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**Extra resources for Stochastic processes without measure theory**

**Sample text**

In this model, Xn is the number of individuals alive at time n.

T = “wait for 3 reds in a row, bet on black. ” We will always assume that P (T < ∞) = 1 in this section. Definition. The value function v : S → [0, ∞) is defined as the most expected profit possible from that starting point; v(x) = sup E(f(XT ) j X0 = x). T There is an optimal strategy Topt so that v(x) = E(f(XTopt ) j X0 = x). Facts about v 1. Consider the strategy T0 ≡ 0 (don’t gamble). Then v(x) = sup E(f(XT ) j X0 = x) ≥ E(f(XT0 ) j X0 = x) = f(x). T That is, v(x) ≥ f(x) for all x ∈ S. 2. Define the strategy T ∗ : play once, then follow the optimal strategy.

A0 a1 a2 a3 a 1 Examples. 1. p0 = 1/4, p1 = 1/4, p2 = 1/2. This gives us µ = 5/4 and φ(s) = 1/4 + s/4 + s2 /2. Solving φ(s) = s gives two solutions f1/2, 1g.