Prouver / réfuter

Prouver / réfuter $E[1_A | \mathscr{F_t}] = 0 \ \text{or} \ 1 \ \text{a.s.} \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}] \ \text{a.s.}$

Etant donné un espace de probabilité filtré , et encore . $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$ $A \in \mathscr{F}$

Supposons que S'ensuit-il que Et ?

\exists t \in N s.t. E [1_{A} | F_{t}] = 1 a.s.

$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | \mathscr{F_t}] = 1 \ \text{a.s.}$

E [1_{A} | F_{s}] = E [1_{A} | F_{t}] a.s. \forall s > t ?

$E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}] \ \text{a.s.} \ \forall s > t \ ?$

\forall s < t

$\forall s < t$

Et si à la place Ou si

\exists t \in N s.t. E [1_{A} | F_{t}] = 0 a.s. ?

$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | \mathscr{F_t}] = 0 \ \text{a.s.} \ ?$

E [1_{A} | F_{t}] = p a.s. for some p \in (0, 1) ?

$E[1_A | \mathscr{F_t}] = p \ \text{a.s.} \ \text{for some} \ p \in (0,1) \ ?$

Ce que j'ai essayé:

$\Bbb E[1_A|\mathscr F_t]=1$ $\Bbb E[1_A]=1$ $1_A=1$ $\Bbb E[1_A|\mathscr F_s]=1$ $s$

$\Bbb E[1_A|\mathscr F_t]=0$ $\Bbb E[1_A]=0$ $1_A=0$ $\Bbb E[1_A|\mathscr F_s]=0$ $s$

$\Bbb E[1_A|\mathscr F_t]=p$ $p\in(0,1)$

$\Bbb E[1_A|\mathscr F_s]=E[E[1_A|\mathscr F_t]|\mathscr F_s] = E[p|\mathscr F_s] = p$ $s>t$

$= p$

$F$ $\mathscr F_t$

E [1_{A} \cdot F] = E [E [1_{A} \cdot F | F_{t}]] = E [F \cdot E [1_{A} | F_{t}]]

$\Bbb E[1_A\cdot F]=\Bbb E[E[1_A\cdot F|\mathscr F_t]]=\Bbb E[F\cdot E[1_A|\mathscr F_t]]$

= E [p \cdot F] = p E [F] = E [1_{A}] \cdot E [F]

$=\Bbb E[p\cdot F]=p\Bbb E[F]=\Bbb E[1_A]\cdot\Bbb E[F]$

$1_A$ $F$ $\sigma(A)$ $\mathscr F_t$ $\sigma(A)$ $\mathscr F_s$ $s<t$ $E[1_A|\mathscr F_s] = E[1_A] = p$ $s>t$

$\mathscr F_s$ $\mathscr F_s$

— BCLC
source

$E[1_A | \mathscr{F_t}] = 1$ $E[1_A | \mathscr{F_t}] \in \{0, 1\}$ $E[1_A | \mathscr{F_t}]$ $\{0, 1\}$ $E[1_A | \mathscr{F_t}]=1_B$ $B\in\mathscr{F_t}$ $\int_F 1_B d\mathbb{P}=\int_F 1_A d\mathbb{P}$ $F\in\mathscr{F_t}$ $F=B$ $P(B)=P(A\cap B)$ $B\subset A$ $E[E[1_A | \mathscr{F_t}]] = E[1_B]$ $P(A)=P(B)$ $A=B$

$s\gt t$ $\mathscr{F_t}\subset\mathscr{F_s}$ $E[1_A | \mathscr{F_t}]=E[E[1_A | \mathscr{F_t}] | \mathscr{F_s}]$ $E[1_A | \mathscr{F_t}]=1_A$ $E[1_A | \mathscr{F_s}]=1_A$ $s>t$ $1_A$ $s<t$ $A\in\mathscr{F_s}$ $E[1_A | \mathscr{F_s}]=1_A$ $A$ $\mathscr{F_s}$ $E[1_A | \mathscr{F_s}]$ $A=\{\omega_2\}\in\mathscr{F_2}\setminus\mathscr{F_1}$ $E[1_A | \mathscr{F_0}]=\frac{1}{8} 1_\Omega$ $E[1_A | \mathscr{F_1}]=\frac{1}{2}1_{\{\omega_1,\omega_2\}}$ $E[1_A | \mathscr{F_2}]=1_{\{\omega_2\}}$ $E[1_A | \mathscr{F_3}]=1_{\{\omega_2\}}$

— S. Catterall réintègre Monica
source

P (B) = P (A \cup B) \to B \subseteq A

$P(B) = P(A \cup B) \to B \subseteq A$

E [1_{A} | F_{t}] = 1_{A}

$E[1_A | \mathscr{F}_t] = 1_A$

A

$A$

F_{0}

$\mathscr{F}_0$

0

$0$

@ocramz et S. Catterall, édition terminée. Comment est-ce pls? ^ - ^

— BCLC

A

$A$

ω_{i}

$\omega_i$

A

$A$

A

$A$

0

$0$

P (B) = P (A \cap B)

$P(B) = P(A \cap B)$

B

$B$

A \cap B

$A\cap B$

A^{c} \cap B

$A^c\cap B$

P (A^{c} \cap B) = 0

$P(A^c\cap B)=0$

B \subset A

$B\subset A$

P (A) = P (B)

$P(A)=P(B)$

A = B \in F_{t}

$A=B\in \mathscr{F_t}$