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 Sn be a symmetric one-dimensional random walk with respect to the standard filtration {Fi} i...
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
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
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.
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
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 (Sn)n=0.12 that with So = 0, show for all n>0 and all -n
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...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Theorem 5.10:
Please answer both parts
Problem 7. Let Sn = X1 + ... + Xn, where Xį are independent with EX; = 0 and Var(X;) = 02. By example 5.3, p. 205, S2 – no2 is a martingale. Let Ta = min{n : Snl > a}. (a) Use Theorem 5.10, to show that E[Ta] > . (b) Show that, for simple random walk, o2 = 1, we have equality. Theorem 5.10. If Mn is a supermartingale with respect to...
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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
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing bound- aries. Suppose X0 = 50. Let T = min{j : Xj = 0 or 100}. Let Fn denote the information contained in X1,··· ,Xn. (1) Verify that Xn is a martingale. (2) Find P (XT = 100). (3) Let Mn = Xn2 − n. Verify that Mn is also a martingale. (4) It is known that Mn and T...