Question

Let \(\left\\{X_{k}\right\\}\) be a moving average process $$ X_{n}=\sum_{j=0}^{\infty} \alpha_{j} \xi_{n-j}, \quad \alpha_{0}=1, \quad \sum_{=0}^{\infty} \alpha_{j}^{2}<\infty $$ where \(\left\\{\xi_{n}\right\\}\) are zero-mean independent random variables having common variance \(\sigma^{2}\). Show that $$ U_{n}=\sum_{k=0}^{n} X_{k-1} \bar{\xi}_{k}, \quad n=0,1, \ldots $$ and $$ V_{n}=\sum_{k=0}^{n} X_{k} \xi_{h}-(n+1) \sigma^{2}, \quad n=0,1, \ldots $$ are martingales with respect to \(\left\\{\zeta_{n}\right\\}\).

Short answer

To show that \(U_n\) and \(V_n\) are martingales with respect to \(\left\\{\zeta_{n}\right\\}\), we need to verify three conditions for each: finiteness of expectations, \(\zeta_n\)-measurability, and conditional expectations equal to the current value. For \(U_n = \sum_{k=0}^{n} X_{k-1} \bar{\xi}_{k}\), expectations are finite, it is \(\zeta_n\)-measurable, and \(\mathbb{E}[U_{n+1} | \zeta_n] = U_n\), since \(X_n\) and \(\xi_{n+1}\) are independent. Thus, \(U_n\) is a martingale. Similarly for \(V_n = \sum_{k=0}^{n} X_{k} \xi_{h} - (n+1) \sigma^2\), expectations are finite, it is \(\zeta_n\)-measurable, and \(\mathbb{E}[V_{n+1} | \zeta_n] = V_n\), since \(X_{n+1}\) and \(\xi_{h}\) are independent. Hence, \(V_n\) is also a martingale.

Step-by-step solution

Analyzing the Moving Average Process

Given the moving average process, \(X_n = \sum_{j=0}^{\infty} \alpha_j \xi_{n-j}\), where \(\alpha_0 = 1\) and \(\sum_{=0}^{\infty} \alpha_{j}^2 < \infty\), with \(\left\\{\xi_{n}\right\\}\) being zero-mean independent random variables having common variance \(\sigma^2\).

Defining Martingales

A sequence of random variables \(\left\\{Y_n\right\\}\) is a martingale with respect to a sequence of sigma algebras \(\left\\{ \zeta_n \right\\}\) if for all \(n\), 1. \( \mathbb{E}[|Y_n|] < \infty \) 2. \(Y_n\) is \(\zeta_n\)-measurable 3. \(\mathbb{E}[Y_{n+1} | \zeta_n] = Y_n\) Our goal is to show that both \(U_n\) and \(V_n\) fulfill these conditions.

Analyzing \(U_n\)

Given \(U_n = \sum_{k=0}^{n} X_{k-1} \bar{\xi}_{k}\) for \(n = 0,1, \ldots\) 1. The expectation of \(U_n\) is finite since \(\left\\{X_{k}\right\\}\) and \(\left\\{\xi_{n}\right\\}\) are finite. 2. \(U_n\) is \(\zeta_n\)-measurable since the expression of \(U_n\) only involves random variables up to the index \(n\). 3. To show \(\mathbb{E}[U_{n+1} | \zeta_n] = U_n\), we have, \[\mathbb{E}[U_{n+1} | \zeta_n] =\mathbb{E}\left[\sum_{k=0}^{n+1} X_{k-1} \bar{\xi}_{k} | \zeta_n\right] = \mathbb{E}[U_n + X_n \bar{\xi}_{n+1} | \zeta_n] = U_n + \mathbb{E}[X_n \bar{\xi}_{n+1} | \zeta_n] = U_n\] Since \(X_n\) and \(\xi_{n+1}\) are independent, we have \(\mathbb{E}[X_n \bar{\xi}_{n+1} | \zeta_n] = 0\). Therefore, \(U_n\) is a martingale with respect to \(\left\\{\zeta_{n}\right\\}\).

Analyzing \(V_n\)

Given \(V_n = \sum_{k=0}^{n} X_{k} \xi_{h} - (n+1) \sigma^2\) for \(n = 0,1, \ldots\) 1. The expectation of \(V_n\) is finite since \(\left\\{X_{k}\right\\}\) and \(\left\\{\xi_{n}\right\\}\) are finite. 2. \(V_n\) is \(\zeta_n\)-measurable since the expression of \(V_n\) only involves random variables up to the index \(n\). 3. To show \(\mathbb{E}[V_{n+1} | \zeta_n] = V_n\), we have, \[\mathbb{E}[V_{n+1} | \zeta_n] = \mathbb{E}\left[\sum_{k=0}^{n+1} X_{k} \xi_{h} - (n+2) \sigma^2 | \zeta_n\right] = \mathbb{E}\left[(V_n + \sigma^2) +(X_{n+1} \xi_{h} - \sigma^2) | \zeta_n\right] = V_n + \mathbb{E}[X_{n+1} \xi_{h} - \sigma^2 | \zeta_n] = V_n\] Since \(X_{n+1}\) and \(\xi_{h}\) are independent, we have \(\mathbb{E}[X_{n+1} \xi_{h} - \sigma^2 | \zeta_n] = 0\). Therefore, \(V_n\) is a martingale with respect to \(\left\\{\zeta_{n}\right\\}\).

Key concepts

Stochastic Processes

A stochastic process is a collection of random variables indexed by time or space, representing the evolution of some system of random values over time. It can be thought of as a random function, with the indexing parameter typically representing time. Examples include stock market movements, fluctuating interest rates, and the random walk of particles in physics.

Stochastic processes are particularly powerful tools in mathematics and statistics for modeling seemingly random phenomena that actually have underlying, but possibly unknown or partially known, probabilistic structures. Understanding stochastic processes enables us to predict future events, value financial derivatives, or assess risks in various scenarios.

The moving average process discussed in the exercise is a specific type of stochastic process often used to smooth out short-term fluctuations and highlight longer-term trends in time series data.

Moving Average Process

A moving average process is a sequence of random variables where each variable is defined as the sum of current and past random shock effects. Each shock effect is weighted by a coefficient, making the process resemble an average. In the context of time series analysis, the moving average model helps to understand and predict future values based on past observations.

The coefficients α_j play a critical role in the moving average process, determining the 'memory' of the process and how much past shocks influence current values. The condition Σα_j² < ∞ ensures that the series converges and the process is well-defined. In financial modeling, this process helps to filter out the 'noise' from the true 'signal' when forecasting time series data.

Random Variables

A random variable is a mathematical function that assigns a real number to each outcome in a sample space of a random experiment. It essentially quantifies the outcomes of randomness. For instance, when tossing a coin, one could assign the number 0 to tails and 1 to heads, forming a simple random variable.

Random variables are broadly classified into two types: discrete, which take on a countable number of distinct values, and continuous, which can assume any value within an interval or collection of intervals. The concept of random variables is fundamental in probability and statistics because it provides a mechanism for quantifying the stochastic nature of processes and for conducting statistical inference.

Sigma Algebras

A sigma algebra (or σ-algebra) is a mathematical structure that formalizes the concept of measurement in probability theory. It is a collection of subsets of a given set that includes the empty set, the set itself, and is closed under complementation and countable unions. This allows for a systematic approach to defining and working with events, especially when dealing with an infinite number of possible outcomes.

In the context of stochastic processes, a sigma algebra represents the information available at a given time, and its evolution over time is what enables us to define concepts like martingales. Martingales are stochastic processes that model fair games, and their conditional expectations with respect to the sigma algebra do not change over time; in other words, future values do not deviate from the current value on average, given the current information.