 1 Ψ δ is the distribution function of π δ. ψ δ (v) dv,

29 Proposition ε 0 2 (0, 1) then π δ w! δ0. If (δ) 1 δ 2 = o (1) and if for some Ψ δ (ε) Ψ δ (ε)! 0 for all ε 2 (0, ε 0)

30 Remark In case c (δ) δ 2 it may happen that π is absolutely continuous on [0, 1].

31 Above it was assumed that l < (in fact, we took l = 1). The following Example has l =. Example For g (t) = t α e λt (α > 1 2 ) it can be shown, using detailed calculations for this special case given in [BNCP08], that π = δ 0.

32 We now seek conditions under which the conditional variance of the normalised realised quadratic variation tends to 0 as δ! 0, i.e. In that case and provided n o Z E [Y δ ]jσ! 0 holds we have that, conditionally on σ, [Y δ ] t p! Z Varf[Y δ ]jσg! 0. (13) 0 σ 2+ t v σ 2+ v π (dv) (14) σ 2+ t v σ 2+ v π (dv). (15)

33 Let n = bt/δc and n j Y = Y jδ Y (j 1)δ. Via the Cauchy-Schwarz inequality and using that for any pair X and Y of mean zero normal random variables we have it can be shown that CovfX 2, Y 2 g = 2CovfX, Y g 2. (16)

34 Varf[Y δ ] t jσg 2K (σ) 2 n 1 δ + 2 δ k=1 k c k (δ) + k=n where K (σ) = sup 0st σ 2 s (as σ is assumed càdlàg, K (σ) < a.s.) and c k (δ) = c k (δ) c (δ) with c k (δ) Z 1 c k (δ) = δ ψ δ ((k + u) δ) du. 0!!

35 Thus, for Varf[Y δ ]jσg! 0 to be valid it su ces to have n 1 δ k=1 k c k (δ)! 0 and c k (δ)! 0. k=n Example If g (t) = t α (1 t) β 1 with 2 < α and β 1 p then the above two conditions hold and [Y δ ]! σ 2+.

36 [Y δ ] t bt/δc \ fvar fy0 gg (1 ˆr (δ)) Z p! 0 σ 2+ t v σ 2+ v π (dv).

37 So far we have assumed that q = 0 and σ??b. In joint (near completion) with José Manuel Corcuera and Mark Podolski these conditions have been substantially weakened. p This more re ned analysis has shown that [Y δ ]! σ 2+ in wider generality and using the theory of multipower variation and recent powerful results of Malliavin calculus, due to Nualart, Peccati et al, a feasible CLT for [Y δ ] has been established. The results are further extended to consistency and feasible CLTs for multipower variations, in particular for bipower variation (as is essential for inference on σ 2+ ).

38 Above only the case of time-wise behaviour at a single point in space was considered. In the general turbulence setting, space and the velocity vector are three dimensional. There the, analogous to those disussed above, are substantially more intricate, major di erences occurring

39 already for the case of a one-dimensional space component. Outline The realised variation ratio (RVR) is de ned by RVR t = π 2 [Y δ ] [1,1] t [Y δ ] t where [Y δ ] [1,1] is the bipower variation, i.e. [Y δ ] [1,1] t = bt/δc 1 j=1 Y(j+1)δ Y jδ Y (j 1)δ Y jδ. The properties of RVR are presently under study in joint with Neil Shephard.

40 I r = g g? I In cases, can one give a natural meaning to dx t such that d [X ] t = (dx t ) 2? I Volatility modulated Volterra Processes (VMVP): Y t (x) = Y t = Z Z I Relevance for Finance? K t (s) σ s B (ds) K t (ξ, s; x) σ s (ξ) W (dξds) Arbitrage?

41 Barndor -Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2007): Power variation for Gaussian processes with stationary increments. (Submitted.) Barndor -Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2008): Power variation for BSS processes. (In preparation. Title preliminary.) Barndor -Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M. and Shephard, N. (2006): A central limit theorem for realised power and bipower variations of continuous semimartingales. In Yu. Kabanov, R. Liptser and J. Stoyanov (Eds.): From Stochastic Calculus to Mathematical Finance. Festschrift in Honour of A.N. Shiryaev. Heidelberg: Springer. Pp Barndor -Nielsen, O.E., Graversen, S.E., Jacod, J., and Shephard, N. (2006): Limit theorems for bipower variation in nancial econometrics. Econometric Theory 22,

42 Barndor -Nielsen, O.E., Blæsild, P. and Schmiegel, J. (2004): A parsimonious and universal description of turbulent velocity increments. Eur. Phys. J. B 41, Barndor -Nielsen, O.E. and Schmiegel, J. (2004): Lévy-based tempo-spatial modelling; with applications to turbulence. Uspekhi Mat. NAUK 59, Barndor -Nielsen, O.E. and Schmiegel, J. (2006b): Time change and universality in turbulence. Research Report 2007/8. Thiele Centre for Applied Mathematics in Natural Science. Barndor -Nielsen, O.E. and Schmiegel, J. (2007): ; with applications to turbulence and cancer growth. In F.E. Benth, Nunno, G.D., Linstrøm, T., Øksendal, B. and Zhang, T. (Eds.): Stochastic Analysis and Applications: The Abel Symposium Heidelberg: Springer. Pp

43 Barndor -Nielsen, O.E. and Schmiegel, J. (2007): A stochastic di erential equation frame for the timewise dynamics of turbulent velocities. Theory Prob. Its Appl. (To appear.) Barndor -Nielsen, O.E., Schmiegel, J. and Shephard, N. (2008): Realised variation ratio. (In preparation.) Barndor -Nielsen, O.E. and Schmiegel, J. (2007): Time change, volatility and turbulence. To appear in Proceedings of the Workshop on Mathematical Control Theory and Finance. Lisbon Barndor -Nielsen, O.E. and Shephard, N. (2001): Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in nancial economics (with Discussion). J. R. Statist. Soc. B 63, Barndor -Nielsen, O.E. and Shephard, N. (2006): Multipower variation and stochastic volatility. In M. do Rosário Grossinho, A.N. Shiryaev, M.L. Esquível and

44 P.E. Oliveira: Stochastic Finance. New York: Springer. Pp Barndor -Nielsen, O.E. and Shephard, N. (2008): Financial Volatility in Continous Time. (2007). Cambridge University Press. (To appear.) Barndor -Nielsen, O.E., Schmiegel, J. and Shephard, N. (2006): Time change and universality in turbulence and nance. (Submitted.) Basse, A. (2007a): Spectral representation of Gaussian semimartingales. (Unpublished manuscript.) Basse, A. (2007b): Gaussian moving averages and semimartingales. (Unpublished manuscript.) Basse, A. (2007c): Representation of Gaussian semimartingales and applications to the covariance function. (Unpublished manuscript.) Baudoin, F. and Nualart, D. (2003): Equivalence of Volterra processes. Stoch. Proc. Appl. 107,

45 Bender, C. and Marquardt, T. (2007): Stochastic calculus for convoluted Lévy processes. (Unpublished manuscript.) Decreusefond, L. (2005): Stochastic integration with respect to Volterra processes. Ann. I. H. Poincaré PR41, Decreusefond, L. and Savy, N. (2006): Anticipative calculus with respect to ltered Poisson processes. Ann. I. H. Poincaré PR42, Guyon, X. (1987): s des champs gaussian stationaires: applications à l identi cation. Proba. Th. Rel. Fields 75, Guyon, X. and Leon, J. (1989): Convergence en loi des H-variation d un processus gaussien stationaire. Ann. Inst. Henri Poincaré 25,

46 Hult, H. (2003): Approximating some Volterra type stochastic intergrals with applications to parameter estimation. Stoch. Proc. Appl. 105, Istas, J. (2007): Quadratic variations of spherical fractional Brownian motions. Stoch. Proc. Appl. 117, Knight, F. (1992): Foundations of the Prediction Process. Oxford: Clarendon Press. Marquardt, T. (2006): Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, Shiryaev, A.N. (2006): Kolmogorov and the turbulence. Research Report Thiele Centre for Applied Mathematics in Natural Science. Shiryaev, A.N. (2007): On the classical, statistical and stochastic approaches to hydrodynamic turbulence.

47 Research Report Thiele Centre for Applied Mathematics in Natural Science. Outline