Convex functions of martingales

TheoremConvex functions of martingales are submartingales#✓fully proved

Let \(\mu\) be a probability measure, let \((\mathcal{F}_n)_{n \in \mathbb{N}}\) be a filtration, and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued martingale with respect to \((\mathcal{F}_n)\) and \(\mu\). Let \(\varphi : \mathbb{R} \to \mathbb{R}\) be convex, and suppose \(\varphi(X_n)\) is integrable for every \(n\). Then \((\varphi(X_n))_{n \in \mathbb{N}}\) is a submartingale with respect to \((\mathcal{F}_n)\) and \(\mu\).

theorem Martingale.submartingale_comp_convex
    [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hmart : Martingale X ℱ μ)
    {φ : ℝ → ℝ} (hφ_convex : ConvexOn ℝ Set.univ φ)
    (hφ_int : ∀ n, Integrable (fun ω => φ (X n ω)) μ) :
    Submartingale (fun n ω => φ (X n ω)) ℱ μ

Sources

• Lecture 26/27, martingale observations.

TheoremConvex nondecreasing functions preserve submartingales#✓fully proved

Let \(\mu\) be a probability measure, let \((\mathcal{F}_n)_{n \in \mathbb{N}}\) be a filtration, and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued submartingale with respect to \((\mathcal{F}_n)\) and \(\mu\). Let \(\varphi : \mathbb{R} \to \mathbb{R}\) be convex and nondecreasing, and suppose \(\varphi(X_n)\) is integrable for every \(n\). Then \((\varphi(X_n))_{n \in \mathbb{N}}\) is a submartingale with respect to \((\mathcal{F}_n)\) and \(\mu\).

Monotonicity is needed: a convex function of a submartingale need not be a submartingale. For example, take \(\varphi(x) = (x - 4)^2\) and a submartingale with \(X_i = 2\) and \(X_j = 3\) almost surely for some \(i < j\); then \(\varphi(X_i) = 4 > 1 = \varphi(X_j)\).

theorem Submartingale.comp_convex_of_monotoneOn
    [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hsub : Submartingale X ℱ μ)
    {φ : ℝ → ℝ} (hφ_convex : ConvexOn ℝ Set.univ φ)
    (hφ_mono : MonotoneOn φ Set.univ)
    (hφ_int : ∀ n, Integrable (fun ω => φ (X n ω)) μ) :
    Submartingale (fun n ω => φ (X n ω)) ℱ μ

TheoremConcave functions of martingales are supermartingales#✓fully proved

Let \(\mu\) be a probability measure, let \((\mathcal{F}_n)_{n \in \mathbb{N}}\) be a filtration, and let \((X_n)_{n \in \mathbb{N}}\) be a real-valued martingale with respect to \((\mathcal{F}_n)\) and \(\mu\). Let \(\varphi : \mathbb{R} \to \mathbb{R}\) be concave, and suppose \(\varphi(X_n)\) is integrable for every \(n\). Then \((\varphi(X_n))_{n \in \mathbb{N}}\) is a supermartingale with respect to \((\mathcal{F}_n)\) and \(\mu\).

theorem Martingale.supermartingale_comp_concave
    [IsProbabilityMeasure μ]
    {ℱ : Filtration ℕ m0} {X : ℕ → Ω → ℝ}
    (hmart : Martingale X ℱ μ)
    {φ : ℝ → ℝ} (hφ_concave : ConcaveOn ℝ Set.univ φ)
    (hφ_int : ∀ n, Integrable (fun ω => φ (X n ω)) μ) :
    Supermartingale (fun n ω => φ (X n ω)) ℱ μ

3 fully proved

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