Generalized fractional Lévy processes 1

Transcription

1 Generalized fractional Lévy processes 1 Claudia Klüppelberg Center for Mathematical Sciences and Institute for Advanced Study Technische Universität München Sandbjerg, January joint work with Muneya Matsui, Keio University, Japan partly extends work by Tina Marquardt; cf. also joint work with Holger Fink

2 Motivation from stochastic volatility models As limit models, for standard Brownian motion B and fractional Brownian motion B H (H (0, 1/2)), we consider types of models and dz(t) = (µ + βv(t))dt + v(t)db(t), v(t) = f (y(t)) for f : + and dy(t) = λy(t)dt + db H (t), λ > 0, dz(t) = (µ + βv(t))dt + v(t)db(t) d(log v(t)) = λ log v(t)dt + σdb H (t). 2

3 Generalized fractional Lévy processes Throughout: L is a two-sided centered, finite variance Lévy process, no Gaussian component and Lévy measure ν. W.l.o.g. : E[(L(t)) 2 ] = te[(l(1)) 2 ] = t x2 ν(dx) = t for all t 0. ecall, E[exp{iθL(t)}] = exp{tψ(θ)} for t 0, where ψ(θ) = (e iθx 1 iθx)ν(dx), θ. (1) Definition Let g : + + with g(0) = 0 and such that {g((t s) +) g(( s) + )} 2 ds < for all t. Then S(t) = {g((t u) + ) g(( u) + )}dl(u), t, (2) is called generalized fractional Lévy process (GFLP). Example Define g(t) = t H 2 1, then S is a fractional Lévy process. 3

4 For x > 0 let σ 2 (x) := Var[S(x)] and define the time scaled GFLP S x (t) := S(xt) σ(x), t. Define Γ x (s, t) = Cov[S(xs), S(xt)] σ 2 (x), s, t > 0. Theorem Let S be as in (2) with derivative g V ρ 1 for ρ (0, 1 2 ) and B H FBM with H = ρ + 1/2. Then S x d B H as x, where convergence holds in the Skorokhod space D() with the metric of uniform convergence on compacta. 4

5 Stochastic integrals with respect to a GFLP Throughout: g : + + has derivative g > 0. Note that for t > 0 g((t u) + ) g(( u) + ) = 1 (0,t] (v)g ((v u) + )dv, u. Use g as extension of the iemann-liouville kernel function and define for appropriate functions h (I h)(u) g := h(v)g (v u)dv = h(v)g ((v u) + )dv, u. u Proposition Let g > 0 and 1 0 g (s)ds + 1 (g (s)) 2 ds <. Define H := { h : + : (I h) g 2 (u)du < }. 5 If h L 1 () L 2 (), then h H.

6 The space H Define H as the completion of L 1 () L 2 () wrt the norm h H := ( E[(L(1)) 2 ] (I h) g 2 (u)du ) 1/2. Theorem Let S be a GFLP with kernel function g and let h H. Then in the L 2 (Ω) sense: h(u)ds(u) = (I h)(u)dl(u) g. Moreover, the following isometry holds: h(u)ds(u) 2 L 2 (Ω) = h 2 H. 6

7 Note: 7 L 2 (Ω) and H are inner product spaces and for h 1, h 2 H, h 1 (u)ds(u), h 2 (u)ds(u) = h 1, h 2 H. L 2 (Ω) Proposition Let S be a GFLP with kernel function g. Then for h 1, h 2 H, Cov [ h 1 (u)ds(u), h 2 (v)ds(v) ] = h 1, h 2 H. Now, Γ(u, v) = 2 Cov[S(u), S(v)] u v and, in particular, h1, h 2 H = = g ((u w) + )g ((v w) + )dw, h 1 (u)h 2 (v) g ((u w) + )g ((v w) + )dwdudv.

8 Fractional Ornstein-Uhlenbeck type processes Definition Let B H be FBM for 0 < H < 1 and let λ, σ > 0. (i) For an initial finite random variable Y(0) a fractional Ornstein-Uhlenbeck process (FOU) is defined as t ) Y H (t) := e (Y λt H (0) + σ e λs db H (s), t. (ii) If the initial random variable is given by 0 Y H (0) = σ e λs db H (s), the FOU is stationary and we denote this stationary version by 0 t Y H (t) = σ e λ(t s) db H (s), t. 8

9 Definition Let S be a GFLP and let λ, σ > 0. (i) For finite V(0) an OU process driven by a GFLP is defined as t ) V(t) := e (V(0) λt + σ e λu ds(u), t. (ii) If the initial random variable is given by 0 0 V(0) = σ e λu ds(u), t, the OU process driven by a GFLP is stationary and we denote this stationary version by t V(t) = σ e λ(t u) ds(u), t. 9

10 Proposition Let S be a GFLP and λ > 0. For all t, in L 2 (Ω), 10 V(t) := t e λ(t u) ds(u) = t (I g e λ(t ) )(u)dl(u). Furthermore, for all s, t we have E[V(t)] = 0 and Cov[V(s), V(t)] = where, we recall, t Γ(u, v) = 2 Cov[S(u), S(v)] u v s = The chf of V(t 1 ),..., V(t m ) for t 1 < < t m is j=1 e λ(t u) e λ(s v) Γ(u, v)dudv, j=1 g ((u w) + )g ((v w) + )dw. E [ exp { m iθ j V(t j ) }] = exp { ψ ( m tj θ j e λ(tj v) g ((v s) + )dv ) ds }, where θ j, j = 1,..., m, and ψ is given in (1).

11 Limit theory for OU processes driven by time scaled GFLPs For x > 0 let σ 2 (x) := Var[S(x)] and recall the time scaled GFLP S x (t) := S(xt) σ(x), t, and note that for x > 0 S(xt) = 1 (0,tx] (v)ds(v) = (I 1 g (0,tx] )(u)dl(u), t 0. 11

12 Theorem Let S be a GFLP with kernel function g and let h H. (i) Then for x > 0, in the L 2 (Ω) sense, h(u)ds x (u) = x h(v)g ((xv u) + )dvdl(u). σ(x) (ii) Assume that h t H for t. Then Cov [ h s (u)ds x (u), h t (u)ds x (u) ] = h t (u)h s (v)γ x (u, v)dudv, where Γ x (u, v) = x2 σ 2 (x) g (x(u w) + )g (x(v w) + )dw. 12

13 Theorem If g V ρ 1 for ρ (0, 1 2 ) and H = ρ + 1 2, then for s, t, lim Γ x 2 x(s, t) = lim g (x(s w) + )g (x(t w) + )dw x x σ 2 (x) = ρ2 (s w)ρ 1 + (t w) ρ 1 + dw {(1 u)ρ + ( u) ρ +} 2 du 2 = t s Cov(BH (t), B H (s)) = H(2H 1) t s 2H 2 lim Γ x (s, t) = {(s u)ρ + ( u) ρ +}{(t u) ρ + ( u) ρ +}du x {(1 u)ρ + ( u) ρ +} 2 du = Cov(B H (s), B H (t)) = 1 2 ( t 2H + s 2H t s 2H ). 13

14 Note that E[S x (t)] = 0 and Var[S x (t)] = 1 for all t. Definition For λ, σ > 0 we define the OU process driven by the time-scaled GFLP t ) V x (t) := e (V λt x (0) + σ e λs ds x (s), t 0. (ii) If the initial random variable is given by 0 V x (0) = σ e λs ds x (s), then V x is stationary and we denote this stationary process by 0 t V x (t) := σ e λ(t s) ds x (s), t. 14

15 Proposition For x > 0 let S x be a time-scaled GFLP. (i) For t, in the L 2 (Ω)-sense, t V x (t) = e λ(t u) (u)ds x (u) = x t e λ(t v) g ((xv u) + )dvdl(u). σ(x) (ii) For s, t, we have E[V x (t)] = 0 and s t Cov[V x (s), V x (t)] = where Γ x (u, v) = x2 σ 2 (x) e λ(t u) (u)e λ(s v) Γ x (u, v)dudv, g (x(u w) + )g (x(v w) + )dw. 15

16 Theorem Let S be a GFLP with kernel function g and g V ρ 1 for ρ (0, 1 2 ). For x > 0, recall that t V ρ x(t) = σ e λ(t s) ds x (s), t, is a stationary OU process driven by the time-scaled GFLP. Then V x ( ) d Y H ( ) = σ e λ(t s) db H (s) as x, where convergence holds in D() with the metric of uniform convergence on compacta. ecall that H = ρ

17 We show joint weak convergence of price and volatility process to a continuous time long memory stochastic volatility model. Theorem [Jacod and Shiryaev] For x > 0 we consider, with H = ρ + 1 2, z x (t) := µt + β t v x (t) := f (V ρ x(t)), 0 t v x (s)ds + vx (s)db s, 0 where f : + is a continuous function. Then for x the bivariate process (z x ( ), v x ( )) converges in D( 2 +), with the mtric of uniform convergence on compacta, to t t z(t) := µt + β v(s)ds + v(s)dbs, 0 0 v(t) := f (Y H (t)). 17