Mathematical Institute University of Munich Stochastic Processes Problem Sheet 3 Prof. Dr. Franz Merkl WS 2014/15 Problem 9 Let Xn , n ∈ N, be independent random variables with standard normal distribution. We recursively define: Y0 := 0, Yn := Yn−1 + Xn 1{Yn−1 ≥0} + sign(Xn ) 1{Yn−1 <0} , n ∈ N, where sign(Xn ) := 1{Xn >0} − 1{Xn <0} denotes the sign of Xn . Show that L Y √n n w −−−→ N (0, 1). n→∞ Problem 10 Let (Xn )n∈N0 be a martingale with respect to a filtration (Fn )n∈N0 . Suppose that the increments of (Xn )n∈N0 are bounded, i.e. there exists a constant C > 0 such that |Xn −Xn−1 | ≤ C holds for all n ∈ N. For n ∈ N, we define σn2 := Var(Xn |Fn−1 ), n X 2 Σn := σk2 . k=1 Assume that Σ2n −−−→ ∞ P -almost surely. For k ∈ N, we additionally define n→∞ Tk := max n ∈ N : Σ2n ≤ k . Show that L XTk √ k w −−−→ N (0, 1). k→∞ Problem 11 Definition: Let d ∈ N. For a finite measure µ on (Rd , B(Rd )), the function Z d µ ˆ : R → C, µ ˆ(t) := eit·x µ(dx), Rd is called the (multidimensional) Fourier transform of µ. Prove the inversion formula for the multidimensional Fourier R transform: If µ is a finite measure µ(t)| dt < ∞, then µ has the on (Rd , B(Rd )) with integrable Fourier transform µ ˆ, i.e. Rd |ˆ density Z 1 d R 3 x 7→ e−it·x µ ˆ(t) dt (2π)d Rd with respect to the d-dimensional Lebesgue measure. Problem 12 Let B := (Bt )t∈R+ be a stochastic process with continuous sample paths and B0 = 0. Suppose 0 that B and (Bt2 − t)t∈R+ are martingales with respect to a filtration (Ft )t∈R+ . Show that B 0 0 is a standard Brownian motion. Hint: Apply the central limit theorem for martingales to the suitably stopped process B. Solutions to this problem sheet cannot be corrected and therefore need not be handed in into the homework bin.