29.1. CONDITIONAL EXPECTATION 783

Lemma 29.1.4 Let φ be a convex real valued function defined on an interval I. Thenfor each x ∈ I, there exists ax such that for all t ∈ I,

φ (t)≥ ax (t− x)+φ (x) .

Also φ is continuous on I.

Proof: Let x ∈ I and let t > x. Then by convexity of φ ,

φ (x+λ (t− x))−φ (x)λ (t− x)

≤ φ (x)(1−λ )+λφ (t)−φ (x)λ (t− x)

=φ (t)−φ (x)

t− x.

Therefore t→ φ(t)−φ(x)t−x is increasing if t > x. If t < x, t− x < 0 so

φ (x+λ (t− x))−φ (x)λ (t− x)

≥ φ (x)(1−λ )+λφ (t)−φ (x)λ (t− x)

=φ (t)−φ (x)

t− x

and so t→ φ(t)−φ(x)t−x is increasing for t ̸= x. Let

ax ≡ inf{

φ (t)−φ (x)t− x

: t > x}.

Then if t1 < x, and t > x,

φ (t1)−φ (x)t1− x

≤ ax ≤φ (t)−φ (x)

t− x.

Thus for all t ∈ I,φ (t)≥ ax (t− x)+φ (x). (29.5)

The continuity of φ follows easily from this and the observation that convexity simplysays that the graph of φ lies below the line segment joining two points on its graph. Thus,we have the following picture which clearly implies continuity. ■

Lemma 29.1.5 Let I be an interval on R and let φ be a convex function defined on I.Then there exists a sequence {(an,bn)} such that

φ (t) = sup{ant +bn,n = 1, · · ·} .

Proof: Let ax be as defined in the above lemma. Let

ψ (x)≡ sup{ar (x− r)+φ (r) : r ∈Q∩ I}.