- #1
WMDhamnekar
MHB
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In this exercise, we consider simple, nonsymmetric random walk. Suppose 1/2 < q < 1 and ##X_1, X_2, \dots## are independent random variables with ##\mathbb{P}\{X_j = 1\} = 1 − \mathbb{P}\{X_j = −1\} = q.## Let ##S_0 = 0## and ##S_n = X_1 +\dots +X_n.## Let ##F_n## denote the information contained in ##X_1, \dots , X_n##
1. Which of these is ##S_n##: martingale, submartingale, supermartingale (more than one answer is possible)?
2. For which values of r is ##M_n = S_n − rn ## a martingale?
3. Let ##\theta = (1 − q)/q## and let ##M_n =\theta^{S_n}## . Show that ##M_n## is a martingale.
4. Let a, b be positive integers, and ##T_{a,b} = \min\{j : S_j = b \text{or} S_j = −a\}.## Use the optional sampling theorem to determine ##\mathbb{P}\{ S_{T_{a,b} }= b\}## .
5. Let ##T_a = T_{a,\infty}.## Find ##\mathbb{P}\{T_a < \infty\}##
My answers:
1. ##S_n## is a submartingale. This is because ##E[S_{n+1} | F_n] \geq qS_n + (1 − q)S_n = S_n##, and ##S_n## is increasing in n.
2. ##M_n## is a martingale if and only if r = 0. This is because ##E[M_{n+1} | F_n] = E[S_{n+1} − r(n+1)| F_n] =(S_n - rn) = q(S_n − rn) + (1 − q)(S_n − rn) = S_n − rn##, so ##r = 0## is required for ##M_n## to be a martingale.
3. We have ##E[\theta^{S_{n+1}} | F_n] = \theta^{qS_n + (1 − q)S_n} = \theta^{S_n} = M_n##, so ##M_n## is a martingale.
4. Using the optional sampling theorem and the fact that ##S_j## is likely to increase by 1 in each step with ##\mathbb{P}[\frac12 < q < 1]##, we have ##\mathbb{P}\{ S_{T_{a,b}} = b\} = q^a##.
5. Since ##S_n## is a submartingale, ##T_a < \infty## is unsure. ##T_a## is the stopping time where ##n## is the first time ##S_n## reaches −a, so ##\mathbb{P}\{T_a < \infty\} = 0 \leq p <q## where (p +q)=1
1. Which of these is ##S_n##: martingale, submartingale, supermartingale (more than one answer is possible)?
2. For which values of r is ##M_n = S_n − rn ## a martingale?
3. Let ##\theta = (1 − q)/q## and let ##M_n =\theta^{S_n}## . Show that ##M_n## is a martingale.
4. Let a, b be positive integers, and ##T_{a,b} = \min\{j : S_j = b \text{or} S_j = −a\}.## Use the optional sampling theorem to determine ##\mathbb{P}\{ S_{T_{a,b} }= b\}## .
5. Let ##T_a = T_{a,\infty}.## Find ##\mathbb{P}\{T_a < \infty\}##
My answers:
1. ##S_n## is a submartingale. This is because ##E[S_{n+1} | F_n] \geq qS_n + (1 − q)S_n = S_n##, and ##S_n## is increasing in n.
2. ##M_n## is a martingale if and only if r = 0. This is because ##E[M_{n+1} | F_n] = E[S_{n+1} − r(n+1)| F_n] =(S_n - rn) = q(S_n − rn) + (1 − q)(S_n − rn) = S_n − rn##, so ##r = 0## is required for ##M_n## to be a martingale.
3. We have ##E[\theta^{S_{n+1}} | F_n] = \theta^{qS_n + (1 − q)S_n} = \theta^{S_n} = M_n##, so ##M_n## is a martingale.
4. Using the optional sampling theorem and the fact that ##S_j## is likely to increase by 1 in each step with ##\mathbb{P}[\frac12 < q < 1]##, we have ##\mathbb{P}\{ S_{T_{a,b}} = b\} = q^a##.
5. Since ##S_n## is a submartingale, ##T_a < \infty## is unsure. ##T_a## is the stopping time where ##n## is the first time ##S_n## reaches −a, so ##\mathbb{P}\{T_a < \infty\} = 0 \leq p <q## where (p +q)=1
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