Stochastic Finance, 4th Edition

Appendix

A.1 Convexity

This section contains a few basic facts on convex functions and on convex sets. For further facts we refer to [232], from where we also took some of the material in this section. In the sequel, E will denote a general real vector space.

Definition A.1. A subset C of E is called convex if λx + (1 − λ)y ∈ C whenever x, y ∈ C and 0 ≤ λ ≤ 1.

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A convex combination of x1, . . . , xn ∈ E is of the form

with real numbers λi ≥ 0 satisfying λ1 + · · · + λn = 1.

Proposition A.2. A subset C of E is convex if and only if it contains all convex combinations of its elements.

Proof. If C contains all convex combinations of its elements, ...