0.1 Stationarity, Ergodicity, Spectral measure and spectral density

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1 Prof. Dr. Raier Dahlhaus Probability Theory 2 Summer term 207 Exam preparatio sheet 0. Statioarity, Ergodicity, Spectral measure ad spectral desity Let (Ω, A, P) be a probability space. All quatities are defied o this probability space. (a) Let B = (B t ) t 0 be a Browia motio ad (ε ) Z a i.i.d. sequece with Eε i = 0 ad Var(ε i ) =. Decide if the followig processes X = (X ) N0 are statioary ad / or ergodic. (i) X := i= ε i (ii) X := ε (iii) X := ε 2 + ε 2 + (iv) X := ε 0 (v) X := ε mod 2. Calculate the spectral measure / spectral desity (if possible). (vi) X := ε + αε. Calculate the spectral measure / spectral desity (if possible). (vii) Additioally assume that ε 0 Exp(α) with α > 0. Defie X := {ε>ε +2 }. Calculate the spectral measure / spectral desity (if possible). (viii) X := B (ix) X := B +2 B (x) X := B 2 B (b) Let (U t ) t N be a i.i.d. sequece of Beroulli distributed radom variables with P(U t = ) = 4. Defie X t = {Ut=U t+2 }. Show that X = (X t ) t N is statioary ad calculate the spectral desity of X. (c) Let (U i ) i Z be a sequece of i.i.d. radom variables. Assume that U i U[ 2a, 2a] is uiformly distributed o the iterval [ 2a, 2a], where a > 0 is some ukow parameter. For i Z, defie { U i, U i a, X i := U i { Ui a} + U i { Ui >a} = U i, U i > a (i) Show that (X i ) i Z is statioary ad ergodic. (ii) Show that (X i ) i Z has a spectral desity f X ad calculate f X. Hit: First show that EX i = 0. (iii) Show that the estimator â := 4 3 is a strogly cosistet estimator for a. X i i=

2 (d) Let (U i ) i Z be a sequece of i.i.d. radom variables. Assume that U i U[0, ] is uiformly distributed o the iterval [0, ]. Let a (0, ) be some ukow parameter. For i Z, defie X i := {Ui <a U i } (i) Show that (X i ) i Z is statioary ad ergodic. (ii) Show that (X i ) i Z has a spectral desity f X ad calculate f X. (iii) Show that the estimator ( 6 â := is a strogly cosistet estimator for a. ) /3 X i X i (e) Assume that E[ε 2 0] = ad E[ε 4 0] = 3. Let α, α 2 (0, ) with α + α 2 <. Let X = (X t ) t Z be the recursively defied process X t = α + α 2 Xt ε 2 t. I the followig we assume that the a.s. uique solutio has the form X t = H(ε t, ε t,...) with some measurable fuctio H : R N R, ad we assume that E[X0] 4 <. (i) Defie m k := E[X k 0 ] for k = 2, 4. Calculate m 2, m 4. i= (ii) Now assume that α is kow. Fid a estimator ˆα 2 for α 2 based o your result for m 2 i (i), ad show that it is strogly cosistet. (f) Let α (, ). Cosider the AR() process X = (X t ) t Z defied via the recursio X t = αx t + ε t. I the followig we cosider the a.s. uique solutio X t = k=0 αk ε t k. (i) Calculate c(0), c() ad fid expressios α = f(c(0), c()) ad σ 2 = g(c(0), c()). (ii) For k = 0,, defie ĉ (k) := k i= X ix i+k. Show that ˆα := f(ĉ (0), ĉ ()) ad ˆσ 2 := g(ĉ (0), ĉ ()) are strogly cosistet estimators for α ad σ 2, respectively. (g) Let (X t ) t Z be a statioary ad ergodic sequece. For x, λ R, defie cosistet estimators ˆF (x) ad ˆφ (λ) based o X,..., X such that ˆF (x) F X (x) a.s. ad ˆφ (λ) φ X (λ) a.s., where φ X, F X deote that characteristic fuctio ad the distributio fuctio of X Browia motio ad its properties Let B = (B t ) t 0 be a Browia motio o a probability space (Ω, A, P). Let F := (F t ) t 0 with F t := σ(b s : s t) be the caoical filtratio. (a) Show that the followig processes X = (X t ) t [0,] are Browia motios. (i) X t := B t B. { B t, t < t 0, (ii) X t :=, where t 0 (0, ) is a fixed costat ad Z is B t0 + Z(B t B t0 ), t t 0 idepedet of B with P(Z = ) = P(Z = ) =. 2 (iii) X t := ab t + bb t, where a, b R with a 2 + b 2 = ad B = (B t) t 0 is aother Browia motio idepedet of B. (iv) X t := W r (t) v(r (t)), where W = (W t) t 0 is a cotiuous cetered Gaussia process with covariace fuctio γ(s, t) := EW s W t = u(s)v(t) for s t ad u, v C[0, ) with v( ) 0, W 0 := 0 ad r(t) := u(t)/v(t) be cotiuous ad strictly icreasig with r(0) = 0. 2

3 (b) Show that X = (X t ) t [0,] defied via X t := ( t)b t t, X := 0 is a Browia Bridge. (c) Show that the followig processes X = (X t ) t 0 are martigales w.r.t. F: (i) X t := B 2 t t (ii) X t := tb t t 0 B s ds tλ2 λbt (iii) X t := e 2, Hit: Ee Y = e σ2 /2 for Y N(0, σ 2 ). (d) For T > 0, let M T := max 0 s T B s ad m T := mi 0 s T B s. (i) Show that Φ : C[0, ] R, Φ(f) := max 0 s T f(s) is B(C[0, ])-B(R)-measurable. Hit: Use that {(, x] : x R} is a geeratig system of B(R) / Alterative: Show that Φ is cotiuous. (ii) Show that M T d = T M. (iii) Show that i case of existece, E [ M T + m T ] = 0. (e) Let τ := if{t 0 : B t < } ad τ 2 := if{t 0 : B t > }. (i) Show that Φ : C[0, ] [0, ] { }, Φ(f) := if{t [0, ] : f(t) < } is B(C[0, ])- B([0, ] { })-measurable (we use the covetio that if = ). Hit: Use that {(x, ] : x [0, ]} {{ }} is a geeratig system of B([0, ] { }). (ii) Show that τ d = τ 2. (f) Let C > 0, α >. A fuctio f : [0, ] R is called Hoelder cotiuous w.r.t. (α, C) if 2 s, t [0, ] : f(s) f(t) C s t α. (i) Show that M := {ω Ω : B(ω) is Hoelder cotiuous w.r.t. (α, C)} A. (ii) Show that P( k {,..., } : B( k k ) B( ) C α ) 0. (iii) Coclude from (i),(ii) that P-a.s., B is ot Hoelder cotiuous w.r.t. (α, C). (g) Defie f(t) : [0, ] R, f(t) = t ad g : [0, ] R, g(t) = 2t. (i) Show that the set M := {f B g} = {ω Ω : t [0, ] : f(t) B t (ω) g(t)} A. (ii) Let N. Show that P( k {,..., } : k 2 B k 2 2 k 2 ) 0 ( ). (iii) Coclude from (i),(ii) that P-a.s., it holds that B caot be surrouded by f ad g (i.e. P(M) = 0). 0.3 Poisso process ad its properties Let N = (N t ) t 0 be a Poisso process o a probability space (Ω, A, P) with itesity λ > 0. Let S i be the sequece of jump times (S 0 = 0) ad T i := S i S i the itervals betwee two jumps. Let F defied via F t := σ(n s : s t) be the caoical filtratio. (a) Calculate E[N t+s N s ] for s, t 0. (b) Show that N t λt ad (N t λt) 2 λt are martigales w.r.t. F. (c) Calculate the followig quatities: 3

4 (i) P(N =, N 2 = 3), P(N =, N 2 3). (ii) P(N s = k N t = ) for s t, k {0,..., }. (iii) P(S x, S 2 x + y) for x, y 0. (iv) P(S x N t = ) for t > x,. (v) P(N x = S 2 = t) for t > x. (d) Let s, x 0. Calculate the probability that there are o jumps i the iterval (x, x + s]. (e) Fix t 0. Let D t := T Nt+ deote the time betwee the N t -th jump ad the (N t + )-th jump. Calculate the distributio of D t ad ED t. (f) Let M = (M t ) t 0 be aother Poisso process with itesity µ > 0 o (Ω, A, P). (i) Calculate the distributio of M T. (ii) Calculate the distributio of N 2T N T. (g) Assume that red cars arrive at a itersectio accordig to a Poisso process with itesity λ > 0 ad blue cars arrive idepedetly of the red cars accordig to a Poisso process with itesity µ > 0. (i) Let X be the umber of red cars that arrive before the first blue car. Determie the distributio of X. (ii) Give that exactly oe car passed the itersectio util time T, what is the probability that it was red? (h) Assume that passegers arrive at a bus statio as a Poisso process with rate λ (time i miutes). The oly bus departs after a determiistic time Z 0. Let W = NZ N Z i= (Z S i ) be the average waitig time for all passegers. Defie W = Z i the case N 2 Z = 0. (i) Compute EW. (ii) Now assume that Z Exp(µ) is idepedet of the Poisso process, where µ > 0. Compute EW i this ew situatio. (j) Suppose cars eter a oe-way ifiite legth, ifiite lae highway accordig to a Poisso process N = (N t ) t 0 with itesity λ > 0. Car i eterig the highway choose a velocity V i ad travel at this velocity. Assume that the V i are i.i.d. with distributio fuctio F ad positive ad idepedet of N. Derive the distributio of the umber of cars that are located i a iterval (0, a) at time t. 0.4 Weak covergece i C[0, ] (a) Let (X () ) N be a sequece of Gaussia stochastic processes o a probability space (Ω, A, P) with values i C[0, ]. Assume that EX () t := µ (t) = si(t) ad Cov(X () t, X (s) t ) = ( s + t (s t) 2 + ) for all N ad s, t [0, ]. 2 Show that X () D B i C[0, ]. Hit: If Y N(µ, σ 2 ), the E[Y 4 ] = µ 4 + 6µ 2 σ 2 + 3σ 4. (b) Let (ε i ) i Z be a i.i.d. sequece of N(0, σ 2 )-distributed radom variables (σ 2 > 0). Let δ (0, ) be fixed (thik of δ to be small). For a [ + δ, δ] =: A, defie X (a) := k=0 ak ε k. 4

5 (i) Show that with some costat C > 0 idepedet of, it holds for all a, a A that E[ X (a) X (a ) 2 ] C a a 2. (ii) Let Z = (Z a ) a A be a cotiuous Gaussia process with covariace fuctio γ(a, a ) = E[Z a Z a ] = σ2. Show that i (C(A), a a ) it holds that X D Z. Hit / remark: Use without proof that the results from the lecture also hold for C(A) istead of C[0, ]. (iii) Let (T ) N be some sequece of radom variables with T A a.s. ad T P τ, where τ A is some costat. Determie the limit distributio of X (T ). Hit: Use the cotiuous mappig theorem with Φ : C(A) A R, Φ(f, x) := f(x). (c) Let X i, i N be a i.i.d. sequece of real radom variables ad L > 0 fixed. For M [0, ], defie the treshold fuctio h M (x) which is i [ M, M], 0 i [ (M + L), (M + L)] c ad liearly iterpolated i betwee. Defie E(M) := E[X h M (X )]. For M [0, ], defie Ê (M) := i= X ih M (X i ) as a estimator of E(t) ad ˆP (M) := ( E (M) E(M) ). (i) Show that there exists some costat C > 0 idepedet of such that E [ ˆP (M) ˆP (M ) 2] C M M 2. (ii) Let Z be a cotiuous cetered Gaussia process with covariace fuctio γ(m, M ) := E[Z M Z M ] (which has to be determied i the followig). Show that i (C[0, ], ) it holds that ˆP D Z. (d) Let X i, i N be a i.i.d. sequece of real radom variables with E(t) := E[si(X t)] ad E[X 2 ] <. For t [0, ], defie Ê(t) := i= si(x it) as a estimator of E(t) ad ˆP (t) := ( Ê (t) E(t) ). (i) Show that there exists some costat C > 0 idepedet of such that E [ ˆP (t) ˆP (s) 2] C s t 2. (ii) Let Z be a cotiuous cetered Gaussia process with covariace fuctio γ(s, t) := E[Z s Z t ] (which has to be determied i the followig). Show that i (C[0, ], ) it holds that ˆP D Z. (iii) Now assume X U[0, ]. Show that sup t [0,] Ê (t) P cos(). Hit: ( cos(t))/t is odecreasig i [0, 2] ad use without proof that Φ : (C[0, ], ) (R, ), Φ(f) := sup t [0,] f(t) is cotiuous. (e) Let X i, i N be a i.i.d. sequece of real radom variables with X i E[X] 5 <. For t 0, defie E(t) := E[X +t ]. For t [0, ], defie Ê(t) := as a estimator of E(t) ad ˆP (t) := ( E (t) E(t) ). (i) Show that there exists some costat C > 0 idepedet of such that Hit: Use a Taylor expasio of order. E [ ˆP (t) ˆP (s) 2] C s t 2. 5 a.s. ad i= X+t i

6 (ii) Let Z be a cotiuous cetered Gaussia process with covariace fuctio γ(s, t) := E[Z s Z t ] = E(t+s+) E(s)E(t). Show that i (C[0, ], ) it holds that ˆP D Z. (f) Let (ε t ) t Z be a i.i.d. sequece with Eε 0 = 0 ad Var(ε 0 ) = > 0 ad E[ε 4 0] <. Let S k := k i= ε i (k =,..., ), S 0 := 0. Let B = (B t ) t 0 be a Browia motio. Show that (i) 3/2 k= S k D 0 B t dt. (ii) mi k=0,..., S k D mi t [0,] B t. Hit: Use without proof that max k=,..., ε k P 0. (iii) Now assume that Eε 0 = µ. Show that i (C[0, ], ), ( (R (t)) t [0,] := (S t t S ) + ) (t t )ε t + where B is a Browia Bridge. t [0,] D B, (g) Assume that (X i ) i N, (Y i ) i N are i.i.d. sequeces of real-valued radom variables with EX = µ, Var(X ) = ad E[X 4 ] <. Furthermore, Y i [0, ] a.s. ad the distributio fuctio of Y is give by F. Defie the empirical distributio fuctio ˆF (y) := i= {Y i y}. (i) Show that Φ : (C[0, ] [0, ], ) R, Φ(f, x) := f(x) is cotiuous. (ii) Use Dosker s theorem to show that for y R, it holds that ˆF(y) (X i µ) D N(0, F (y)). i= P Hit: You may use without proof that max k=,..., X i 0 to deal with the residual terms. (iii) Show that for all y [0, ], ˆF(y) i= X P i µf (y). (h) Let X i, i N be a i.i.d. sequece with EX i = 0, Var(X i ) = ad E[X 4 i ] <. Defie S := k= X k, S 0 := 0 ad N := max{k {,..., } : S k S k 0} which is the idex of the last sig chage of S k, k =,..., before. Show that for all x [0, ], P( N x) P(sup{t [0, ] : B t = 0} x). Hit: Apply the geeralized cotiuous mappig theorem to the fuctioal Φ : C[0, ] R, Φ(f) := sup{t [0, ] : f(t) = 0} which is cotiuous i a set D with P(B D) =. (j) Let (ε t ) t Z be a i.i.d. sequece with Eε 0 = 0 ad Var(ε 0 ) = σ 2 > 0. Let us cosider a special case of the AR() model, where we defie X 0 = 0, X t := αx t + ε t, t N with α :=, i.e. X t = t k= ε k. Show that i this case, the estimator ˆα := of α fulfills (ˆα α) D B 0 s db s 0 B2 s ds, t= XtX t+ t= X2 t 6

7 where B = (B t ) t 0 is a Browia motio. Hit: First discuss the deomiator. For the omiator, use the formula x(y x) = 2 (y2 x 2 ) 2 (x y)2 ad a telescope sum argumet. The argue why omiator ad deomiator coverge together. 0.5 Ito calculus Let B = (B t ) t 0 be a Browia motio o a probability space (Ω, A, P) ad F = (F t ) t 0 its caoical filtratio F t := σ(b s : s t). Let T > 0. Ofte used formulas: For Z N(0, ), it holds that E[e σz ] = e σ2 /2 ad E[Z 4 ] = 3. (a) For t [0, T ], let X t := t 0 s2 B s db s ad Y t := e λt t 0 B2 s db s with some λ > 0. Calculate EX t, E[Xt 2 ], E[Y t ], E[Yt 2 ] ad E[X t Y t ]. (b) For t [0, T ], let X t := + t 0 eλbs db s for λ > 0. Calculate E[X t ] ad E[X 2 t ]. (c) Show that the followig processes are martigales w.r.t. F by writig them as a appropriate Ito itegral: (i) X t = 5 + B 4 t 6tB 2 t + 3t 2, (ii) X t = e λ2 t/2 si(λb t ) (iii) X t := e λ2 t/2 cosh(λb t ) (it holds that cosh(x) = (e x + e x )/2) (iv) X t = t 2 B 3 t t 0 sb s(2b 2 s + 3s) ds. (d) (Combiatio of Ito formula / isometry): (i) Show that E [ e Bt t 0 e Bs db s ] = 2(e t/2 ). (ii) Calculate E[B t t 0 B2 s db s ]. (e) Suppose that f : [0, T ] R is a determiistic twice cotiuously differetiable fuctio. Prove the itegratio by parts formula t f(s) db 0 s = f(t)b t t B 0 sf (s) ds for all t [0, T ]. (f) Fix T > 0. Let h : [0, T ] R be some cotiuously differetiable determiistic fuctio. Fid a solutio of the stochastic differetial equatio dx t = h(t)x t dt + σx t db t, X 0 = 0. Hit: Look for a solutio of the form X t = f(t)e σbt σ2 t/2, where f : [0, T ] R has to be determied. (g) Fix σ > 0. Give T > 0, let X = (X t ) t [0,) deote the solutio of the stochastic differetial equatio dx t = σ db t X t t dt, X 0 = e. (i) Show that X t = σ( t) t db 0 s s, t [0, ) is a solutio of the above SDE. I the followig, let X deote the correspodig cotiuous versio. (ii) Show that E[X t ] = 0 for all t [0, ). (iii) Compute Var(X t ) for t [0, ) ad show that it is equal to the variace fuctio of a (ostadard) Browia Bridge which has covariace fuctio γ(s, t) = σ 2 (mi{s, t} st) Remark: ideed oe ca show that X t is a Browia Bridge. 7

8 (h) Cosider the Orstei-Uhlebeck process X t = xe λt + ν( e λt ) + e λt t 0 σeλs db s for some x R, ν, λ, σ > 0. Show that X = (X t ) t [0,T ] satisfies the SDE dx t = λ(ν X t ) dt + σ db t, X 0 = x. (i) Let h : [0, T ] R be a determiistic cotiuous fuctio. Cosider the SDE dx t = λx t dt + h(t) db t, X 0 = x R. Show that X = (X t ) t [0,T ] with X t = e λt (x + t 0 h(s)eλs db s ) is a solutio. Homepage of the lecture: 8