1. Definition of the critical event

Consider the event \( A = \{\max_{1 \leq i \leq n} X_i \geq C\} \), which signifies that \( X_t \) reaches or exceeds \( C \) at least once between \( t = 1 \) and \( t = n \). The goal is to bound \( P(A) \).

2. Transformation into an indicator variable

Introduce the indicator variable \( \mathbf{1}_A \), which equals 1 if the event \( A \) occurs and 0 otherwise:

\[ \mathbf{1}_A = \begin{cases} 1, & \text{if } \max_{1 \leq i \leq n} X_i \geq C, \\ 0, & \text{otherwise.} \end{cases} \]

By definition, \( P(A) = \mathbb{E}[\mathbf{1}_A] \). Bounding \( P(A) \) is equivalent to bounding the expectation of \( \mathbf{1}_A \).

3. Modified process construction

Construct a modified version of the process \( X_t \), denoted \( Y_t \), defined as:

\[ Y_t = \begin{cases} C, & \text{if } \max_{1 \leq i \leq t} X_i \geq C, \\ X_t, & \text{otherwise.} \end{cases} \]

The process \( Y_t \) replaces \( X_t \) with \( C \) as soon as \( X_t \) reaches or exceeds \( C \).

4. Properties of \( Y_t \)

By construction, \( Y_t \) is a submartingale, like \( X_t \), since replacing values by \( C \) does not decrease the conditional expectation. Formally:

\[ \mathbb{E}[Y_{t+1} \mid \mathcal{F}_t] \geq Y_t. \]

5. Final expectation of \( Y_t \)

Since \( Y_t \) is identical to \( X_t \) as long as \( \max_{1 \leq i \leq t} X_i < C \) and equals \( C \) otherwise, the following holds for any \( t \):

\[ Y_t \leq C \cdot \mathbf{1}_A + X_t \cdot (1 - \mathbf{1}_A). \]

In particular, for \( t = n \), we obtain:

\[ \mathbb{E}[Y_n] \leq \mathbb{E}[C \cdot \mathbf{1}_A + X_n \cdot (1 - \mathbf{1}_A)]. \]

6. Expanding the inequality

Using the linearity of expectation:

\[ \mathbb{E}[Y_n] \leq C \cdot \mathbb{E}[\mathbf{1}_A] + \mathbb{E}[X_n] - \mathbb{E}[X_n \cdot \mathbf{1}_A]. \]

Since \( \mathbb{E}[\mathbf{1}_A] = P(A) \), this becomes:

\[ \mathbb{E}[Y_n] \leq C \cdot P(A) + \mathbb{E}[X_n] - \mathbb{E}[X_n \cdot \mathbf{1}_A]. \]

7. Bounding \( Y_t \)

By definition, \( Y_t \leq C \), and hence:

\[ \mathbb{E}[Y_n] \leq \mathbb{E}[X_n^+]. \]

Combining this with the previous inequality:

\[ \mathbb{E}[X_n^+] \geq C \cdot P(A). \]

Rearranging terms gives the desired result:

\[ P(A) \leq \frac{\mathbb{E}[X_n^+]}{C}. \]

The inequality demonstrates that the probability of \( X_t \) reaching or exceeding \( C \) is constrained by the expected maximum positive value of \( X_n \).

The demonstration of Doob's inequality hinges on a key idea: linking the probability that a submartingale \( X_t \) exceeds a given threshold \( C \) to its final expectation. This is achieved through the construction of a modified process \( Y_t \), which mimics \( X_t \) until \( C \) is reached and then remains fixed at \( C \). The analysis begins by defining the event \( A \), where \( X_t \) surpasses \( C \), and expressing its probability \( P(A) \) in terms of an indicator variable \( \mathbf{1}_A \). This reformulation allows for mathematical manipulation. A modified process \( Y_t \) is then introduced: it mirrors \( X_t \) as long as \( X_t < C \), but once \( C \) is reached, \( Y_t \) is set to \( C \).

This construction simplifies the analysis while preserving the submartingale property of the original process. The process \( Y_t \) is shown to remain controlled in expectation, and its final expectation \( \mathbb{E}[Y_n] \) provides an upper bound for the extreme values of \( X_t \). By combining these properties, the probability \( P(A) \), representing extreme excursions, is bounded by \( \mathbb{E}[X_n^+] / C \). In summary, the demonstration highlights that while a submartingale can grow and fluctuate, the probability of reaching high values is fundamentally tied to its overall characteristics, particularly its final expectation. This relationship ensures that extreme events remain rare and governed by the process's average behavior.