Math Problems Vault

Problem 30.1

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=0\right) = \mathbb {P}\left(\xi_{i}=2\right) = 1/2$ . Prove that the sequence of products $\eta_{n} = \prod_{i=1}^{n} \xi_{i}$ forms a martingale.

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Remark.

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Let $(\Omega , \mathcal{F}, \mathbb {P}\left(\right))$ be a probability space, $T \subseteq \mathbb {R}$ , and let $\{ \mathcal{A}_{t}\}$ be a filtration of $\sigma$ -algebras, i.e. $\mathcal{A}_{t} \subseteq \mathcal{F}$ and $\mathcal{A}_{t} \subseteq \mathcal{A}_{s}$ for all $t,s \in T$ , $t<s$ . A random process $\xi (t)$ , $t \in T$ , is called a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ if $\xi (t)$ is measurable with respect to $\mathcal{A}_{t}$ , $\mathbb {E}\left[\left|\xi (t)\right|\right] < \infty$ , and $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] = \xi (t)$ almost surely for all $t<s$ . If for every $t$ , $\mathcal{A}_{t}$ is generated by the family $\{ \xi (s), s \leq t\}$ and $\xi (t)$ is a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ , one simply says that $\xi (t)$ is a martingale. The process $\xi (t)$ is called a submartingale (supermartingale) with respect to $\{ \mathcal{A}_{t}\}$ if the first two properties in the definition of a martingale hold and $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] \geq \xi (t)$ almost surely (respectively, $\mathbb {E}\left[\xi (s) \mid \mathcal{A}_{t}\right] \leq \xi (t)$ almost surely) for all $t<s$ .

Problem 30.2

(a)(b)

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=1\right) = 1 - \mathbb {P}\left(\xi_{i}=-1\right) = p$ , $p \neq 1/2$ . Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that the following sequence is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ :

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(a)

$\eta_{n} = S_{n} - n(2p-1)$ ;

(b)

$\eta_{n} = \left(\dfrac {1-p}{p}\right)^{S_{n}}$ .

Problem 30.3

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {P}\left(\xi_{i}=1\right) = \mathbb {P}\left(\xi_{i}=-1\right) = 1/2$ . Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that for any numbers $A \in (-\infty ,\infty )$ and $\lambda \in (0,\pi /2)$ , the sequence

\eta _{n} = \frac{e^{i\lambda (S_{n}+A)}}{\cos ^{n}\lambda }

is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ .

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Problem 30.4

(a)(b)

Let $\xi_{1}, \xi_{2}, \ldots$ be independent, identically distributed random variables with zero mean. Set $S_{n} = \xi_{1} + \ldots + \xi_{n}$ and $\mathcal{A}_{n} = \sigma (S_{1}, \ldots , S_{n})$ . Prove that the following sequence is a martingale with respect to the filtration $\{ \mathcal{A}_{n}\}$ :

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(a)

$\eta_{n} = (S_{n})^{2} - n\mathbb {E}\left[\xi_{1}^{2}\right]$ , if $\mathbb {E}\left[\xi_{1}^{2}\right] < \infty$ ;

(b)

$\eta_{n} = \dfrac {e^{\lambda S_{n}}}{(\mathbb {E}\left[e^{\lambda \xi_{1}}\right])^{n}}$ , if $\mathbb {E}\left[e^{\lambda \xi_{1}}\right] < \infty$ .

Problem 30.5

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with zero expectations. Denote $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence of sums $\{ S_{n}\}$ forms a martingale.

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Problem 30.6

Let $\xi_{1}, \xi_{2}, \ldots$ be nonnegative random variables with finite mean values, $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence of sums $\{ S_{n}\}$ forms a submartingale.

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Problem 30.7

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with zero expectations and finite variances. Denote $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence $\{ S_{n}^{2} - B_{n}\}$ forms a martingale, where $B_{n} = \operatorname {Var}\left[\xi_{1}\right] + \ldots + \operatorname {Var}\left[\xi_{n}\right]$ .

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Problem 30.8

Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables with $\mathbb {E}\left[\xi_{n}\right] = 1$ for every $n$ . Prove that the sequence of products $\eta_{n} = \prod_{i=1}^{n} \xi_{i}$ forms a martingale.

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Problem 30.9

Let $\xi$ be a random variable with finite expectation, and let $\{ \mathcal{A}_{n}\}_{n \geq 0}$ be a nondecreasing sequence of $\sigma$ -algebras. Prove that the sequence $\xi_{n} = \mathbb {E}\left[\xi \mid \mathcal{A}_{n}\right]$ forms a martingale.

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Problem 30.10

Let $\xi_{1}, \xi_{2}, \ldots$ be independent, identically distributed random variables with finite expectation, and let $S_{n} = \xi_{1} + \ldots + \xi_{n}$ . Prove that the sequence

\ldots , \frac{S_{n}}{n}, \ldots , \frac{S_{3}}{3}, \frac{S_{2}}{2}, S_{1}

forms a martingale.

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Problem 30.11

Let $\xi (t)$ be a martingale with respect to the filtration $\{ \mathcal{A}_{t}\}$ . Prove that the process $\xi (t)$ has uncorrelated increments, i.e. $\operatorname {Cov}\left[ \xi (s)-\xi (t), \xi (v)-\xi (u) \right] = 0$ for $t<s<u<v$ .

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Problem 30.12

Let $\xi (t)$ be a homogeneous Poisson process with parameter $\lambda$ . Prove that $\zeta (t) = \mathrm{exp}\left(\xi (t)-at\right)$ is a submartingale for $a \leq \lambda (e-1)$ and a supermartingale for $a \geq \lambda (e-1)$ .

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Problem 30.13

Prove that a Wiener process started at zero is a continuous-time martingale.

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Problem 30.14

Let $\{ \xi_{n}\}$ be a martingale with $\mathbb {E}\left[(\xi_{n}-\xi_{n-1})^{2}\right] \leq c$ for some $c<\infty$ . Prove that $\xi_{n}/n \to \mathbb {E}\left[\xi_{1}\right]$ in probability.

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Problem 30.15

Let $\xi_{0}, \xi_{1}, \ldots$ be a branching process, $\xi_{0}=1$ and $\mathbb {E}\left[\xi_{1}\right]=m$ . Set $\eta_{n} = \xi_{n}/m^{n}$ . Prove that $\{ \eta_{n}\}$ forms a martingale.

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Problem 30.16

(a)(b)

Let $w(t)$ be a standard Wiener process. Determine which of the following processes are martingales:

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(a)

$e^{cw(t)}$ ;

(b)

$tw(t) - \displaystyle \int_{0}^{t} w(s)\, ds$ .