Question

2Let \(M=N-\Lambda\) be the counting process local martingale. (a) Show that \(\mathrm{E} N(t)=\mathrm{E} \Lambda(t)\) (hint: use the monotone convergence theorem). (b) If \(\mathrm{E} \Lambda(t)<\infty\), then show that \(M\) is a martingale by verifying the martingale conditions. (c) If \(\sup _{t} \mathrm{E} \Lambda(t)<\infty\), then show that \(M\) is a square integrable martingale.

Short answer

Question: Given a counting process local martingale \(M = N - \Lambda\), prove the three following statements. a) \(\mathrm{E}N(t) = \mathrm{E}\Lambda(t)\) using the monotone convergence theorem. b) If \(\mathrm{E} \Lambda(t) < \infty\), then M is a martingale. c) If \(\sup_{t} \mathrm{E} \Lambda(t) < \infty\), then M is a square-integrable martingale. Answer: a) To show that \(\mathrm{E}N(t) = \mathrm{E}\Lambda(t)\), we first define the sequences of random variables \((N_n)\) and \((\Lambda_n)\), where \(N_n = N(t_n)\) and \(\Lambda_n = \Lambda(t_n)\) for some increasing sequence of times \(t_n\). Since \(N(t)\) and \(\Lambda(t)\) are non-negative and increasing in \(t\), it is clear that \((N_n)\) and \((\Lambda_n)\) are non-negative and increasing sequences that converge almost surely to \(N(t)\) and \(\Lambda(t)\) respectively. Then, using the monotone convergence theorem, we have \(\mathrm{E} N(t) = \lim_{n\to\infty} \mathrm{E} N_n = \lim_{n\to\infty} \mathrm{E} \Lambda_n = \mathrm{E} \Lambda(t)\). b) To show that M is a martingale when \(\mathrm{E} \Lambda(t) < \infty\), we must verify the martingale conditions: (i) \(M(t)\) is adapted, (ii) \(M(t)\) is integrable, and (iii) \(M(t)\) has the martingale property. Since \(M = N - \Lambda\) is the counting process local martingale, conditions (i) and (ii) hold by definition. Moreover, we have \(\mathrm{E}[M(t) | \mathcal{F}_s] = N(s)-\mathrm{E}[\Lambda(t)-\Lambda(s)|\mathcal{F}_s] + \mathrm{E}[\Lambda(t)-\Lambda(s)] - \mathrm{E}[\Lambda(t) | \mathcal{F}_s] = M(s)\). Hence, all martingale conditions are satisfied, and \(M\) is a martingale. c) To show that \(M\) is a square-integrable martingale if \(\sup_{t} \mathrm{E} \Lambda(t) < \infty\), we must verify the following conditions: (i) \(M(t)\) is a martingale and (ii) \(\mathrm{E}[M^2(t)] < \infty\) for all \(t\). From step 2, we know that \(M(t)\) is a martingale. Additionally, we have \(\mathrm{E}[M^2(t)] = \mathrm{E}[(N(t) - \Lambda(t))^2] \leq \mathrm{E}[N^2(t)] + 2\mathrm{E}[N(t)\Lambda(t)] + \mathrm{E}[\Lambda^2(t)]\). Since \(\sup_{t} \mathrm{E} \Lambda(t) < \infty\) and \(N(t)\) is a counting process, the expectation of each term in the inequality is finite. Therefore, \(\mathrm{E}[M^2(t)] < \infty\) for all t. Hence, \(M\) is indeed a square-integrable martingale.

Step-by-step solution

Proving \(\mathrm{E}N(t) = \mathrm{E}\Lambda(t)\) using Monotone Convergence Theorem

Let's define the sequence of random variables \((N_n)\) and \((\Lambda_n)\), where \(N_n = N(t_n)\) and \(\Lambda_n = \Lambda(t_n)\) for some increasing sequence of times \(t_n\). The monotone convergence theorem states that if \((X_n)\) is a sequence of non-negative measurable functions that converges almost surely to a measurable function \(X\) and if \((X_n)\) increases almost surely to \(X\), then \(\lim_{n\to\infty}\mathrm{E} X_n = \mathrm{E} X\). Since \(N(t)\) and \(\Lambda(t)\) are non-negative and increasing in \(t\), it is clear that \((N_n)\) and \((\Lambda_n)\) are non-negative and increasing sequences that converge almost surely to \(N(t)\) and \(\Lambda(t)\) respectively. Then, using the monotone convergence theorem, we have that $$ \mathrm{E} N(t) = \lim_{n\to\infty} \mathrm{E} N_n, $$ and $$ \mathrm{E} \Lambda(t) = \lim_{n\to\infty} \mathrm{E} \Lambda_n. $$ Now, since \(M = N - \Lambda\), we have $$ N_n = M_n + \Lambda_n, $$ where \(M_n = M(t_n)\). Taking expectations, we get $$ \mathrm{E} N_n = \mathrm{E} (M_n+\Lambda_n) = \mathrm{E} M_n + \mathrm{E} \Lambda_n. $$ Note that \(M_n\) is a local martingale, so by its definition, \(\mathrm{E} M_n = \mathrm{E} M_0 = 0\) for all \(n\). Then, $$ \mathrm{E} N_n = \mathrm{E} \Lambda_n. $$ Therefore, by the monotone convergence theorem, we have $$ \mathrm{E} N(t) = \lim_{n\to\infty} \mathrm{E} N_n = \lim_{n\to\infty} \mathrm{E} \Lambda_n = \mathrm{E} \Lambda(t). $$

Verifying Martingale Conditions for M

To show that \(M\) is a martingale when \(\mathrm{E} \Lambda(t) < \infty\), we must verify the martingale conditions: (i) \(M(t)\) is adapted, i.e., \(M(t)\) is \(\mathcal{F}_t\)-measurable. (ii) \(M(t)\) is integrable, i.e., \(\mathrm{E}|M(t)| < \infty\). (iii) \(M(t)\) has the martingale property, i.e., \(\mathrm{E}[M(t) | \mathcal{F}_s] = M(s)\) for \(0 \leq s \leq t\). Since \(M = N - \Lambda\) is the counting process local martingale, conditions (i) and (ii) hold by definition. Now, to verify condition (iii), we have: $$ \mathrm{E}[M(t) | \mathcal{F}_s] = \mathrm{E}[N(t) - \Lambda(t) | \mathcal{F}_s] = \mathrm{E}[N(t) | \mathcal{F}_s] - \mathrm{E}[\Lambda(t) | \mathcal{F}_s] $$ Since \(N\) is a counting process, \(\mathrm{E}[N(t)|\mathcal{F}_s] = \mathrm{E}[N(t)-N(s)|\mathcal{F}_s] + N(s)\). Also, from step 1, we know that \(\mathrm{E}[N(t)-N(s)] = \mathrm{E}[\Lambda(t)-\Lambda(s)]\), so $$ \mathrm{E}[M(t) | \mathcal{F}_s] = N(s)-\mathrm{E}[\Lambda(t)-\Lambda(s)|\mathcal{F}_s] + \mathrm{E}[\Lambda(t)-\Lambda(s)] - \mathrm{E}[\Lambda(t) | \mathcal{F}_s]. $$ Since \(\mathrm{E}[\Lambda(t)-\Lambda(s)|\mathcal{F}_s]=\mathrm{E}[\Lambda(t)|\mathcal{F}_s]-\Lambda(s)\), the above expression simplifies to $$ \mathrm{E}[M(t) | \mathcal{F}_s] = N(s)-\Lambda(s) = M(s). $$ Hence, all martingale conditions are satisfied, and \(M\) is a martingale.

Verifying Square-integrable Martingale Conditions for M

To show that \(M\) is a square-integrable martingale when \(\sup_{t} \mathrm{E} \Lambda(t) < \infty\), we must verify the following conditions: (i) \(M(t)\) is a martingale. (ii) \(\mathrm{E}[M^2(t)] < \infty\) for all \(t\). From step 2, we already know that \(M(t)\) is a martingale. Now we need to verify the second condition. Note that \(M^2(t) = (N(t) - \Lambda(t))^2\). By the Cauchy-Schwarz inequality, we have that $$ \mathrm{E}[M^2(t)] = \mathrm{E}[(N(t) - \Lambda(t))^2] \leq \mathrm{E}[N^2(t)] + 2\mathrm{E}[N(t)\Lambda(t)] + \mathrm{E}[\Lambda^2(t)]. $$ Since \(\sup_{t} \mathrm{E} \Lambda(t) < \infty\) and \(N(t)\) is a counting process, the expectation of each term in the inequality is finite. Therefore, we have that $$ \mathrm{E}[M^2(t)] < \infty, $$ for all t. Therefore, \(M\) is indeed a square-integrable martingale.

Key concepts

Monotone Convergence Theorem

The Monotone Convergence Theorem is a fundamental result in measure theory which plays a critical role in problems involving infinite limits. Imagine you're stacking blocks, one on top of the other, each block representing the value of a non-negative random variable at a specific point in time. As time goes on, provided you keep adding blocks without end, the height of the stack grows without bounds. This theorem assures you that if you carefully measure the height of this ever-growing stack at each time interval, the expected height at infinity will match the height you'd predict by looking at how it's grown so far.

More formally, the theorem says if \(X_n\) is a sequence of non-negative, increasing random variables that converge towards \(X\), then the limit of the expected values of \(X_n\) as \(n\) approaches infinity is equal to the expected value of \(X\). In the case of our exercise, the stacking blocks can be seen as counts in a counting process or accumulated intensity in a random process, represented by \(N(t)\) and \(\Lambda(t)\) respectively.

When you apply this theory to statistics or probability, it becomes incredibly handy for dealing with complex stochastic processes and ensures that intuitive results hold even as our processes stretch towards the infinite.

Counting Process

A Counting Process can be visualized like a digital clicker that sports fans might use to count the number of spectators entering a stadium. It represents the total count that increases by one with each new arrival or event. In mathematics, we model events that occur randomly over time using this concept. When we talk about a counting process in the context of stochastic or random processes, we're looking at variables that represent the number of events that have occurred up to time \(t\).

A classical example is the number of customers arriving at a bank, the frequency of buses at a station, or the number of particles decaying in a unit of time in physics. The key thing to remember is that these can be unpredictable, yet their progression over time has a certain structure that we can analyze. Counting processes help us build models of these phenomena, often leading to insights into their underlying patterns and probabilities of future events.

Local Martingale

A Local Martingale, a concept from the mathematical field of stochastic processes, is a bit like a tightrope walker who, despite wobbling, has an uncanny ability to keep their average position over time. Whilst the martingale represents a fair game, such as flipping a fair coin, a local martingale is similar but it allows for some temporary deviation from fairness under certain conditions.

In rigorous terms, a local martingale is a process that, on a local level, seems to possess the martingale property — meaning its conditional expectation, given the past, is equal to its present value. However, the catch is that it doesn't have to maintain this property in the long run or under all circumstances. Mathematically, there exists a sequence of stopping times with certain properties that when applied to the process turn it into a bona fide martingale. This is like saying our tightrope walker may need to catch their balance occasionally, relying on a safety net (the stopping times), to maintain the act of a fair balance over time.

Square-Integrable Martingale

A Square-Integrable Martingale represents a type of fair game in gambling or decision-making that not only maintains its fairness over time but also assures that the average of the squares of the outcomes remains finite. This condition is essential in the mathematical world because it guarantees that the game doesn't get out of hand — for example, the bets don't grow so large that they're no longer manageable.

In technical terms, a square-integrable martingale is a martingale process \(M(t)\) where not only does the expected value of \(M(t)\) given the past up to time \(s\) equal \(M(s)\), but also the expected value of \(M(t)^2\) is finite for all time points \(t\). This puts a cap on the variability of the martingale, ensuring it won't behave too wildly. In practice, this means that our mathematical model — the martingale — remains useful and meaningful, allowing us to make accurate predictions and analyses. In the context of our example, this condition ensures that the counting process martingale remains controlled, predictable, and does not accumulate infinite variance over time.