Question

Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

score 166 · Answer 1 · 2022-03-15 23:52:40

166

Views

Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

Published on 15 Mar 2022 - 23:52

score 68 · Answer 2 · 2022-03-17 06:52:14

68

Views

With stopping time $T$ and positive constant $t$, show that $T \land t$ is $\mathscr{F}_t$ measurable.

Published on 17 Mar 2022 - 6:52

score 171 · Answer 3 · 2022-03-21 14:09:19

171

Views

Karatzas + Shreve 2.19: Show that $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable

Published on 21 Mar 2022 - 14:09

score 104 · Answer 4 · 2022-03-22 05:37:20

104

Views

Karatzas + Shreve 2.22: For optional time $S$, stopping time $T$, prove that $\mathscr{F}_{S+} \subset \mathscr{F}_T$

Published on 22 Mar 2022 - 5:37

score 20 · Answer 5 · 2022-03-22 14:21:36

20

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Showing natural filtration of counting process is the same as filtration at $n-1$'st jump $T_{n-1}$ given the event $\{T_{n-1}\le t < T_n\}$

Published on 22 Mar 2022 - 14:21

score 70 · Answer 6 · 2022-03-23 10:36:35

70

Views

Why is the natural filtration of Brownian motion not $\mathcal{B}(\mathbb{R})$ for all $t$?

Published on 23 Mar 2022 - 10:36

score 82 · Answer 7 · 2022-03-26 21:20:07

82

Views

Given sequence of optional times $T_n$ and optional time $T = \inf_n T_n$, show $\mathscr{F}_{T+} = \cap_n \mathscr{F}_{T_n+}$

Published on 26 Mar 2022 - 21:20

score 35 · Answer 8 · 2022-03-27 04:15:09

35

Views

With optional time $T$ on $\mathscr{F}_t$, and $T_n = \frac{k}{2^n}$ on $\{ \frac{k-1}{2^n} \le T < \frac{k}{2^n} \}$, show $T_n$ is a stopping time

Published on 27 Mar 2022 - 4:15

score 68 · Answer 9 · 2022-03-29 09:58:38

68

Views

Composition factors of intersection of modules

Published on 29 Mar 2022 - 9:58

score 108 · Answer 10 · 2022-04-12 07:36:40

108

Views

Does $\sigma(\bigcap_{n=1}^\infty \mathcal{F}_n)=\bigcap_{n=1}^\infty \sigma(\mathcal{F}_n)$ hold?

Published on 12 Apr 2022 - 7:36

score 68 · Answer 11 · 2022-04-30 13:31:13

68

Views

Filtrations of stopping times.

Published on 30 Apr 2022 - 13:31

score 14 · Answer 12 · 2026-03-01 17:08:05

14

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For every $n \geq 0$ and $A_{n} \in \Omega_{n}$, $\mathbb{E} \left [X_{n+1} \mathit{1}_{A_{n}} \right ]=0$. Explain.

Published on 01 Mar 2026 - 17:08

score 112 · Answer 13 · 2022-05-18 20:21:23

112

Views

Problem with intuition regarding sigma algebras and information

Published on 18 May 2022 - 20:21

score 145 · Answer 14 · 2022-05-19 10:46:45

145

Views

Meaning of measurableness

Published on 19 May 2022 - 10:46

score 46 · Answer 15 · 2022-05-23 19:52:15

46

Views

What is the correct filtration?

Published on 23 May 2022 - 19:52

List Question

Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

With stopping time $T$ and positive constant $t$, show that $T \land t$ is $\mathscr{F}_t$ measurable.

Karatzas + Shreve 2.19: Show that $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable

Karatzas + Shreve 2.22: For optional time $S$, stopping time $T$, prove that $\mathscr{F}_{S+} \subset \mathscr{F}_T$

Showing natural filtration of counting process is the same as filtration at $n-1$'st jump $T_{n-1}$ given the event $\{T_{n-1}\le t < T_n\}$

Why is the natural filtration of Brownian motion not $\mathcal{B}(\mathbb{R})$ for all $t$?

Given sequence of optional times $T_n$ and optional time $T = \inf_n T_n$, show $\mathscr{F}_{T+} = \cap_n \mathscr{F}_{T_n+}$

With optional time $T$ on $\mathscr{F}_t$, and $T_n = \frac{k}{2^n}$ on $\{ \frac{k-1}{2^n} \le T < \frac{k}{2^n} \}$, show $T_n$ is a stopping time

Composition factors of intersection of modules

Does $\sigma(\bigcap_{n=1}^\infty \mathcal{F}_n)=\bigcap_{n=1}^\infty \sigma(\mathcal{F}_n)$ hold?

Filtrations of stopping times.

For every $n \geq 0$ and $A_{n} \in \Omega_{n}$, $\mathbb{E} \left [X_{n+1} \mathit{1}_{A_{n}} \right ]=0$. Explain.

Problem with intuition regarding sigma algebras and information

Meaning of measurableness

What is the correct filtration?

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