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Let ${\tau }_n\uparrow \infty$ be a localizing sequence for the discrete-time local martingale $X$. For $t\ge 1$, the increment of the stopped martingale isand $\{{\tau }_n\ge t\}\in {\mathcal{F}}_{t-1}$. Thus, for any $A\in {\mathcal{F}}_{t-1}$,The assumed integrability gives the dominating random variable $|X_t|+|X_{t-1}|$. The dominated convergence theorem now yields $\mathbb{E}[1_A(X_t-X_{t-1})]=0$, or $\mathbb{E}[X_t\mid {\mathcal{F}}_{t-1}]=X_{t-1}$. Iterating the tower property of conditional expectation proves an integrable discrete-time local martingale is a true martingale. The proof uses the discrete-time increment indicator; the conclusion does not extend to continuous time.

Under the standard local martingale definition, $X_0$ is integrable. For a nonnegative $X$, the Fatou lemma and a localizing sequence giveHence every $X_t$ is integrable, and part (a) applies. Therefore a nonnegative discrete-time local martingale is a true martingale. This is stronger than the continuous-time supermartingale conclusion.

Take an increasing localizing sequence ${\tau }_n$ for $X$ and define the stopping timesThe predictable process condition $K_{u+1}\in {\mathcal{F}}_u$ makes $R_n$ a stopping time. For each fixed finite time range, the finitely many coefficients $K_1,\dots ,K_t$ are finite almost surely; hence $T_n\uparrow \infty$ almost surely.

The stopped martingale transform can be writtenIts coefficient is ${\mathcal{F}}_{s-1}$-measurable and bounded in absolute value by $n$. Each summand is consequently integrable with zero conditional expectation given ${\mathcal{F}}_{s-1}$, since $X^{{\tau }_n}$ is a true martingale. Thus $M^{T_n}$ is a true martingale andThe original PDF fixes $M_0=0$; the TeX extraction omits this equality.

Let ${\theta }_t$ and ${\eta }_t$ be the predictable holdings of the stock and the bank account. With no consumption, the self-financing portfolio has $X=\theta S+\eta B$ and $dX=\theta dS+\eta dB$. The Itô product rule, together with the dynamics from part (a), givesThe drift vanishes because $X=\theta S+\eta B$ and $\sigma \lambda =\mu -r$. Thus $XY$ is a local martingale. It is nonnegative by the assumed nonnegative wealth and strict positivity of $Y$, and henceFor example, this last fact follows directly from Conditional Fatou lemma applied to a localizing sequence; the initial capital is assumed finite. As usual, holdings must be integrable against the asset semimartingales for the self-financing portfolio equation to be defined.

Let $({\tau }_n)$ be a localizing sequence for $M$. For $s\le t$, the optional sampling theorem givesBoth sides converge almost surely to $M_t$ and $M_s$, and $|M_{u\wedge {\tau }_n}|\le Z$. Conditional dominated convergence therefore yields $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$, so $M$ is a martingale. Moreover, the family $(M_t{)}_{t\ge 0}$ is dominated by the integrable random variable $Z$, hence is uniformly integrable. Thus $M$ is a uniformly integrable martingale.