Let be a martingale with , where is a constant. Prove that can be expressed as the difference of two nonnegative martingales.

Stochastic Processes Solution – Problem 7: Prove that can be expressed as the difference of two nonnegative martingales

Question

Let $\{Y_{n},\mathcal{F}_{n},n\geq1\}$ be a martingale with $E|Y_{n}|\leq K,n\geq1$ , where $K$ is a constant. Prove that $Y$ can be expressed as the difference of two nonnegative martingales.

Step-by-step solution

Step 1. Define $M_n = E[Y_\infty^+ \mid \mathcal{F}_n], \quad N_n = E[Y_\infty^- \mid \mathcal{F}_n]$ where $Y_\infty$ is the $L^1$ limit of $Y_n$ as $n \to \infty$ (by the $L^1$ -bounded martingale convergence theorem, the martingale converges to some $Y_\infty \in L^1$ ).

Step 2. Since $\sup_n E|Y_n| \le K < \infty$ , by Doob's martingale convergence theorem, there exists a random variable $Y_\infty \in L^1$ such that $Y_n \to Y_\infty$ a.s. and in $L^1$ . Moreover, $Y_n = E[Y_\infty \mid \mathcal{F}_n]$ .

Step 3. Let $M_n = E[Y_\infty^+ \mid \mathcal{F}_n], \quad N_n = E[Y_\infty^- \mid \mathcal{F}_n].$ Clearly $M_n, N_n$ are nonnegative martingales. Furthermore, $Y_n = E[Y_\infty \mid \mathcal{F}_n] = E[Y_\infty^+ - Y_\infty^- \mid \mathcal{F}_n] = M_n - N_n.$

Step 4. Verification: $M_n, N_n$ are martingales since they are sequences of conditional expectations. Nonnegativity follows from $Y_\infty^+ \ge 0$ and $Y_\infty^- \ge 0$ , giving $M_n \ge 0$ and $N_n \ge 0$ . Their difference equals $Y_n$ .

Thus $Y_n = M_n - N_n$ is the difference of two nonnegative martingales.

Final answer

QED.

Marking scheme

This rubric is designed strictly according to the official solution (assuming the closed martingale/UI property) and provides a parallel grading chain for the general Krickeberg decomposition method that does not rely on the UI assumption.

1. Checkpoints (max 7 pts total)

Score exactly one chain; take the maximum subtotal among chains; do not add points across chains.

Chain A: Official Solution Path (Based on the closed martingale property of $Y_\infty$ )

Existence of the limit: Cite Doob's martingale convergence theorem or use the $L^1$ -bounded condition ( $E|Y_n| \le K$ ) to state that $Y_n$ converges to a random variable $Y_\infty$ (a.s. or $L^1$ ). [1 pt]
Closed martingale representation (core step): Write or assert that $Y_n$ can be represented as the conditional expectation of $Y_\infty$ , i.e., $Y_n = E[Y_\infty \mid \mathcal{F}_n]$ . [2 pts]
Construction of nonnegative components: Using the positive and negative parts of $Y_\infty$ , define $M_n = E[Y_\infty^+ \mid \mathcal{F}_n]$ and $N_n = E[Y_\infty^- \mid \mathcal{F}_n]$ . [2 pts]
Verification and conclusion: Verify the following properties and draw the conclusion: (1) $M_n, N_n$ are martingales (by properties of conditional expectation); (2) $M_n, N_n \ge 0$ (from $Y_\infty^\pm \ge 0$ ); (3) $Y_n = M_n - N_n$ . [2 pts, additive]

Chain B: General Krickeberg Decomposition (Based on the limit construction from $Y_n^+$ )

Submartingale property: State that $Y_n^+$ is a submartingale. [1 pt]
Construction of the dominating martingale (core step): Define $M_n = \lim_{k \to \infty} E[Y_{n+k}^+ \mid \mathcal{F}_n]$ (or take $\sup_k$ ), and state that this limit exists due to $L^1$ boundedness. [3 pts]
Verification of $M_n$ : Show that the constructed $M_n$ is a martingale. [1 pt]
Construction of $N_n$ and nonnegativity: Define $N_n = M_n - Y_n$ , use $M_n \ge Y_n^+$ (or the submartingale property) to prove $N_n \ge 0$ and that it is a martingale, thereby completing the decomposition. [2 pts]

Total (max 7)

2. Zero-credit items

Simple algebraic decomposition: Only writing $Y_n = Y_n^+ - Y_n^-$ and claiming $Y_n^+, Y_n^-$ are martingales (in fact they are typically submartingales), receives 0 pts.
Unsupported assertion: Only restating the problem conditions or claiming "by the Krickeberg decomposition theorem the conclusion holds" without any concrete construction process, receives 0 pts.
Symbol accumulation: Listing general properties of $E[Y|\mathcal{F}_n]$ without applying them to the specific $Y_\infty$ or construction process in this problem, receives 0 pts.

3. Deductions

Logic gap (Limit Gap): If using Chain A but not mentioning the convergence of $Y_n$ or Doob's theorem before using $Y_\infty$ , deduct 1 pt.
Notational error: Confusing conditional expectation $E[\cdot \mid \mathcal{F}_n]$ with full expectation $E[\cdot]$ , or omitting the $\sigma$ -algebra $\mathcal{F}_n$ , deduct 1 pt.
Incomplete process: Completing the definition of $M_n, N_n$ but entirely failing to mention/verify either their "nonnegativity" or "martingale property," deduct 1 pt.

Stochastic Processes – Problem 7: Prove that can be expressed as the difference of two nonnegative martingales