Discrete-Time Martingales

Abstract

As mentioned in the introduction, stochastic processes can be classified into two main families, namely Markov processes on the one hand, and martingales on the other hand. Markov processes have been our main focus of attention so far, and in this chapter we turn to the notion of martingale. In particular we will give a precise mathematical meaning to the description of martingales stated in the introduction, which says that when $(X_n)_{n\in {\mathord {\mathbb {N}}}}$ is a martingale, the best possible estimate at time $n\in {\mathord {\mathbb {N}}}$ of the future value $X_m$ at time $m>n$ is $X_n$ itself. The main application of martingales will be to recover in an elegant way the previous results on gambling processes of Chap. 2. Before that, let us state many recent applications of stochastic modeling are relying on the notion of martingale. In financial mathematics for example, the notion of martingale is used to characterize the fairness and equilibrium of a market model.

Exercises

Exercise 10.1

Consider a sequence $(X_n)_{n \ge 1}$ of independent Bernoulli random variables with

$$ \mathbb {P}( X_n = 1 ) = \mathbb {P}( X_n = -1 ) = 1/2, \qquad n\ge 1, $$

and the process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ defined by $M_0 := 0$ and

$$ M_n := \sum _{k=1}^n 2^{k-1} X_k, \qquad n \ge 1. $$

See (Fig. 10.2), Note that when $X_1=X_2=\cdots =X_{n-1}=-1$ and $X_n=1$, we have

$$ M_n = - \sum _{k=1}^{n-1} 2^{k-1} + 2^{n-1} = - \frac{1-2^{n-1}}{1-2} + 2^{n-1} = 1, \qquad n \ge 1. $$

• (a) Show that the process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is a martingale.

• (b) Is the random time

  $$ \tau : = \inf \{ n \ge 1 \ : \ M_n = 1 \} $$

  a stopping time?

• (c) Consider the stopped process

  $$ M_{\tau \wedge n} := M_n \mathbbm {1}_{ \{ n< \tau \} } + \mathbbm {1}_{ \{ \tau \le n\} } = \left\{ \begin{array}{ll} M_n = 1-2^n &{} \text{ if } n < \tau , \\ \\ M_\tau = 1 &{} \text{ if } n \ge \tau , \end{array} \right. $$

  $n \in {\mathord {\mathbb {N}}}$, See (Fig. 10.3). Give an interpretation of $(M_{n\wedge \tau })_{n\in {\mathord {\mathbb {N}}}}$ in terms of betting strategy for a gambler starting a game at $M_0=0$.

• (d) Determine the two possible values of $M_{\tau \wedge n}$ and the probability distribution of $M_{\tau \wedge n}$ at any time $n\ge 1$.

• (e) Show, using the result of Question (d), that we have

  $$\begin{aligned} \mathrm{I}\! \mathrm{E}[ M_{\tau \wedge n} ] = 0, \qquad n \in {\mathord {\mathbb {N}}}. \end{aligned}$$

• (f) Show that the result of Question (e) can be recovered using the stopping time theorem.

Fig. 10.2

Possible paths of the process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$

Fig. 10.3

Possible paths of the stopped process $(M_{\tau \wedge n})_{n\in {\mathord {\mathbb {N}}}}$

Exercise 10.2

Let $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ be a discrete-time submartingale with respect to a filtration $(\mathcal{F}_n)_{n\in {\mathord {\mathbb {N}}}}$, with $\mathcal{F}_0 = \{ \emptyset , \varOmega \}$, i.e. we have

$$ M_n \le \mathrm{I}\! \mathrm{E}[ M_{n+1} \mid \mathcal{F}_n ], \qquad n \ge 0. $$

• (a) Show that we have $\mathrm{I}\! \mathrm{E}[ M_{n+1} ] \ge \mathrm{I}\! \mathrm{E}[ M_n]$, $n \ge 0$, i.e. a submartingale has an increasing expectation.

• (b) Show that independent increment processes whose increments have nonnegative expectation are examples of submartingales.

• (c)  (Doob-Meyer decomposition) Show that there exists two processes $(N_n)_{n\in {\mathord {\mathbb {N}}}}$ and $(A_n)_{n\in {\mathord {\mathbb {N}}}}$ such that

• (i) $(N_n)_{n\in {\mathord {\mathbb {N}}}}$ is a martingale with respect to $(\mathcal{F}_n)_{n\in {\mathord {\mathbb {N}}}}$,

• (ii) $(A_n)_{n\in {\mathord {\mathbb {N}}}}$ is non-decreasing, i.e. $A_n\le A_{n+1}$, a.s., $n\in {\mathord {\mathbb {N}}}$,

• (iii) $(A_n)_{n\in {\mathord {\mathbb {N}}}}$ is predictable in the sense that $A_n$ is $\mathcal{F}_{n-1}$-measurable, $n\in {\mathord {\mathbb {N}}}$, and

• (iv) $M_n = N_n + A_n$, $n\in {\mathord {\mathbb {N}}}$.

Hint: Let $A_0=0$ and

$$ A_{n+1} := A_n + \mathrm{I}\! \mathrm{E}[M_{n+1}-M_n \mid \mathcal{F}_n ], \quad n\ge 0, $$

and define $(N_n)_{n\in {\mathord {\mathbb {N}}}}$ in such a way that it satisfies the four required properties.

• (d) Show that for all bounded stopping times $\sigma$ and $\tau$ such that $\sigma \le \tau$ a.s., we have

  $$ \mathrm{I}\! \mathrm{E}[M_\sigma ] \le \mathrm{I}\! \mathrm{E}[M_\tau ]. $$

  Hint: Use the Doob stopping time Theorem 10.6 for martingales and (10.3.3).

Exercise 10.3

Consider an asset price $(S_n)_{n=0,1,\ldots , N}$ which is a martingale under the risk-neutral measure $\mathbb {P}^*$, with respect to the filtration $(\mathcal{F}_n)_{n\in {\mathord {\mathbb {N}}}}$. Given the (convex) function $\phi (x) := (x-K)^+$, show that the price of an Asian option with payoff

$$ \phi \left( \frac{S_1 + S_2 +\cdots + S_N}{N} \right) $$

is upper bounded by the price of the European call option with maturity N, i.e. show that

$$ \mathrm{I}\! \mathrm{E}^* \left[ \phi \left( \frac{S_1 + S_2 + \cdots + S_N}{N} \right) \right] \le \mathrm{I}\! \mathrm{E}^* [ \phi (S_N ) ]. $$

Hint: Use in the following order:

• (i) the convexity inequality

  $$ \phi \left( \frac{x_1 + x_2 + \cdots + x_n}{n} \right) \le \frac{\phi ( x_1)+\phi ( x_2)+\cdots + \phi (x_n)}{n}, $$

• (ii) the martingale property of $(S_k)_{k\in {\mathord {\mathbb {N}}}}$,

• (iii) the conditional Jensen inequality $\phi ( \mathrm{I}\! \mathrm{E}[ F \mid \mathcal{G} ] ) \le \mathrm{I}\! \mathrm{E}[ \phi ( F ) \mid \mathcal{G} ]$,

• (iv) the tower property of conditional expectations.

Exercise 10.4

A process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is a submartingale if it satisfies

$$ M_k \le \mathrm{I}\! \mathrm{E}[ M_n \mid \mathcal{F}_k ] , \qquad k=0,1,\ldots , n. $$

• (a) Show that the expectation $\mathrm{I}\! \mathrm{E}[M_n]$ of a submartingale increases with time $n\in {\mathord {\mathbb {N}}}$.

• (b) Consider the random walk given by $S_0: =0$ and

  $$ S_n := \sum _{k=1}^n X_k = X_1+ X_2+ \cdots + X_n, \qquad n \ge 1, $$

  where $(X_n)_{n\ge 1}$ is an i.i.d. Bernoulli sequence of $\{0,1\}$-valued random variables with $\mathbb {P}(X_n = 1 ) =p$, $n \ge 1$. Under which condition on $\alpha \in {\mathord {\mathbb {R}}}$ is the process $(S_n - \alpha n )_{n\in {\mathord {\mathbb {N}}}}$ a submartingale?

Exercise 10.5

Recall that a process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is a submartingale if it satisfies

$$ M_k \le \mathrm{I}\! \mathrm{E}[ M_n \mid \mathcal{F}_k ] , \qquad k=0,1,\ldots , n. $$

• (a) Show that any convex function $(\phi (M_n))_{n \in {\mathord {\mathbb {N}}}}$ of a martingale $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is itself a submartingale. Hint: Use Jensen’s inequality.

• (b) Show that any convex nondecreasing function $\phi (M_n)$ of a submartingale $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ remains a submartingale.

Problem 10.6

• (a) Consider $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ a nonnegative martingale. For any $x >0$, let

  $$ \tau _x := \inf \{ n \ge 0 \ : \ M_n \ge x \}. $$

  Show that the random time $\tau _x$ is a stopping time.

• (b) Show that for all $n\ge 0$ we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} M_k \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[M_n]}{x}. \end{aligned}$$

  (10.5.1)

  Hint: Use the Markov inequality and the Doob stopping time Theorem 10.6 for the stopping time $\tau _x$.

• (c) Show that (10.5.1) remains valid when $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is a nonnegative submartingale.

  Hint: Use the Doob stopping time theorem for submartingales as in Exercise 10.2-(d).

• (d) Show that for any $n\ge 0$ we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} M_k \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[(M_n)^2]}{x^2}, \qquad x>0. \end{aligned}$$

• (e) Show that more generally we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} M_k \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[ | M_n |^p]}{x^p}, \qquad x>0, \end{aligned}$$

  for all $n\ge 0$ and $p \ge 1$.

• (f) Given $(Y_n)_{n\ge 1}$ a sequence of centered independent random variables with same mean $\mathrm{I}\! \mathrm{E}[Y_n] =0$ and variance $\sigma ^2 = {\mathrm {Var}}[ Y_n]$, $n \ge 1$, consider the random walk $S_n = Y_1+Y_2+\cdots +Y_n$, $n\ge 1$, with $S_0=0$.

  Show that for all $n\ge 0$ we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} | S_k | \ge x \right) \le \frac{n\sigma ^2}{x^2}, \qquad x>0. \end{aligned}$$

• (g) Show that for any (not necessarily nonnegative) submartingale we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} M_k \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[ M_n^+ ]}{x}, \qquad x>0, \end{aligned}$$

  where $z^+ = \max ( z , 0)$, $z\in {\mathord {\mathbb {R}}}$.

• (h) A process $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ is a supermartingale⁵ if it satisfies

  $$ \mathrm{I}\! \mathrm{E}[ M_n \mid \mathcal{F}_k ] \le M_k, \qquad k=0,1,\ldots , n. $$

  Show that for any nonnegative supermartingale we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} M_k \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[ M_0 ]}{x}, \qquad x>0. \end{aligned}$$

• (i) Show that for any nonnegative submartingale $(M_n)_{n\in {\mathord {\mathbb {N}}}}$ and any convex nondecreasing nonnegative function $\phi$ we have

  $$\begin{aligned} \mathbb {P}\left( \max _{k=0,1,\ldots , n} \phi ( M_k ) \ge x \right) \le \frac{\mathrm{I}\! \mathrm{E}[\phi ( M_n )]}{x}, \qquad x>0. \end{aligned}$$

  Hint: Consider the stopping time

  $$ \tau ^\phi _x := \inf \{ n \ge 0 \ : \ M_n \ge x \}. $$

• (j) Give an example of a nonnegative supermartingale which is not a martingale.