Square-integrable martingales

TheoremConditional variance theorem for martingales#✓fully proved

Let \((\Omega, \mu)\) be a probability space with a filtration \((\mathcal{F}_n)_{n \in \mathbb{N}}\), and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued martingale with respect to \((\mathcal{F}_n)\) and \(\mu\) such that \(X_n\) is square-integrable for every \(n\). Then for all \(m \le n\), almost surely

\[ \mathbb{E}[(X_n - X_m)^2 \mid \mathcal{F}_m] = \mathbb{E}[X_n^2 \mid \mathcal{F}_m] - X_m^2. \]

This expresses the conditional variance of the increment \(X_n - X_m\) in terms of the conditional second moment of \(X_n\).

theorem Martingale.condExp_sub_sq_eq
    {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hmart : Martingale X ℱ μ)
    (hL2 : ∀ n, MemLp (X n) 2 μ)
    {m n : ℕ} (hmn : m ≤ n) :
    μ[fun ω => (X n ω - X m ω) ^ 2 | ℱ m] =ᵐ[μ]
      fun ω => (μ[fun ω' => (X n ω') ^ 2 | ℱ m]) ω - (X m ω) ^ 2

Sources

• Lecture 28, conditional variance theorem.

TheoremOrthogonality of martingale increments#✓fully proved

Let \((\Omega, \mu)\) be a probability space with a filtration \((\mathcal{F}_n)_{n \in \mathbb{N}}\), and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued martingale with respect to \((\mathcal{F}_n)\) and \(\mu\) such that \(\mathbb{E}[X_k^2] < \infty\) for every \(k\). Let \(m \le n\) and let \(Y\) be an \(\mathcal{F}_m\)-measurable real random variable with \(\mathbb{E}[Y^2] < \infty\). Then \(\mathbb{E}[(X_n - X_m) Y] = 0\). In other words, the increment \(X_n - X_m\) is orthogonal in \(L^2\) to every square-integrable \(\mathcal{F}_m\)-measurable random variable.

theorem Martingale.integral_sub_mul_eq_zero
    [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hmart : Martingale X ℱ μ)
    {m n : ℕ} (hmn : m ≤ n)
    {Y : Ω → ℝ} (hY_meas : StronglyMeasurable[ℱ m] Y)
    (hY_int : Integrable (fun ω => (Y ω) ^ 2) μ)
    (hX_sq : ∀ k, Integrable (fun ω => (X k ω) ^ 2) μ) :
    ∫ ω, (X n ω - X m ω) * Y ω ∂μ = 0

Sources

• Lecture 28, orthogonality of martingale increments.

Uses 1

Proof usesOrthogonality of the conditional-expectation residual

TheoremPythagorean identity for martingale increments#✓fully proved

Let \((\Omega, \mu)\) be a probability space with a filtration \((\mathcal{F}_n)_{n \in \mathbb{N}}\), and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued martingale with respect to \((\mathcal{F}_n)\) and \(\mu\) such that \(\mathbb{E}[X_k^2] < \infty\) for every \(k\). Then for all \(m \le n\),

\[ \mathbb{E}[(X_n - X_m)^2] = \mathbb{E}[X_n^2] - \mathbb{E}[X_m^2]. \]

This is the orthogonality of martingale increments applied to the \(\mathcal{F}_m\)-measurable random variable \(X_m\), expanded.

theorem Martingale.integral_sub_sq_eq
    [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hmart : Martingale X ℱ μ)
    {m n : ℕ} (hmn : m ≤ n)
    (hX_sq : ∀ k, Integrable (fun ω => (X k ω) ^ 2) μ) :
    ∫ ω, (X n ω - X m ω) ^ 2 ∂μ =
      ∫ ω, (X n ω) ^ 2 ∂μ - ∫ ω, (X m ω) ^ 2 ∂μ

Uses 1

Proof usesPythagorean identity for the conditional mean

3 fully proved

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