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Martingales - Exercises

Exercise 1

  1. Let \(X\) and \(Y\) be two super-martingales. Show that \(X\wedge Y\) is a super-martingale;

  2. Let \(X_1, X_2, \ldots\) be independent random variables (iid) such that \(P[X_n =1] =P[X_1 =1]=p\) and \(P[X_n=-1]=P[X_1 = -1]=1-p\) with \(0<p<1\). In other terms \(X_1, X_2, \ldots\) defines a sequence of iid Bernoulli random variables. We define \(\mathcal{F}_0 = \{\emptyset, \Omega\}\) and \(\mathcal{F}_t = \sigma(X_1, \ldots, X_t)\) for every \(t\geq 1\). We consider the random walk

    \[S_0 = x, \quad \text{and} \quad S_t = S_{t-1} + X_t = x + \sum_{s=1}^t X_s\]

    Find conditions on \(p\) such that - \(S\) is a super-martingale; - \(S\) is a sub-martingale; - \(S\) is a martingale.

  3. Show, or find a counter example for the following assertion: Let \(X\) be an adapted process such that \(X_t\) is integrable and \(E[X_0]=E[X_t]\) for every \(t\). Then \(X\) is a martingale.

  4. Let \(X\) be an adapted and predictable process. Show that \(X\) is a martingale if and only if \(E[X_\tau]=E[X_0]\) for every bounded stopping time \(\tau\).

Exercise 2

  1. Suppose that \(X\) is a square integrable martingale. Show that \(X^2=(X^2_t)\) is a sub martingale and compute the Doob-Meyer decomposition of \(X^2\).

  2. Suppose that \(X\) is an adapted process such that \(C>X_t>c>0\) for every \(t\) where \(C,c\) are some constants. Show that there exists a unique decomposition

    \[X=MA\]

    where \(M\) is a bounded martingale and \(A\) is a predictable bounded process with \(A_0=1\).

Exercise 3

For bounded random variables \(X\) (or with any epononential moment) define

\[m(\lambda)=E\left[ e^{\lambda X} \right], \quad \text{and}\quad\phi(\lambda)=\ln\left( E\left[ e^{\lambda X} \right] \right)=\ln(m(\lambda)), \quad \lambda \in \mathbb{R}\]
  1. Show that \(\phi\) is a convex function which is twice differentiable.
  2. Show that \(\phi^\prime(0)=E\left[ X \right]\) and \(\phi^{\prime\prime}(0)=\text{VaR}(X)=E\left[ X^2 \right]-E[X]^2\).

From now on, let \(R_1,\ldots,R_t,\ldots\) be a sequence of independent random variables such that \(R_t\sim X\) for every \(t\) and let

\[S_t=S_{t-1}+R_t,\quad\text{with}\quad S_0=0\]

Define the process \(L(\lambda)\) -- depending on \(\lambda\) -- given by

\[L_0(\lambda)=0,\quad\text{and}\quad L_t(\lambda)=\exp\left( \lambda S_t-t\phi(\lambda) \right)=\prod_{s=1}^t \frac{\exp(\lambda R_t)}{m(\lambda)}, \quad t\geq 1\]

and the filtration \(\mathcal{F}_0=\{\emptyset, \Omega\}\) and \(\mathcal{F}_t=\sigma( R_s\colon s\leq t)\).

  1. Show that for every \(\lambda \in \mathbb{R}\), the process \(L(\lambda)\) is a martingale.
  2. We denote by \(L^\prime_t=\frac{dL_t}{d\lambda}(0)\). Deduce that \(L^\prime\) is a Martingale for which holds \(L^\prime_t=S_t-t\phi^\prime(0)=S_t-t E[X]\).

Exercise 4

let \(X\) be an adapted stochastic process such that \(E[\sup_t |X_t|]<\infty\). Denote by \(\mathcal{T}\) the set of all stopping times and let \(T \in \mathbb{N}_0\) be an arbitrary time horizon. We define the process \(S\) recursively by

\[S_T=X_T\quad \text{and}\quad S_t=\max \left\{ E\left[ S_{t+1}|\mathcal{F}_t \right];X_t \right\}, \quad t\leq T-1.\]

Define further \(\tau^t=\inf\{s\colon s\geq t \text{ and }S_s=X_s\}\) for every \(t=0,\ldots, T\). From step \((a)\) to \((c)\) we assume that we are given the finite time horizon \(T\).

  1. Show that \(S\) is a super-martingale such that \(S_t\geq X_t\) for every \(t=0,\ldots, T\);
  2. Let \(U\) be a super-martingale such that \(U\geq X\). Show that \(U\geq S\).

  3. Show that \(E[X_{\tau^t}|\mathcal{F}_s]=E[S_t|\mathcal{F}_s]\) for every \(s\leq t\leq T\) and conclude that

    \[E\left[ X_{\tau^t} \right]=E\left[ S_t \right]=\max_{\{\tau \in \mathcal{T}\colon t\leq \tau\leq T\}} E\left[ X_\tau \right]\]

  4. We denote by \(S^T=(S_t^T)_{t=0,\ldots, T}\) the process defined in (a) whereby, we stress the dependence on the time horizon due to its recursive definition. Clearly, \(S_t=\lim_{T\to \infty} S^T_t\) defines a process \(S\). Show that \(S\) is a super-martingale for which holds \(S\geq X\). Show further that for every other super-martingale \(U\) such that \(U\geq X\) it follows \(U\geq S\).

Exercise 5

Let \((\Omega, \mathcal{F}, P)\) be a probability space and \(\mathbb{F}=(\mathcal{F}_t)\) be a filstration. For a stochastic process \(X\) we denote by \(\mathbb{F}^X\) the filtration generated by \(X\), that is

\[\mathcal{F}_t^X = \sigma\left( X_s \colon s\leq t \right), \quad t=0, 1, \ldots\]
  1. Let \(X\) be a \(\mathbb{F}^X\) martingale and \(Y\) a \(\mathbb{F}^Y\) martingale. Suppose that \(XY = (X_tY_t)\) is integrable. Prove -- or provide a counter example -- that \(XY\) is a \(\mathbb{F}^{XY}\) martingale.
  2. Let \(X\) be a \(\mathbb{F}^X\) martingale and \(Y\) a \(\mathbb{F}^Y\) martingale. Suppose that \(\mathbb{F}^X\) and \(\mathbb{F}^Y\) are independent. Prove -- or provide a counter example -- that \(XY\) is a \(\mathbb{F}^{XY}\) martingale.
  3. Provide an example of a martingale \(X=(X_t)\) such that \(\sup_t E[|X_t|]<\infty\) and \(X_t\to X_{\infty}\) \(P\)-almost surely for some \(X_{\infty}\) but for which however it does not hold \(E[|X_t-X_{\infty}|]\to 0\).
  4. Let \((X^n)\) be a sequence of martingales, that is \(X^n = (X^n_t)\) is a martingale for each \(n\). We suppose that \(X^n_t \to X_t\) almost surely for every \(t\) where \(X\) is an integrable stochastic process. Prove -- or provide a counter example -- that \(X\) is a martingale.
  5. Let \((X^n)\) be a sequence of martingale, that is \(X^n = (X^n_t)\) is a martingale for each \(n\). We suppose that \(X^n_t \to X_t\) almost surely for every \(t\) where \(X\) is a stochastic process. Prove that there exists an increasing sequence of stopping times \((\tau^m)\) with \(\tau^m \nearrow \infty\) such that \(X^{\tau^m}\) is a martingale for every \(m\).

Exercise 6

Let \(R=(R_t)\) be a sequence of independent and identically distributed random variables with \(P[R_t=1]=P[R_1=1]=p \in (0,1)\) and \(P[R_t=-1]=P[R_1=-1]=1-p\). We define \(\mathcal{F}_0=\{\emptyset,\Omega\}\) and \(\mathcal{F}_t=\sigma(R_s\colon s\leq t)\) for \(t\geq 1\). We consider the random walk

\[S_t=S_{t-1}+R_t, \quad S_0=0\]

Define \(\tau =\inf\{t\colon S_t=a\text{ or }S_t=-b\}\) for two integers \(a,b\).

  1. Show that

    • If \(p=1/2\), then \(P[ \liminf X_t=-\infty\text{ and } \limsup X_t =\infty]=1\);
    • If \(p>1/2\), then \(P\left[ \lim X_t=\infty \right]=1\);

  2. Suppose that \(p=1/2\).

    • Show that \(P[\tau<\infty]=1\).

    • Show that

      \[P\left[ S_{\tau}=a \right]=\frac{b}{a+b}\]

      This is the probability that \(S\) reach the value \(a\) before hitting \(-b\).

    • Show that \(M=S^2_t-t\) is a martingale and deduce that

      \[E\left[ \tau \right]=ab\]

  3. Suppose that \(p> 1/2\). Show that that \(M_t:=(q/p)^{S_t}\) is a martingale and deduce that

    \[P[S_{\tau}=a]=\frac{(q/p)^b-1}{(q/p)^{a+b}-1}\]

  4. Suppose that \(p>1/2\) and show that

    \[E[\tau]=\frac{a+b}{p-q}\frac{(q/p)^{b}-1}{(q/p)^{a+b}-1}-\frac{b}{p-q}\]

Exercise 7

Let \((Y_t)\) be a sequence of independent identically distributed random variables such that \(E[Y_t]=0\) for every \(t\) and not identically constant on some probability space \((\Omega,\mathcal{F}, P)\). We consider the filtration \(\mathcal{F}_0=\{\emptyset,\Omega\}\) and \(\mathcal{F}_t=\sigma(Y_1,\ldots,Y_t)\) and process

\[X_0:=1,\quad X_t:=X_0+\sum_{s=1}^t Y_s.\]

We interpret this process as a stock price which is fair in the sense that it is a martingale and therefore does not brings any gain or loss in expectation. And for every strategy \(H\), that is predictable process, the investment gain \(H\bullet X_T\) at time \(T\) does not brings in average more than \(H_0X_0\) due to Doob's optional sampling theorem. We extend the filtration with the information provided by \(X_T\), that is

\[\tilde{\mathcal{F}}_t=\sigma(\mathcal{F}_t,X_T), \quad t=0,\ldots,T\]

This can be interpreted as the information of an insider knowing for whatever reason the terminal value of the price at time \(T\). We denote the non-insider filtration \(\mathbb{F}\) and the insider filtration \(\tilde{\mathbb{F}}\). Show that

  1. \(X\) is a martingale under the filtration \(\mathbb{F}\). Show that \(X\) can not be a martingale under the insider filtration \(\tilde{\mathbb{F}}\). However, the process

    \[\tilde{X}_t=X_t-\sum_{s=0}^{t-1}\frac{X_T-X_s}{T-s}, \quad t=0,\ldots, T\]

    is a martingale under \(\tilde{\mathbb{F}}\).

  2. With the information about the terminal value \(X_T\) of the stock, it is possible to realize arbitrage gains. It means that you can find a predictable process but with respect to \(\tilde{\mathbb{F}}\) such that starting with \(0\) money, that is \(H_0=0\), you end up with positive gains and even strict gains with strict positive probability. That is

    \[P\left[ H\bullet X_T \geq 0 \right]=1\quad\text{and}\quad P\left[ H\bullet X_T> 0 \right]>0\]

    Find the best "insider strategy" -- that is \(\tilde{\mathbb{F}}\)-predictable process \(H\) with \(H_0=0\) -- that brings the maximum of gains among the insider strategies such that \(|H_s|\leq 1\) for every \(s=1,\ldots,T\).

Exercise 8

Let \((Y_t^n)\) be a family of iid random variables taking value in \(\{0, 1, \ldots\}\). We denote by \(p_i = P[Y_1^1=i]\), for \(i \in \{0,1,\ldots\}\) and \(\mu=\sum p_i=E[Y_1^1]\) and suppose that \(0<\mu<\infty\). Let \(\mathcal{F}_0=\{\emptyset,\Omega\}\) and \(\mathcal{F}_t=\sigma(Y^k_s\colon k \in \mathbb{N}, 1\leq s\leq t)\). Define

\[ X_0=x \quad\text{and}\quad X_{t+1}= \begin{cases} Y_{t+1}^1+\ldots+Y_{t+1}^{X_t} &\text{if }X_t>0\\ \\ 0 &\text{otherwise} \end{cases} \]

This process describes the evolution of a population where each person \(k=1,\ldots X_t\) gives birth independently to \(Y^{k}_{t+1}\) number of descendant.

  1. Show that \(X_t/\mu^t\) is a martingale.
  2. Show that if \(\mu<1\), then \(X_t=0\) for all \(t\) sufficiently large, and \(X_t/\mu^t \to 0\).
  3. Show that if \(\mu=1\) and \(P[Y^1_1=1]<1\), then \(X_t=0\) for all \(t\) sufficiently large. (Hint, then \(X\) is a positive martingale so that you can use martingale convergence theorems).