Prove X is a martingale

Question

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by:

$X_0 = 1$

and for $n \geq 1$

$X_{n+1} = 2X_n$ w/ prob 1/2

$X_{n+1} = 0$ w/ prob 1/2

and $\mathscr{F_n} = \mathscr{F_n}^{X} \doteq \sigma(X_0, X_1, ..., X_n)$.

I think that $X_{n} = 2^{n} \prod_{i=1}^{n} 1_{A_i} \forall n \geq 0$ where $A_1 = \{ \omega \in \Omega | X_{2}(\omega) = 2 X_1(\omega) \} \in \mathscr{F}$ as follows:

$X_{1} = 2X_0* 1_{A_1} + 0*1_{A_1^c}$

$X_{2} = 2X_1* 1_{A_2} + 0*1_{A_2^c}$

$=2(2X_0* 1_{A_1} + 0*1_{A_1^c})*1_{A_2} + 0*1_{A_2^c}$

Is that right? If not, why? If so, here is my attempt:

(leaving out adaptability and integrability stuff)

We must show that $E[X_{n+1}|\mathscr{F_n}] \equiv E[2^{n+1} \prod_{i=1}^{n+1} 1_{A_i}|\mathscr{F_n}] = 2^n \prod_{i=1}^{n} 1_{A_i}$? Is that right?

help please?

Answer 1

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$ where $\mathscr{F_n} = \mathscr{F_n}^{X} \doteq \sigma(X_0, X_1, \ldots, X_n)$, show that $X = (X_n)_{n \geq 0}$ is a $(\mathscr{F}_n^X, \mathbb{P})$-martingale where $X$ is given by:

$X_{n+1} = 2X_n$ w/ prob 1/2

$X_{n+1} = 0$ w/ prob 1/2

and $X_0 = 1$.

Define the iid random variables $V_0 = 1$,

$V_1, V_2, \ldots \sim P(V_i = 0) = P(V_i = 2) = 1/2$. Then, $X_n = \prod_{i=0}^{n} V_i$.

1. $X_n$'s are bounded and hence integrable.
2. $X_n$'s are adapted to their natural filtration.
3. $E[X_n \mid \mathscr{F_m}] = X_m$

\begin{align} \text{LHS} & = E\left[\prod_{i=0}^{n} V_i \mid \mathscr{F_m}\right] \\ & = E\left[\prod_{i=0}^{n} V_i \mid \mathscr{F_m}\right] \\ & = E\left[\prod_{i=0}^{m} V_i \prod_{i=m+1}^{n} V_i\mid \mathscr{F_m}\right] \\ & = E\left[X_m \prod_{i=m+1}^{n} V_i\mid \mathscr{F_m}\right] \\ & = X_m E\left[\prod_{i=m+1}^{n} V_i\mid \mathscr{F_m}\right] \\ & = X_m E\left[\prod_{i=m+1}^{n} V_i\right] \tag{*} \\ & = X_m \prod_{i=m+1}^{n} E\left[V_i\right] \text{ by the independence of the $V_i$'s} \\ & = X_m \prod_{i=m+1}^{n} E\left[V_i\right] \\ & = X_m \prod_{i=m+1}^{n} (1) \\ & = \text{RHS} \quad \text{QED} \end{align}

(*)

$\mathscr G_m = \sigma(V_1,\ldots,V_m) \supset \mathscr F_m$

Being independent of $\mathscr G_m$ implies being independent of $\mathscr F_m$

Answer 2

There are 3 things to check:

1. $X_n$ is measurable with respect to $F_n$: This is ok by definition of $F$.
2. $X_n$ is integrable: as $X_n$ is bounded, so that is ok.
3. (the big part) $E[X_{n+1}|X_n] = X_n$:

conditionally to $X_n$, there are two options:

• with probability 1/2, $X_{n+1} = 2X_n$
• with probability 1/2, $X_{n+1} = 0$

hence $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n $$