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Doob's Forward Convergence Theorem

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Definition
StochasticDiffsStochasticProcesses

Let (Xn)n∈N(X_{n})_{n\in\mathbb{N}} be a (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}}-supermartingale and assume (Xn)n∈N(X_{n})_{n\in\mathbb{N}} is bounded in L1L^{1} (i.e. sup⁡n∈NE[∣Xn∣]<∞\sup_{n\in\mathbb{N}}E[|X_{n}|]<\infty). Then ∃l∈L1(Ω,F,P)\exists \mathscr{l}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P) such that Xn→lX_{n}\to \mathscr{l} a.s. as n→∞n\to\infty.

Let (Xn)n∈N(X_{n})_{n\in\mathbb{N}} be (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}}-martingale such that Xn≥0X_{n}\ge 0 ∀n∈N\forall n\in\mathbb{N}. Then ∃l∈L1(Ω,F,P)\exists \mathscr{l}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P) such that Xn→lX_{n}\to \mathscr{l} a.s. as n→∞n\to\infty.