How to show $σ =inf(t ≥ 0 : |B_t| =sqrt{3})$ is optimal for $M_t = B_t^4 − 6tB^2_t +e^{2B_t−2t}+3t^2$
Given $σ =inf(t ≥ 0 : |B_t| =sqrt{3})$, show $sigma$ is optimal for $M_t = B_t^4 − 6tB^2_t +e^{2B_t−2t}+3t^2$ for $t ≥ 0$.
Struggling to show that it is bounded, my attempt so far is:
$$|M_{t wedge sigma}|=|B_{t wedge sigma}^4 − 6({t wedge sigma})B^2_{t wedge sigma} +e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|leq|B_{t wedge sigma}^4+e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|$$
Unsure how to proceed.
probability-theory brownian-motion martingales
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Given $σ =inf(t ≥ 0 : |B_t| =sqrt{3})$, show $sigma$ is optimal for $M_t = B_t^4 − 6tB^2_t +e^{2B_t−2t}+3t^2$ for $t ≥ 0$.
Struggling to show that it is bounded, my attempt so far is:
$$|M_{t wedge sigma}|=|B_{t wedge sigma}^4 − 6({t wedge sigma})B^2_{t wedge sigma} +e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|leq|B_{t wedge sigma}^4+e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|$$
Unsure how to proceed.
probability-theory brownian-motion martingales
edited 17 hours ago
Did
246k23221455
asked Jan 1 at 11:38
Zugzwangerz
32
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Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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2
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55
add a comment |
Given $σ =inf(t ≥ 0 : |B_t| =sqrt{3})$, show $sigma$ is optimal for $M_t = B_t^4 − 6tB^2_t +e^{2B_t−2t}+3t^2$ for $t ≥ 0$.
Struggling to show that it is bounded, my attempt so far is:
$$|M_{t wedge sigma}|=|B_{t wedge sigma}^4 − 6({t wedge sigma})B^2_{t wedge sigma} +e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|leq|B_{t wedge sigma}^4+e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|$$
Unsure how to proceed.
probability-theory brownian-motion martingales
edited 17 hours ago
Did
246k23221455
asked Jan 1 at 11:38
Zugzwangerz
32
New contributor
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given $σ =inf(t ≥ 0 : |B_t| =sqrt{3})$, show $sigma$ is optimal for $M_t = B_t^4 − 6tB^2_t +e^{2B_t−2t}+3t^2$ for $t ≥ 0$.
Struggling to show that it is bounded, my attempt so far is:
$$|M_{t wedge sigma}|=|B_{t wedge sigma}^4 − 6({t wedge sigma})B^2_{t wedge sigma} +e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|leq|B_{t wedge sigma}^4+e^{2B_{t wedge sigma}−2({t wedge sigma})}+3({t wedge sigma})^2|$$
Unsure how to proceed.
probability-theory brownian-motion martingales
probability-theory brownian-motion martingales
edited 17 hours ago
Did
246k23221455
asked Jan 1 at 11:38
Zugzwangerz
32
New contributor
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 17 hours ago
Did
246k23221455
asked Jan 1 at 11:38
Zugzwangerz
32
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Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 17 hours ago
Did
246k23221455
edited 17 hours ago
Did
246k23221455
edited 17 hours ago
Did
246k23221455
246k23221455
asked Jan 1 at 11:38
Zugzwangerz
32
New contributor
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 1 at 11:38
Zugzwangerz
32
asked Jan 1 at 11:38
Zugzwangerz
32
32
New contributor
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Zugzwangerz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55
add a comment |
2
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55
2
2
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55
add a comment |
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2
What do you mean by "$sigma$ is optimal" ...? When do you call a stopping time optimal?
– saz
Jan 1 at 16:55