Problem 5.2 (10 points) For the simple symmetric random walk (Sn)n=0.12 that with So = 0, show for all n>0 and all -n<k<n Problem 5.2 (10 points) For the simple symmetric random walk (S...
Problem 5.4 (10 points) Let (Sn)n-01. be a simple, symmetric random walk with starting value So-s e R. (a) Show that ES for alln0 b) Show that ElSn+1 Sn] Sn for 0. (c)Suppose that (Sn)n-0,12,. . denotes the profit and loss from $1 bets of a gambler with initial capital So-s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)?
Problem 5.4...
4. Let {Sn,n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y, max{Sk, 1 3 k S nt, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain
4. Let {Sn, n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y max{Sk, 1 Sk n, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain.
DO NOT COPY OTHER ANSEWERS!!!!
2. (10 points) Let (%)n>o be a simple symmetric random walk. Compute P(Sn-y|S,n-x) for the two cases n > m and n < m
3. Let(Sn, n > 0} be a symmetric Random Walk on Z. Defined To-inf(n-1 : Sn-0) the time of first passage to state 0, prove that PlT, = 2nlSo = 0] = 2n.plsøn = 이So = 0] for any n 2 1
3. Let {Sn, n > 0} be a symmetric Random Walk on Z. Defined To inf(n > 1 : Sn-0} the time of first passage to state 0, prove that 2n - 1 for any n 2 1
Let Sn be a symmetric one-dimensional random walk with respect to the standard filtration {Fi} i >= 0. Show that Mn= S2n - n is a martingale.
Let us start with the usual conditional probability exercise Let Sn be the random walk S, -So + 61 +...+ En such that Ei €{+1} are iid with P(Ei=1) So = x (0,N) Z. p. Let 1. Show that P(S In S0, S1, ..., Sn 1) P(Sn In Sn 1) Hint Start with P(S, Ir. S DO, S I1...,S-I I -1) / 0 ill I; - I;+11 1. Thal is, if the sequence of steps is not possible for the...
Sn denotes the position is at
during the nth time of the random walk. For example, if you're on
the 3rd position at the 10th time, S10 = 3.
0) c 10.10 Show that un = P(S,-So) satisfies un adeduce that an asymmetric random walk revisits its starting point only finitely often w a will need Stirling's formula (see TheoremA 4): n (n/e) /2π n as n You Exercise 10.11 Consider a two-dimo
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with P(X1 = +1) = 1. Let Sn = 2h=1 Xk (i.e. Sn, n > 1 is a symmetric simple random walk with steps Xk, k > 1). (a) Compute E[S+1|X1, ... , Xn] for n > 1. Hint: Check out Example 3.8 in the lecture notes (Version Mar/04/2020) for inspiration. (b) Find deterministic coefficients an, bn, Cn possibly depending on n so that Mn...