PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

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1 PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please give complete solutions, all claims need to be justified. Late homework will not be accepted. Please let me know if you find any misprints or mistakes. 1. Due by March 4, 11:am 1. Let (X n ) n N be an i.i.d. positive sequence, and S n = X X n. Let N t = sup{n : S n t}. Prove that (N t ) t R+ is a stochastic process.. Let (W t ) t R+ be a Wiener process. Find cov(w s, W t ). 3. Prove that every Borel set B in R d is regular, i.e., for every probability Borel measure µ, every ε >, there is a compact set K and open set U such that K B U and µ(u \ K) < ε. 4. Prove that cylinders C(t 1,..., t n, B), B B(R n ), t 1,..., t n, n N form an algebra. 5. We defined the cylindrical σ-algebra as the σ-algebra generated by elementary cylinders. Prove that if we replace elementary cylinders by cylinders we obtain an equivalent definition. 6. Let F T = σ{c(t 1,..., t n, B) : t 1,..., t n T } for T T. Prove that B(R T ) = F T. countable T T 7. Use characteristic functions to prove the existence of a Wiener process (up to continuity of paths). 8. Let (X t ) t [,1] be an (uncountable) family of i.i.d. r.v. s with nondegenerate distribution. Prove that no modification of this process can be continuous. 9. A multidimensional version of the Kolmogorov Chentsov theorem. Suppose d 1, and there is a stochastic field X : [, 1] d Ω R that satisfies E X(s) X(t) α C s t d+β for some α, β, C > and all t, s [, 1] d. Prove that there is a continuous modification of X on [, 1] d. 1. Prove the following statement. Suppose there is a family of ch.f. (φ s,t ( )) s<t such that for all λ R and all t 1 < t < t 3, φ t1,t (λ)φ t,t 3 (λ) = φ t1,t 3 (λ). 1

2 PROF. YURI BAKHTIN Then for every distribution function F, there is a stochastic process (X t ) t R+ with independent increments such that X F and Ee iλ(xt Xs) = φ s,t (λ) for all s < t and λ R. 11. Show that the Kolmogorov Chentsov theorem cannot be relaxed: inequality E X t X s C t s is not sufficient for existence of a continuous modification. Hint: consider the following process: let τ be a r.v. with exponential distribution, and define X t = 1 {τ t}. 1. Prove that there exists a Poisson process such that: (a) its realizations are nondecreasing, taking only whole values a.s. (b) its realizations are continuous on the right a.s. (c) all the jumps of the realizations are equal to 1 a.s. 13. Give an example of a non-gaussian -dimensional random vector with Gaussian marginal distributions. 14. Let Y N (a, C) be a d-dimensional random vector. Let Z = AY where A is an n d matrix. Prove that Z is Gaussian and find its mean and covariance matrix. 15. Prove that an R d -valued random vector X is Gaussian iff for every vector b R d, the r.v. b, X is Gaussian. 16. Prove that (s, t) t s defined for s, t is positive semi-definite. Hint: 1 [,t], 1 [,s] L (R + ) = t s. 17. Prove that (s, t) e t s is positive semi-definite. 18. Prove that if X is a Gaussian vector in R d with parameters (a, C) and C is non-degenerate, then the distribution of X is absolutely continuous w.r.t. Lebesgue measure and the density is 1 p X (x) = det(c) 1/ (π) d/ e 1 C 1(x a),(x a). 19. Find a condition on the mean a(t) and covariance function r(s, t) that guarantees existence of a continuous Gaussian process with these parameters.. Suppose (X, X 1,..., X n ) is a (not necessarily centered) Gaussian vector. Show that there are constants c, c 1,..., c n such that E(X X 1,..., X n ) = c + c 1 X c n X n. Your proof should be valid even if the covariance matrix of (X 1,..., X n ) is degenerate. 1. Consider the standard Ornstein Uhlenbeck process X (Gaussian process with mean and covariance function r(s, t) = e t s ). (a) Prove that X has a continuous modification. (b) Find E(X 4 X 1, X, X 3 ).. Prove that for every centered Gaussian process X with independent increments on R + = [, ), there is a nondecreasing nonrandom function f : R + R + such that X has the same f.d.d. s as Y defined by Y (t) = W (f(t)), for a Wiener process W.

3 PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS 3. Due by March 5, 11:am 1. Suppose the process X t is a stationary Gaussian process, and let H be the Hilbert space generated by (X t ) t R, i.e., the space consisting of L -limits of linear combinations of values of X t. Prove that every element in H is a Gaussian r.v.. Find the covariance function of a stationary process such that its spectral dx 1+x. measure is ρ(dx) = 3. Give an example of a weakly stationary stochastic process (X n ) n N such that (X X n )/n converges in L to a limit that is not a constant. 4. Let (X t ) t R be a weakly stationary centered process with covariance function C and spectral measure ρ. Find the covariance function and spectral measure for process (Y t ) t R defined by Y t = X t. 5. Let (X n ) n Z be a weakly stationary process. Prove that for any K N and any numbers a K, a K+1,..., a K 1, a K, the process (Y n ) n Z defined by Y n = K k= K a k X n+k is weakly stationary. Express the spectral measure of Y in terms of the spectral measure for X. 6. Let stationary process (X n ) n Z satisfy E X <. Prove that with probability 1, lim n (X n /n) =. 7. Consider a map θ : Ω Ω. A set A is called (backward) invariant if θ 1 A = A, forward invariant if θa = A. Prove that the collection of backward invariant sets forms a σ-algebra. Give an example of Ω and θ such that the collection of forward invariant sets does not form a σ- algebra. 8. Consider the transformation θ : ω {ω + λ} on [, 1) equipped with Lebesgue measure. Here {...} denotes fractional part of a number. This map can be interpreted as rotation of the circle parametrized by [, 1) with endpoints and 1 identified. Prove that this dynamical system is ergodic if and only if λ / Q. Hint: take the indicator of an invariant set and write down the Fourier series for it (w.r.t. e πinx ). What happens to this expansion under θ? 9. Prove that every Gaussian martingale is a process with independent increments. 1. Let (X t, F t ) t be a continuous process. Let a >, and let τ = inf{t : X(t) > a}. Show that τ is a stopping time w.r.t (F t+ ) t, where F t+ = ε> F t+ε. 11. Show that if τ 1 τ... are stopping times w.r.t. to a filtration (F t ), then τ = lim n τ n is also a stopping time w.r.t. to (F t ). 1. Let F τ = {A : A {τ t} F t } for a filtration (F t ) and a stopping time τ. Show that F τ is a σ-algebra.

4 4 PROF. YURI BAKHTIN 13. Give an example of the following: a random variable τ is not a stopping time, F τ is not a σ-algebra. 14. Suppose τ is a stopping time w.r.t. (F t+ ). Let us define τ n = n τ n = k N k n 1 {τ ( k 1 τ n = [n τ] + 1 n = k N n, k k n 1 {τ [ k 1 n, k n ]}, n N. n )}, n N. Prove that for every n N, τ n is a stopping time w.r.t. (F t ) t, F τn F τ+, and τ n τ. 15. Prove: if (X t, F t ) t is a continuous process, then for any stopping time τ, X τ is a r.v. measurable w.r.t. F τ. 16. Let (X t, F t ) be a continuous martingale and let τ be a stopping time w.r.t. F t. Prove that the stopped process (Xt τ, F t ) t, where Xt τ = X τ t, is also a martingale.

5 PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS 5 3. Due by April 15, 11:am 1. Find the density of the distribution of τ b = inf{t : W (t) b}, where b >, and W is the standard Wiener process. Hint: use the reflection principle to find P{τ b x} first. (Optional: prove that (τ b ) b is a process with independent increments.). Let W 1 and W be two independent Wiener processes w.r.t. a filtration (F t ) t, and let X be a bounded process adapted to (F t ) t. For a partition t of time interval [, T ] (i.e., a sequence of times t = (t, t 1,..., t n ) such that = t < t 1 < t <... < t n = T ), we define Q(t) = j A tj (W 1 t j+1 W 1 t j )(W t j+1 W t j ). Prove: lim Q(t) = in max(t j+1 t j ) L. 3. Let (F t ) be a filtration. Suppose that = A (t) + A 1 (t)w (t) for all t, where (A, F t ) and (A 1, F t ) are C 1 processes, and W (t) is a Wiener process w.r.t. (F t ). Prove that A and A Show that the function 1 P (s, x, t, Γ) = P (t s, x, Γ) = e (x y) (t s) dy π(t s) is a Markov transition probability function for the standard Wiener process. 5. Prove the following theorem (Kolmogorov, 1931) using Taylor expansions of test functions: Suppose (P x ) x R d is a (homogeneous) Markov family on R d with transition probabilities P (,, ). Suppose there are continuous functions a ij (x), b i (x), i, j = 1,..., d, such that for every ε >, the following relations hold uniformly in x: B ε(x) B ε(x) Γ P (t, x, B c ε(x)) = o(t), t, (y i x i )P (t, x, dy) = b i (x)t + o(t), t, (y i x i )(y j x j )P (t, x, dy) = a ij (x)t + o(t), t. where B ε (x) is the Euclidean ball of radius ε centered at x. Then the infinitesimal generator A of the Markov semigroup associated to the Markov family is defined on all functions f such that f itself and all its partial

6 6 PROF. YURI BAKHTIN derivatives of first and second order are bounded and uniformly continuous. For such functions Af = 1 a ij ij f + b i i f. ij i 6. Consider a Markov process X in R given by X 1 (t) = X 1 () + W (t), X (t) = X () + t X 1 (s)ds. Find its generator on C -functions with compact support. 7. Consider the Poisson transition probabilities, i.e., fix a number λ > and for i Z and t, let P (i, t, ) be the distribution of i + π λt, where π s denotes a random variable with Poisson distribution with parameter s >. In other words, λt (λt)j i P (i, t, {j}) = e, i Z, j {i, i + 1,..., }, t >. (j i)! Find the generator of the Markov semigroup on all bounded test functions f : Z R. 8. Find the transition probabilities and generator associated to the OU process (see, e.g., the second HW assignment or lecture notes for a definition of OU process). 9. Let W be a standard Wiener process. Prove that Wt t is a martingale. 1. The so-called Stratonovich stochastic integral may be defined for a broad class of adapted processes X t via T def X t dw t = lim max(t j+1 t j ) j X tj+1 + X tj (W tj+1 W tj ) in L. Impose any conditions you need on X and express the difference between the Itô and Stratotovich integrals in terms of quadratic covariation between X and W. Compute T W t dw t. Is the answer a martingale?

7 PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS 7 4. Due by May 6, 11:am M c = {square-integrable martingales with continuous paths} 1. Prove that if (M t, F t ) M c, then E[(M t M s ) F s ] = E[M t M s F s ] = E[ M t M s F s ], s < t.. Suppose (M t, F t ) M c, X is a simple process, and (X M) t = t X sdm s. Prove that [ t ] E[((X M) t (X M) s ) F s ] = E Xr d M r F s, s < t. 3. Let M M c. Prove that Y (X M) = (Y X) M for simple processes X, Y. Find reasonable weaker conditions on X and Y guaranteeing the correctness of this identity in the sense of square integrable martingales. 4. Suppose (M t, F t ) M c, and X, Y are bounded processes. Prove that X M, Y M t = t s X s Y s d M s. Here, for two processes M, N M c the cross-variation M, N t is defined by M, N t = M + N t M N t Let us define the process X by t X t = e λt X + εe λt e λs dw s, t. Here λ R, ε >, W is a standard Wiener process, and X is a squareintegrable r.v., independent of W. Prove that dx t = λx t dt + εdw t. 6. Prove that if f : [, ) is a deterministic function, bounded on any interval [, t], then X t = t f(s)dw s, t, is a Gaussian process. Find its mean and covariance function. 7. In the context of Problem 5, find all the values of λ with the following property: there are a and σ such that if X N (a, σ ), then X t is a stationary process. 8. Suppose u : R [, ) and φ : [, ) R R are smooth bounded functions. Suppose that u : [, ) R R is a smooth function satisfying t u(t, x) = 1 xxu(t, x) + φ(t, x)u(t, x), u(, x) = u (x).

8 8 PROF. YURI BAKHTIN Prove that u(t, x) = Ee t φ(t s,x+ws)ds u (x + W t ), t >, x R, where W is a standard Wiener process. 9. Suppose a R, σ >, x >, and W is the standard Wiener process. Find constants A, B R such that the process S defined for all t by S t = x exp(at + σw t ) (and often called the geometric Brownian motion ) satisfies the following stochastic equation ds t = AS t dt + BS t dw t, t. Find necessary and sufficient conditions on a and σ for (S t ) to be a martingale. 1. Suppose (W t, F t ) is a Wiener process and (X t, F t ) is a bounded process. Use the Itô formula to prove that [ t Z t = exp X s dw s 1 t ] Xs ds, t, is a local martingale w.r.t. (F t ). (In fact, it is a true martingale)