Self-similar Markov processes

Transcription

1 Self-similar Markov processes Andreas E. Kyprianou 1 1 Unversity of Bath

2 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

3 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

4 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

5 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

6 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

7 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

8 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

9 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

10 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

11 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

12 Which are our favourite stochastic processes? Markov chains Diffusions Brownian motion Cts-time Markov processes with jumps Lévy processes \ Self-similar Markov processes

13 Lévy processes Stick to one-dimension A Lévy process is an R-valued random trajectory {X t : t 0} issued from the origin with paths that are right-continuous and left limits and which has stationary and independent increments. More formally stationary and independent increments means: for 0 s t <, X t X s = X t s for 0 s t <, X t X s = X t s. It can be shown that this means the entire process is characterised by its position at time 1 E[e iθxt ] = e Ψ(θ)t for some appropriate function Ψ (the characteristic exponent).

14 Lévy processes Stick to one-dimension A Lévy process is an R-valued random trajectory {X t : t 0} issued from the origin with paths that are right-continuous and left limits and which has stationary and independent increments. More formally stationary and independent increments means: for 0 s t <, X t X s = X t s for 0 s t <, X t X s = X t s. It can be shown that this means the entire process is characterised by its position at time 1 E[e iθxt ] = e Ψ(θ)t for some appropriate function Ψ (the characteristic exponent).

15 Lévy processes Stick to one-dimension A Lévy process is an R-valued random trajectory {X t : t 0} issued from the origin with paths that are right-continuous and left limits and which has stationary and independent increments. More formally stationary and independent increments means: for 0 s t <, X t X s = X t s for 0 s t <, X t X s = X t s. It can be shown that this means the entire process is characterised by its position at time 1 E[e iθxt ] = e Ψ(θ)t for some appropriate function Ψ (the characteristic exponent).

16 Lévy processes Stick to one-dimension A Lévy process is an R-valued random trajectory {X t : t 0} issued from the origin with paths that are right-continuous and left limits and which has stationary and independent increments. More formally stationary and independent increments means: for 0 s t <, X t X s = X t s for 0 s t <, X t X s = X t s. It can be shown that this means the entire process is characterised by its position at time 1 E[e iθxt ] = e Ψ(θ)t for some appropriate function Ψ (the characteristic exponent).

17 Brownian motion

18 Compound Poisson process

19 Brownian motion + compound Poisson process

20 Unbounded variation paths

21 Bounded variation paths

22 Space exploration: some successes and dissatisfaction Fundamentally we want to understand how Lévy processes explore space. 25 years of research has been very successful in giving an (relatively) complete theoretical description......with the caveat that the database of tractable examples for the aforesaid theory is uncomfortably small (relative to Markov chains and diffusions).

23 Space exploration: some successes and dissatisfaction Fundamentally we want to understand how Lévy processes explore space. 25 years of research has been very successful in giving an (relatively) complete theoretical description......with the caveat that the database of tractable examples for the aforesaid theory is uncomfortably small (relative to Markov chains and diffusions).

24 Space exploration: some successes and dissatisfaction Fundamentally we want to understand how Lévy processes explore space. 25 years of research has been very successful in giving an (relatively) complete theoretical description......with the caveat that the database of tractable examples for the aforesaid theory is uncomfortably small (relative to Markov chains and diffusions).

25 Space exploration: some successes and dissatisfaction Example 1: P(Process first exceeds level x by an amount y) = U(dz) ν(z x +y) [0,x) where Ψ(θ) = κ + ( iθ)κ (iθ), θ R, κ + (λ) = q + δλ + (1 e λx )ν(dx), λ 0, ν(x) = ν(x, ) (0, ) and [0, ) e λx U(dx) = 1 κ + (λ)

26 Space exploration: some successes and dissatisfaction Example 1: P(Process first exceeds level x by an amount y) = U(dz) ν(z x +y) [0,x) where Ψ(θ) = κ + ( iθ)κ (iθ), θ R, κ + (λ) = q + δλ + (1 e λx )ν(dx), λ 0, ν(x) = ν(x, ) (0, ) and [0, ) e λx U(dx) = 1 κ + (λ)

27 Space exploration: some successes and dissatisfaction Example 2: Under appropriate assumptions, P(Process ever hits a point x) = u(x) u(0), x R, where R e iθx u(x)dx = 1 Ψ(θ), θ R.

28 Self-similar Markov processes on R α-ssmp R-valued Markov process, equipped with initial measures P x, x R\{0}, with 0 an absorbing state, satisfying the scaling property ( d cxc α t) = X Pcx, x, c > 0 Px t 0

29 Space-time changes and modulation It turns out that everyn R-valued ssmp can be characterised using polar coordinates in C (think re iθ ) as follows: X t = x exp { ξ ϕ( x α t) + iπ(j ϕ( x α t) + 1) }, t 0, x 0, where (ξ, J) is a so-called Markov modulated Lévy process and { s } ϕ(t) = inf s > 0 : e αξu du > t. 0 (ξ, J): J = {J t : t 0} is a Markov chain on {1, 2} with intensity matrix Q. When J t = i, ξ moves as a Lévy process of type i. dξ t = dξ (i) t When J makes a jump at time t, e.g. 1 2, then ξ experiences an additional jump ξ t which is an i.i.d. copy of some pre specified r.v. U 1,2.

30 Space-time changes and modulation It turns out that everyn R-valued ssmp can be characterised using polar coordinates in C (think re iθ ) as follows: X t = x exp { ξ ϕ( x α t) + iπ(j ϕ( x α t) + 1) }, t 0, x 0, where (ξ, J) is a so-called Markov modulated Lévy process and { s } ϕ(t) = inf s > 0 : e αξu du > t. 0 (ξ, J): J = {J t : t 0} is a Markov chain on {1, 2} with intensity matrix Q. When J t = i, ξ moves as a Lévy process of type i. dξ t = dξ (i) t When J makes a jump at time t, e.g. 1 2, then ξ experiences an additional jump ξ t which is an i.i.d. copy of some pre specified r.v. U 1,2.

31 Space-time changes and modulation It turns out that every ssmp can be characterised using polar coordinates (think re iθ ) as follows: X t = x exp { ξ ϕ( x α t) + iπ(j ϕ( x α t) + 1) }, t 0, x 0, where (ξ, J) is a so-called Markov modulated Lévy process and { s } ϕ(t) = inf s > 0 : e αξu du > t. 0 (ξ, J): Markov modulated Lévy processes can also be characterised by a characteristic exponetnt. where Ψ(θ) = E i [e iθxt ; J t = j] = (exp{ Ψ(θ)t}) i,j ( Ψ1 (θ) 0 0 Ψ 2 (θ) ) ( Q 1 E(e iθu1,2 ) E(e iθu2,1 ) 1 )

32 Space-time changes and modulation If the Markov chain has an absorbing state, then the ssmp is in effect a positive self-similar Markov process (pssmp) where ξ is a Lévy process. X t = x exp { ξ ϕ( x α t)}, t 0, x 0,

33 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

34 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

35 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

36 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

37 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

38 Positive feedback There is one class of Lévy processes which has always been considered to be the next best thing after Brownian motion : the (α, ρ)-stable ( process. ) Ψ(θ) = θ α e πiα( 1 2 ρ) 1 {θ>0} + e πiα( 1 2 ρ) 1 {θ<0}, θ R, Only allowed to take α (0, 2], ρ [0, 1]. In fact stable processes are also self-similar Markov processes: E[e iθxt ] = e Ψ(θ)t and E[e iθcx c α t ] = e Ψ(θ)t for all c > 0. So α is the index of self-similarity, but ρ [0, 1] is a symmetry index. When ρ = 1/2, Ψ(θ) = θ α so X = d X. When ρ (0, 1) (keep away from complete asymmetry!) and α (0, 2) then the underlying Markov modulated Lévy process has exponent Ψ(θ) = Γ(α iθ)γ(1+iθ) Γ(αˆρ iθ)γ(1 αˆρ+iθ) Γ(α iθ)γ(1+iθ) Γ(αˆρ)Γ(1 αˆρ) Γ(α iθ)γ(1+iθ) Γ(αρ)Γ(1 αρ) Γ(α iθ))γ(1+iθ) Γ(αρ iθ)γ(1 αρ+iθ).

39 Positive feedback Take the trajectory of an (α, ρ), α (0, 1), ρ (0, 1) and imagine cutting out all the negative parts of its trajectory and then shunting up the remaining bits of path the resulting object is a positive self-similar Markov process. The underlying Lévy process, ξ, has exponent Ψ(θ) = Γ(αρ iθ) Γ( iθ) Γ(1 αρ + iθ) Γ(1 α + iθ), θ R.

40 Positive feedback Take the trajectory of an (α, ρ), α (0, 1), ρ (0, 1) and imagine cutting out all the negative parts of its trajectory and then shunting up the remaining bits of path the resulting object is a positive self-similar Markov process. The underlying Lévy process, ξ, has exponent Ψ(θ) = Γ(αρ iθ) Γ( iθ) Γ(1 αρ + iθ) Γ(1 α + iθ), θ R.

41 Space exploration: some successes and dissatisfaction P(Process first exceeds level x by an amount y) = U(dz) ν(z x +y) [0,x) where Ψ(θ) = κ + ( iθ)κ (iθ), θ R, κ + (λ) = q + δλ + (1 e λx )ν(dx), λ 0, ν(x) = ν(x, ) (0, ) and [0, ) e λx U(dx) = 1 κ + (λ) where P(Process ever hits a point x) = u(x) u(0), x R, R e iθx u(x)dx = 1 Ψ(θ), θ R.

42 Positive feedback α (0, 1) P(Stable process first enters [0, 1] in dy) = P(ξ first enters (, 0] in d(log y)) = sin(παˆρ) x αρ y αρ (x 1) αˆρ (1 y) αˆρ (x y) 1 dy π α (1, 2) P(Stable process hits 1 before 0 when starting from x> 0) = P(ξ ever hits 0 when starting from log x) = sin(πρα) x 1 α 1 [1 (x>1) sin(π ˆρα) + 1 (0<x<1) sin(πρα)] + x α 1 sin(π ˆρα). (sin(πρα) + sin(π ˆρα))

43 Positive feedback α (0, 1) P(Stable process first enters [0, 1] in dy) = P(ξ first enters (, 0] in d(log y)) = sin(παˆρ) x αρ y αρ (x 1) αˆρ (1 y) αˆρ (x y) 1 dy π α (1, 2) P(Stable process hits 1 before 0 when starting from x> 0) = P(ξ ever hits 0 when starting from log x) = sin(πρα) x 1 α 1 [1 (x>1) sin(π ˆρα) + 1 (0<x<1) sin(πρα)] + x α 1 sin(π ˆρα). (sin(πρα) + sin(π ˆρα))

44 A bigger picture It s possible to extend the notion of both Lévy processes and ssmp to higher dimensions For example, a d-dimensional isotropic stable Lévy process is also a ssmp: E[e iθ Xt ] = exp{ θ α t}, t 0, θ R d, necessarily α (0, 2]. The radial distance of such a process from the origin, X t, t 0, is a pssmp. Its underlying Lévy process has characteristic exponent Ψ(θ) = Γ( 1 2 ( iθ + α)) Γ( 1 2 iθ) Γ( 1 2 (iθ + d)) Γ( 1 θ R. 2 (iθ + d α)),

45 A bigger picture It s possible to extend the notion of both Lévy processes and ssmp to higher dimensions For example, a d-dimensional isotropic stable Lévy process is also a ssmp: E[e iθ Xt ] = exp{ θ α t}, t 0, θ R d, necessarily α (0, 2]. The radial distance of such a process from the origin, X t, t 0, is a pssmp. Its underlying Lévy process has characteristic exponent Ψ(θ) = Γ( 1 2 ( iθ + α)) Γ( 1 2 iθ) Γ( 1 2 (iθ + d)) Γ( 1 θ R. 2 (iθ + d α)),

46 A bigger picture It s possible to extend the notion of both Lévy processes and ssmp to higher dimensions For example, a d-dimensional isotropic stable Lévy process is also a ssmp: E[e iθ Xt ] = exp{ θ α t}, t 0, θ R d, necessarily α (0, 2]. The radial distance of such a process from the origin, X t, t 0, is a pssmp. Its underlying Lévy process has characteristic exponent Ψ(θ) = Γ( 1 2 ( iθ + α)) Γ( 1 2 iθ) Γ( 1 2 (iθ + d)) Γ( 1 θ R. 2 (iθ + d α)),

47 Bogdan-Zak transform The Kelvin transform is an inversion of R d through the unit sphere: Kx = x x 2, x Rd.

48 Bogdan-Zak transform Bogdan-Zak transform Suppose that X is a d-dimensional isotropic stable process with d 2. Define η(t) = inf{s > 0 : s 0 X u 2α du > t}, t 0. Then, for all x R d \{0}, {KX η(t) : t 0} under P x is equal in law to (X, P h Kx ), where dp h x = X t α d, t 0, dp x x α d σ(xs :s t)

49 Bogdan-Zak transform The Kelvin transform maps the ball {x R d : x 1/2 1/2} to a half-space.

50 Bogdan-Zak transform How does an isotropic stable process first enter/exit a ball? how does an isotropic stable process cross a hyperplane use rotational symmetry to the orthogonal projection Where does an isotropic stable process first hit the surface of a sphere? where does an isotropic stable process hit a hyperplane use rotational symmetry to the orthogonal projection

51 Bogdan-Zak transform How does an isotropic stable process first enter/exit a ball? how does an isotropic stable process cross a hyperplane use rotational symmetry to the orthogonal projection Where does an isotropic stable process first hit the surface of a sphere? where does an isotropic stable process hit a hyperplane use rotational symmetry to the orthogonal projection

52 Bogdan-Zak transform How does an isotropic stable process first enter/exit a ball? how does an isotropic stable process cross a hyperplane use rotational symmetry to the orthogonal projection Where does an isotropic stable process first hit the surface of a sphere? where does an isotropic stable process hit a hyperplane use rotational symmetry to the orthogonal projection

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