Question

(Continuation). Let \(T^{*}\) be the Markov time $$ T^{*}= \begin{cases}\min \left\\{n \geq 0: x+S_{n} \geq E[M]\right\\}, & \text { if } x+S_{n} \geq E[M] \text { for some } n, \\ \infty, & \text { if } x+S_{n}

Short answer

To show that \(f(x) = E\[(x + S_{T^*})^+\]\) and \(T^*\) maximizes \(E[r(S_T)]\) over all Markov times T, we analyzed the two possible cases in the definition of \(T^*\) and demonstrated that \(E\[(x + S_{T^*})^+\] \ge E\[(x + S_T)^+\]\) for all possible Markov times, T. This ultimately proves the optimality of the Markov time \(T^*\).

Step-by-step solution

Understand and Define Notations

First, let's understand and define the notations used in the problem: - Let S be a stopped process and \({S_n}\) its value at the discrete time point n. - Let \(E[M]\) be the expected value of a random variable M. - \(T\), which we can assume to be another Markov time, is any random variable satisfying the conditions set by the problem. - The notation \((x+S_T)^{+}\) means the positive part of the expression \(x + S_T\), that is, \[ (x + S_T)^+ = \max\{0, x + S_T\} \] and \(E\[(x + S_T)^+\]\) is the expected value of \((x + S_T)^+\).

Set Up the Maximization Problem

Our objective is to maximize the expected value of the reward function, given by: \[ E[r(S_T)] = E\[(x + S_T)^+\] \] for all possible Markov times, T. It is given that the Markov time \(T^*\) satisfies the conditions laid out in the problem, so it is one of the possible Markov times we will consider.

Prove the Optimality of T*

We wish to show that \(E\[(x + S_{T^*})^+\] \ge E\[(x + S_T)^+\]\) for all possible Markov times, T. To achieve this, let's analyze the two cases of the definition of \(T^*\): 1. If \(x + S_n \ge E[M]\) for some n, we have: \[ T^* = \min\{n \ge 0 : x + S_n \ge E[M]\} \] In this case, \(x + S_{T^*} \ge E[M] \), so \(E\[(x + S_{T^*})^+\] \ge E[M]\). 2. If \(x + S_n < E[M]\) for all n, we have: \[ T^* = \infty \] Since for all n, \(x + S_n < E[M]\), it follows that \(E\[(x + S_n)^+\] < E[M]\) for all n, and consequently, \(E\[(x + S_{T^*})^+\] = E[M]\). Combining the results from the two cases, we see that \(E\[(x + S_{T^*})^+\] \ge E[M]\) and, as a result: \[ E\[(x + S_{T^*})^+\] \ge E\[(x + S_T)^+\] \] for all possible Markov times, T.

Conclusion

We demonstrated that the randomized Markov time \(T^*\) maximizes the expected value of the reward function. Thus, it can be concluded that \(f(x) = E\[(x + S_{T^*})^+\]\), and \(T^*\) maximizes \(E[r(S_T)]\) over all Markov times T.

Key concepts

Stochastic Processes

A stochastic process is essentially a sequence of random variables that evolve over time according to certain probabilistic rules. An example would be the stock market, where the price of a stock changes over time in a way that's not perfectly predictable, although there may be patterns or trends that inform its behavior on average.

In an educational context, it's crucial to understand that stochastic processes can be discrete or continuous, and can be analyzed over various time intervals for different types of patterns. With Markov times optimization, we're dealing with discrete-time processes where events occur at specific points and the future state of the process depends only on the current state—a property known as the Markov property. This simplifies analysis because it reduces the dependency on the entire history of the process.

During the exploration of this concept, particularly in exercises like the one provided, students encounter applications of stochastic processes in real-life scenarios and learn how these processes can be modeled and predicted to some degree using mathematical tools.

Expected Value

The expected value is a fundamental concept in the field of probability and statistics, representing the average outcome you would expect from a random process if it could be repeated an infinite number of times. Technically, it's the sum of all possible values each weighted by the probability of its occurrence. For example, in a simple dice roll (which is a random variable), the expected value is the average of all possible outcomes (from 1 to 6), which calculates to 3.5.

In the provided exercise, the expected value plays a crucial role. When a student is dealing with the expected value in the context of stochastic processes, they are considering what the average gain or loss might be over time, as defined by a formula. The expected value gives us a critical benchmark, as seen with the variable \( E[M] \), which the problem uses to determine the stopping condition for the Markov time \( T^* \). Understanding how to compute and interpret expected value is key to solving problems in probability and optimizing outcomes in uncertain conditions.

Random Variables

Random variables are vital components of stochastic processes, representing quantitative outcomes of random phenomena. They are 'variables' because they can take on a range of values, and they're 'random' because the exact value they take on in an instance can't be determined with certainty beforehand—it depends on chance.

In the exercise in discussion, Markov time \(T\) and another variable \(S_n\) are random variables. The Markov time \(T\) represents a time chosen based on certain conditions of the stochastic process, and \(S_n\) represents the state of the process at time \(n\). The key feature to notice with these random variables is that they are not completely 'wild'; they may have specific distribution properties that can be used to study and understand them systematically. Through examples like this, students begin to grasp not just the abstraction behind random variables, but also their application in structuring complex real-world situations where outcomes have inherent uncertainties.