Convoluted Brownian motions: a class of remarkable Gaussian processes

Transcription

1 Convoluted Brownian motions: a class of remarkable Gaussian processes Sylvie Roelly Random models with applications in the natural sciences Bogotá, December 11-15, 217 S. Roelly (Universität Potsdam) 1 / 3

2 Content of the course 1 Brownian motion 2 The Ornstein-Uhlenbeck process(es) 3 Interlude: who is Gaussian and Markov? 4 About bridges of Gaussian Markov processes 5 Convoluted Brownian motions S. Roelly (Universität Potsdam) 2 / 3

3 Brownian motion Brownian motion: historical remarks The botanist, Robert Brown, noted (1828) a chaotic, ubiquitous, not life-related motion of pollen grains suspended in water. Albert Einstein (195) published a physical explanation: the pollen motion results from its collisions with the fast-moving individual water molecules. This explanation served as convincing evidence that atoms and molecules exist: Atomic theory, attacked by a.o. Mach, Ostwald. Jean Perrin (198 ) verified experimentally the quantitative analysis of Einstein, based on Fick s law. Nobel Prize in Physics in 1926 for his work on the discontinuous structure of matter. S. Roelly (Universität Potsdam) 3 / 3

4 Brownian motion First basic physical properties The mean displacement of the pollen grain is proportional to the square root of the time: B(t + t) B(t) 6D t lim B(t)/ t +. t The diffusion constant satisfies D = N 6πkρ, with R: universal gas constant, k: viscosity of the fluid, ρ: the particle radius, N: Avogadro s number, T : absolute temperature The trajectories of B(t) are fractal and random RT S. Roelly (Universität Potsdam) 4 / 3

5 Brownian motion... but, statistically, the density of presence of the particle at time t at the point y, if the path started in x, is: P (B(t) y if B() = x) = k e 1 x y 2 2 2Dt =: pt B (x, y) This is a Gauß bell-shaped curve centred in x with variance 2Dt. Diffusion at time t, 4t and 16t. The density p B t (x, y) = p B t (x y) solves the heat equation PDE: t p t = D ( 2 x x x3 2 )p t S. Roelly (Universität Potsdam) 5 / 3

6 Brownian motion Paths are continuous but they do not admit any tangent at any point (infinite velocity) On the 2D-plane any trajectory visits any fixed bounded region at some time (Recurrence). But in the 3D-space it is probable that a path after some time never come back to a fixed bounded region (Transience) : P( t : B(t) y r B() = x) = r x y < 1 As function of x this map is harmonic at the exterior of the ball. Paths have independent increments. Fine properties of Brownian paths where first studied by Paul Lévy ( ). S. Roelly (Universität Potsdam) 6 / 3

7 Brownian motion Gaussian processes Reminder A Gaussian process (X t ) t I is a collection of rv such that for any < t 1 < < t n in I, the random vector (X t1,, X tn ) is a n-dimensional Gaussian rv. Here I = [, 1] or [, [. Equivalently, for any α 1,, α n R, the R-valued rv α 1 X t1 + α n X tn is Gaussian. The law of the Gaussian process (X t ) t I is characterized by its mean function m m(t) := E(X t ), t I and its covariance function K: K(s, t) := Cov(X s, X t ) = E(X s X t ) E(X s )E(X t ), s, t I. S. Roelly (Universität Potsdam) 7 / 3

8 Brownian motion The Brownian motion as Gaussian process (Mathematical) Definition: A standard Brownian motion is a centered Gaussian process (B(t)) t I characterized by m(t), t I and the covariance function K B (s, t) = min{s, t}, s, t [, [. It then satisfies: B(). (x + B(t)) t I is a Brownian motion starting at x. It is a Gaussian process with the same covariance function K B but with mean m(t) x, t I. S. Roelly (Universität Potsdam) 8 / 3

9 Brownian motion Markov processes Define, for any time t [, 1] F [,s] := σ{x r, r s}, the σ-algebra of the past of the time s F [s,1] := σ{x r, s r 1}, the future of the time s. Reminder: X = (X t ) t [,1] is an (homogeneous) Markov process on Ω if and only if F [,s] and F [s,1] are independent conditionally to F {s} for any f : R R bounded and s t, E ( f (X t ) F [,s] ) = St s f (X s ) where S t f (x) := E ( f (X t ) X = x ) satisfies the semigroup property S t S s f (x) = S t+s f (x), x R, s, t. The associated infinitesimal generator is defined as S t f (x) S f (x) Lf (x) := lim. t t S. Roelly (Universität Potsdam) 9 / 3

10 Brownian motion The Brownian motion as Markov process The semigroup of the Brownian motion satisfies, for any t >, E ( f (B(t)) B() = x ) = = ( f (y) exp 1 2πt R f (y)pt B (x, y) dy R = p B t f (x) (y ) x)2 dy 2t where p B t (x, y) is the transition probability density of the Brownian motion from x to y. Its infinitesimal generator is the Laplace operator L B f (x) = 1 2 f (x). S. Roelly (Universität Potsdam) 1 / 3

11 Brownian motion The Brownian motion as unique solution of a duality formula Stein s formula: By integration by parts R ϕ (x) e x2 /2 dx = ϕ(x) x e x2 /2 dx 2π R 2π for any ϕ Cb (R; R). This identity even characterizes the Gaussian law N (, 1). What about a generalisation on the path space Ω = C([, 1]; R) to characterize the Brownian motion: D g Φ(B) dp = Φ(B)? dp. Ω for which functions g, Φ and which kind of derivation operator D g? Ω S. Roelly (Universität Potsdam) 11 / 3

12 Brownian motion Test functionals: smooth cylindrical functionals on Ω S = {Φ : Φ(ω) = ϕ(ω t1,..., ω tn ) for some ϕ C b (Rn ; R), n N}. Derivation operator D g, g L 2 ([, 1]; R): 1 ( D g Φ(ω) := lim Φ(ω + ε ε ε n tj = j=1. ) g(t)dt) Φ(ω) g(t) ϕ x j (ω t1,..., ω tn ) dt D g Φ :Gâteaux/Malliavin -derivative of Φ in the direction. g(t)dt Stochastic integration: δ g (ω) := 1 g(t) dω t. Theorem Let Q be a Probability measure on Ω such that E Q ( ω t ) < + for all t. Q is the law of a Brownian motion for all Φ S and g step function, E Q (D g Φ) = E Q ( Φ δg ). (IBP) S. Roelly (Universität Potsdam) 12 / 3

13 Brownian motion Remarks: Paths are not perturbed at time, thus (IBP) characterizes only the dynamics but not the law of the initial condition under Q. (IBP) holds for a much larger class of g and Φ: for any Skorohod integrable g L 2 (Ω [, 1]; R) and for any Φ D 1,2, closure of S under the norm Φ 2 1,2 := E(Φ2 + 1 D tφ 2 dt). (IBP) duality between Malliavin derivative D and Skorohod integral δ under the Wiener measure (see e.g. Bismut, 81). Sketch of the proof: Using Girsanov theorem, E(D g Φ(B)) = ( E lim ε Φ(B + ε. g(t)dt) Φ(B) ) ( = E ε Φ(B) lim ε E(εg) 1 ε with E(εg) := exp(ε 1 ε2 1 g(t)db(t) 2 g(t)2 dt) exp(εδ g ). Using Lévy s theorem, since (IBP) leads to the martingale property of ω t and ω 2 t t under Q. (See R.-Zessin, 91). ) S. Roelly (Universität Potsdam) 13 / 3

14 Ornstein-Uhlenbeck processes The Ornstein-Uhlenbeck process What happens if a Brownian particle is submitted additionally to an attractive linear force towards? If it starts in x, its paths would solve the Stochastic Integral Equation in R X t = x + B(t) λ t X s ds, t. This motion is the prototype of a noisy relaxation process, which returns into equilibrium after a relaxation time. Application in physics: Time evolution of the velocity of a particle under the influence of friction due to the medium Application in financial mathematics: Random model for interest rates, currency exchange rates and commodity prices. This linear equation can be solved explicitly : X ou t = xe λt + t e λ(t s) db(s). S. Roelly (Universität Potsdam) 14 / 3

15 Ornstein-Uhlenbeck processes The Ornstein-Uhlenbeck process as... X ou is a Gaussian process with mean function and with covariance function m ou (t) := E(X ou t ) = xe λt, K ou (s, t) = e λ(t s) e λ(t+s), s t. 2λ In particular VarX t = 1 e 2λt 2λ. X ou is a Markov process whose semigroup satisfies St ou (x, f ) := E ( f (Xt ou ) X ou = x ) = 1 2πt R f (y) exp with t := 1 e 2λt 2λ. Its generator is: L ou f (x) = 1 2 f (x) λxf (x). ( ( y xe λt ) 2 ) 2t dy, S. Roelly (Universität Potsdam) 15 / 3

16 Ornstein-Uhlenbeck processes The stationary Ornstein-Uhlenbeck process Can one randomize the initial value x of the OU-process in order to obtain a stationary process? Yes! Taking X N (, 1/2λ) independent of the Brownian motion B in the integral representation t X t ou = X e λt + e λ(t s) db(s) one obtains and X ou t Cov( X ou s (L) = X N (, 1/2λ) t,, X ou t ) = e λ(t s), s t. 2λ S. Roelly (Universität Potsdam) 16 / 3

17 Gauß-Markov Interlude: who is Gaussian and Markov? Let (X t ) t [,1] be a centered L 2 -continuous Gaussian process with X t. It is Markov if and only if, equivalently (Neveu, 68) its covariance function is triangular, i.e. K(s, t) = ϕ(min{s, t}) ψ(max{s, t}) for some positive continuous functions ϕ and ψ such that ϕ/ψ Brownian motion: ϕ = Id and ψ 1 Ornstein-Uhlenbeck process: K ou (s, t) = sinh(λs) e λt, s t stationary Ornstein-Uhlenbeck: K ou (s, t) = eλs its covariance function satisfies λ e λt 2λ 2λ K(s, u) K(t, t) = K(s, t) K(t, u), s t u 1 There exists a Brownian motion B such that X t ψ(t) B( ϕ ψ (t)). stationary Ornstein-Uhlenbeck: X ou t e λt 2λ B(e λ(s+t) ) S. Roelly (Universität Potsdam) 17 / 3

18 Bridges Pinning a Markov process at initial time Let (X t ) t 1 be a Markov process. Its law on the path space is given by its law at time and by its dynamics (i.e. its semigroup S). Pinning at initial time. Let µ be the law of X. The law of the process can be disintegrated as follows: P(X X µ) = P(X X = x)µ(dx). R In particular E ( f (X t ) X µ ) = R S t f (x) µ(dx). S. Roelly (Universität Potsdam) 18 / 3

19 Bridges About bridges Let (X t ) t 1 be a "nice" Markov process with dynamics described by its semigroup S, associated to a density probability p t. Pinning at initial and terminal time. Let ν be the joint law of the pair (X, X 1 ). The law of the process can be disintegrated as follows: P(X ) = P(X X = x, X 1 = y) ν(dx, dy). R 2 Under P(X X = x, X 1 = y), the paths have fixed initial and terminal values. This process is called a xy-bridge, denoted by X xy. Any bridge of X is a time-inhomogeneous Markov process. If X admits a transition density p t then X xy admits the transition probability: P(X xy t (z, z + dz) Xs xy = w) p t s(w, z)p 1 t (z, y) dz. p 1 s (w, y) S. Roelly (Universität Potsdam) 19 / 3

20 Bridges Brownian bridges The -Brownian bridge is a centered Markov Gaussian process with covariance K B (s, t) = s(1 t), s t 1. It has the same law than ( B t t B 1 ) t 1, where B is a Brownian motion. More generally, the xy-brownian bridge is a non centered Markov Gaussian process which can be represented as B xy t x + t(y x) + B t t B 1, t 1 It is a random perturbation of the linear interpolation. B xy solves the stochastic integral equation X t = x + B t + where B is a Brownian motion. t y X s ds, t [, 1[ 1 s S. Roelly (Universität Potsdam) 2 / 3

21 Bridges Ornstein-Uhlenbeck bridges The xy-ornstein-uhlenbeck bridge is a non centered Markov Gaussian process which solves the stochastic integral equation: for t [, 1[, t t X t = x + B t λ X s ds + where B is a Brownian motion. When λ, one recovers the Brownian bridge! λ ( sinh ( λ(1 s) ) y e λ(1 s) X s )ds, What happens if, instead to force the OU-process to reach y, one forces it to be periodic, that is to satisfy X 1 = X? S. Roelly (Universität Potsdam) 21 / 3

22 Convoluted Brownian motion PerOU The periodic Ornstein-Uhlenbeck process, called PerOU and denoted by X ou, solves the following stochastic integral equation with periodic boundary conditions: X t t = X + B t λ X 1 = X, where B is a standard Brownian motion. X s ds, t [, 1], Does X ou exist? If yes is it still Markov, still Gaussian? Attention! Due to the fact that its initial condition involves its final one, the PerOU process can not be adapted to the natural filtration induced by the Brownian motion. (*) S. Roelly (Universität Potsdam) 22 / 3

23 Convoluted Brownian motion Existence and Gaussianity of the PerOU 1 The unique solution of (*) is given, for any t [, 1], by X t ou 1 t = 1 e λ e λ(t s) 1 1 db s + 1 e λ e λ(1+t s) db s = X ou e λt + t e λ(t s) db s. Therefore X ou 1 = X ou = 1 1 e λ 1 e λ(1 s) db s N (, coth(λ/2)/2λ) 2 X ou is a stationary centered Gaussian process whose covariance function K(s, t) := Cov( X s ou, X t ou ) satisfies K(s, s + h) = 1 2λ cosh ( λ(h 1/2) ) sinh(λ/2) The proof in Ocone,Pardoux 89 involves generalized stochastic calculus to get d( X t ou e λt ) = e λt db t. Its Gaussian properties are used in Kwakernaak, 75 to filtering problems. S. Roelly (Universität Potsdam) 23 / 3 t.

24 Convoluted Brownian motion PerOU as Markov field Theorem 1 X ou is a ν-mixture of (X ou ) xx -bridges. 2 It belongs to the reciprocal class of the Ornstein-Uhlenbeck process. It is not Markov but satisfies the time-markov field property: s t, F [s,t] and F [s,t] c are independent conditionally to F {s,t} 2) is mentioned by Carmichael & al. and proved in R.-Thieullen, 2 thanks a duality formula. 1) is proved in R.-Vallois, 16 using an initial enlargement of filtration by a Gaussian rv, to obtain a semi-martingale decomposition of the PerOU. Reminder: Any Markov process is a time-markov field, but the inverse is false. See the survey paper (Léonard-R.-Zambrini 14) S. Roelly (Universität Potsdam) 24 / 3

25 Convoluted Brownian motion Convoluted Brownian motion All the next results are based on a joint paper with Pierre Vallois (Nancy): Convoluted Brownian motion: a semimartingale approach, Theory of Stochastic Processes Vol. 21 (216). Definition For any fixed ϕ L 2(, 1 ), the convoluted Brownian motion is the process X ϕ defined by: X ϕ t := Remarks t ϕ(t s) db s + 1 t ϕ(1 + t s) db s, t [, 1]. (ConvBM) We recover the PerOU process taking ϕ(t) := 1 1 e λ e λt. We have X ϕ = X ϕ 1 = 1 ϕ(1 s) db s. S. Roelly (Universität Potsdam) 25 / 3

26 Convoluted Brownian motion Proposition 1 The process (X ϕ t ) t 1 is a periodic, stationary, centered Gaussian process with covariance function R ϕ (h) := Cov ( X ϕ s, X ϕ s+h) given by R ϕ (h) = h ϕ(1 u)ϕ(h u)du + 2 One has the following representation 1 h X ϕ (t) = ϕ db (t) + ϕ d B (1 t) ϕ(u)ϕ(h + u)du. where B t := B 1 t B 1 and ϕ(t) := ϕ(1 t), t [, 1]. Therefore X ϕ is invariant under time reversal: ( X ϕ 1 t, t 1) (L) = ( X ϕ t, t 1 ). S. Roelly (Universität Potsdam) 26 / 3

27 Convoluted Brownian motion Remark: If ϕ is differentiable with ϕ L 2 (, 1), Xt ϕ = X ϕ + ( ϕ() ϕ(1) ) t B t + Xs ϕ ds, t 1. Example (Trigonometric convoluted Brownian motions) The vector-valued process X trigo := (X cos, X sin ) satisfies the autonomous system 1 Xt cos = 1 Xt sin = which can be solved. cos(1 s)db s + ( 1 cos 1 ) t B t sin(1 s)db s (sin 1) B t + t Xs cos ds, Xs sin ds, S. Roelly (Universität Potsdam) 27 / 3

28 Convoluted Brownian motion Example (Monomial convoluted Brownian motion) We denote by X k the convoluted BM corresponding to ϕ(u) = u k, k N. Xt B 1, Xt 1 = 1 B r dr + tb 1 B t. More generally X k t = X k B t + k t Xs (k 1) ds. Thus consider the R k+1 -valued process X k := (X k,, X 1, X ). It satisfies the autonomous linear integral system X k t 1 where C :=. 1, A := t = X k C B t + A X k s ds, k... k S. Roelly (Universität Potsdam) 28 / 3

29 Convoluted Brownian motion Is X ϕ a Markov field? 1 Trigonometric convoluted Brownian motion Let µ be the Gaussian law of the random vector X trigo = (X cos, X sin). Then, the law of the process X trigo is a µ-mixture of its bridges, where the x x-bridge solves the affine Stochastic Differential Equation in R 2 dx t ( ( 1 = exp 1 X = x. ) ) Id (1, ) d B t + (Λ t x + Λ1 t X t ) dt, where B is a one-dimensional Brownian motion, and Λ t, Λ 1 t are explicit 2 2-matrices. Therefore X trigo is in the reciprocal class of a Markov affine process, and is a time-markov field. S. Roelly (Universität Potsdam) 29 / 3

30 Convoluted Brownian motion 2 Monomial convoluted Brownian motion X 1 := (X 1, X ) [ ] 1 Problem: For A := the matrix exp A Id is not invertible! Therefore the bridges of X 1 are not Markovian. But if we consider an additional coordinate X t := t X s 1 ds, then the R 3 -dimensional process ( X 1 t, X t ) pinned at time solves an autonomous Markov system. Thus ( X 1 t, X t ) = (Xt 1, B 1, X t ) is a time-markov field. S. Roelly (Universität Potsdam) 3 / 3