Question

Prove: if \(\left\\{X_{n}\right\\}\) is a submartingale and \(\varphi(x)\) is a convex, increasing function. then \(\left\\{\varphi\left(X_{n}\right)\right\\}\) is a submartingale whenever \(\boldsymbol{E}\left|\varphi^{+}\left(\boldsymbol{X}_{n}\right)\right|<\infty\) for all \(n\) (cf. Lemma 2.2).

Short answer

Given that \(\{X_n\}\) is a submartingale and \(\varphi(x)\) is a convex, increasing function, we have used Jensen's inequality to obtain the inequality \(\mathbb{E}[\varphi(X_{n+1}) | X_1, ..., X_n] \geq \varphi(X_n)\). With the given condition, \(\mathbb{E}\left|\varphi^+\left(X_{n}\right)\right| < \infty\) for all \(n\), we can conclude that \(\{\varphi(X_n)\}\) is a submartingale.

Step-by-step solution

Recall the definition of a submartingale

A sequence of random variables \(\{X_n\}\) is a submartingale if for every \(n \in \mathbb{N}\), $$ \mathbb{E}[X_{n+1} | X_1, ..., X_n] \geq X_n. $$ Our goal is to show that \(\{\varphi(X_n)\}\) satisfies the same property.

Use Jensen's inequality

Since \(\varphi(x)\) is convex and increasing, we can apply Jensen's inequality. For any \(n \in \mathbb{N}\), we have $$ \mathbb{E}[\varphi(X_{n+1}) | X_1, ..., X_n] \geq \varphi(\mathbb{E}[X_{n+1}| X_1, ..., X_n]). $$

Apply the submartingale property

As \(\{X_n\}\) is a submartingale, we have $$ \mathbb{E}[X_{n+1} | X_1, ..., X_n] \geq X_n. $$ Combining this with the result from Step 2 using the increasing property of \(\varphi\), we get $$ \mathbb{E}[\varphi(X_{n+1}) | X_1, ..., X_n] \geq \varphi(X_n). $$

Check the condition on the expected value of the positive part

We need to verify that \(\mathbb{E}\left|\varphi^+\left(X_{n}\right)\right| < \infty\) for all \(n\). As \(\varphi(x)\) is increasing, its positive part \(\varphi^+(x)\) is also increasing. So, for any non-negative random variable \(X_n\), the positive part of \(\varphi\left(X_{n}\right)\), \(\varphi^+(X_n)\), will also be non-negative. We are given that \(\mathbb{E}\left|\varphi^+\left(X_{n}\right)\right| < \infty\) for all \(n\), which means that \(\mathbb{E}[\varphi^+(X_n)] < \infty\) for all \(n\). Since \(\varphi^+(X_n)\) is non-negative, this implies that \(\mathbb{E}[\varphi(X_n)] < \infty\) for all \(n\).

Conclude that \(\{\varphi(X_n)\}\) is a submartingale

Since the positive part of \(\varphi\left(X_{n}\right)\) is non-negative and has a finite expected value, and the property $$ \mathbb{E}[\varphi(X_{n+1}) | X_1, ..., X_n] \geq \varphi(X_n) $$ holds for all \(n\), we can conclude that \(\{\varphi(X_n)\}\) is a submartingale.

Key concepts

Jensen's Inequality

At the heart of our submartingale problem lies Jensen's Inequality, a fundamental concept in probability and statistics. This inequality tells us about the behavior of a convex function when applied to expectations.

Specifically, if you have a convex function, \( \phi(x) \), and a random variable, \( X \), Jensen's Inequality guarantees that the expected value of \( \phi(X) \) is greater than or equal to \( \phi(\text{E}[X]) \). In mathematical terms:
\[ \mathbb{E}[\phi(X)] \geq \phi(\mathbb{E}[X]) \].

This is crucial when dealing with submartingales because it allows us to compare the expected value of the transformed sequence with the original sequence. By ensuring that the function \( \phi \) does not 'reverse' the inequality present in the submartingale definition, we can retain the submartingale property even after the transformation.

Convex Function

Understanding what makes a function convex is critical for applying Jensen's inequality. A function \( f(x) \) is considered convex if, for any two points on the graph of \( f \) and any \( \lambda \) such that \( 0 \leq \lambda \leq 1 \), the following inequality holds:
\[ f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda) f(y) \].

Visually, this means that if you take any two points on the graph of \( f \), the line segment connecting them lies above or on the graph. In the context of our submartingale problem, we leverage the property that a convex function \( \phi \) turns line segments into arcs that do not fall below the line—this is what allows us to use Jensen's Inequality and maintain the condition needed for a submartingale.

Expected Value

The expected value, often denoted as \( \mathbb{E}[X] \), is the probability-weighted average of all possible values for a random variable \( X \). It's a measure of the 'center' of the random variable's distribution, or intuitively, what you would expect on average if you could repeat an experiment an infinite number of times.

In submartingales, the expected value plays a central role. By definition, for a sequence of random variables to be a submartingale, the expected value of the next variable in the sequence given all previous values must be at least as large as the present value. This implies a sort of 'increasing trend' on average, which is formalized as:
\[ \mathbb{E}[X_{n+1} | X_1, ..., X_n] \geq X_n \].

When assessing if a transformed process via a convex function retains the submartingale property, we examine if this inequality remains true under the transformation. Hence, the expected value provides a convenient and insightful comparison point.

Stochastic Processes

A stochastic process is a sequence of random variables representing a system evolving over time. These variables can represent anything from stock prices to weather patterns, as long as there's inherent randomness in their evolution.

Submartingales are a special kind of stochastic process. They are characterized by the property that, given the present state, the expected future value is not less than the current value. It reflects a tendency for the process to 'drift' upwards over time.

The concept of a submartingale is deeply connected to the expectations and the conditions a random process must satisfy. To determine whether a transformed process \( \{\phi(X_n)\} \) is a submartingale, we ensure that each term, when given previous information, carries the expectation of being larger or equal to the current term, maintaining the essential characteristic of a submartingale even under transformation.