Let A ∈ N+. Let X0 = A and (Xn)n≥1 be a sequence of i.i.d. random variables such that P(X1 = 1) = 1 −...
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Let A ∈ N+. Let X0 = A and (Xn)n≥1 be a sequence of i.i.d. random variables such that
P(X1 = 1) = 1 − P(X1 = −1) = p, with p ∈ (0, 1/2).
Consider a random walk
Sn =
Xn
i=0
Xi
.
Let Fn = σ(X1, . . . , Xn). Define the stopping time
τ = min{n ∈ N : Sn = 0}.
Compute E[τ ]
probability
asked Nov 24 at 8:34
Francesco Ricapito
2
closed as off-topic by Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh Nov 26 at 16:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 5 more comments
up vote
-1
down vote
favorite
Let A ∈ N+. Let X0 = A and (Xn)n≥1 be a sequence of i.i.d. random variables such that
P(X1 = 1) = 1 − P(X1 = −1) = p, with p ∈ (0, 1/2).
Consider a random walk
Sn =
Xn
i=0
Xi
.
Let Fn = σ(X1, . . . , Xn). Define the stopping time
τ = min{n ∈ N : Sn = 0}.
Compute E[τ ]
probability
asked Nov 24 at 8:34
Francesco Ricapito
2
closed as off-topic by Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh Nov 26 at 16:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
Make a martingale
– Makina
Nov 24 at 8:53
1
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
1
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05
|
show 5 more comments
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let A ∈ N+. Let X0 = A and (Xn)n≥1 be a sequence of i.i.d. random variables such that
P(X1 = 1) = 1 − P(X1 = −1) = p, with p ∈ (0, 1/2).
Consider a random walk
Sn =
Xn
i=0
Xi
.
Let Fn = σ(X1, . . . , Xn). Define the stopping time
τ = min{n ∈ N : Sn = 0}.
Compute E[τ ]
probability
asked Nov 24 at 8:34
Francesco Ricapito
2
Let A ∈ N+. Let X0 = A and (Xn)n≥1 be a sequence of i.i.d. random variables such that
P(X1 = 1) = 1 − P(X1 = −1) = p, with p ∈ (0, 1/2).
Consider a random walk
Sn =
Xn
i=0
Xi
.
Let Fn = σ(X1, . . . , Xn). Define the stopping time
τ = min{n ∈ N : Sn = 0}.
Compute E[τ ]
probability
probability
asked Nov 24 at 8:34
Francesco Ricapito
2
asked Nov 24 at 8:34
Francesco Ricapito
2
asked Nov 24 at 8:34
Francesco Ricapito
2
asked Nov 24 at 8:34
Francesco Ricapito
2
asked Nov 24 at 8:34
Francesco Ricapito
2
2
closed as off-topic by Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh Nov 26 at 16:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh Nov 26 at 16:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, amWhy, Davide Giraudo, user10354138, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
Make a martingale
– Makina
Nov 24 at 8:53
1
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
1
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05
|
show 5 more comments
refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
Make a martingale
– Makina
Nov 24 at 8:53
1
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
1
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05
refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
Make a martingale
– Makina
Nov 24 at 8:53
Make a martingale
– Makina
Nov 24 at 8:53
1
1
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
1
1
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05
|
show 5 more comments
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refer math.meta.stackexchange.com/questions/5020/… for proper mathjax formatting
– idea
Nov 24 at 8:50
How do you solve it?
– Francesco Ricapito
Nov 24 at 8:51
Make a martingale
– Makina
Nov 24 at 8:53
1
@Makina What kind of trouble? If $p=1/2$ then $c=0$. If $p neq 1/2$ then $c neq 0$.... which is good because this will allow the OP to calculate $mathbb{E}(tau)$.
– saz
Nov 24 at 15:41
1
@Makina Why...? Since $S_0 = A >0$ and $S_{tau}=0$, it follows from $mathbb{E}(S_{tau})-mathbb{E}(S_0) = c mathbb{E}(tau)$ that $$mathbb{E}(tau) = - frac{A}{c}$$ I agree with you that we need a different approach if $S_0 = 0$ but the OP is assuming that $S_0=A>0$.
– saz
Nov 24 at 16:05