An Introduction to Financial Mathematics, 2nd Edition

Chapter 10

1.  For n = 1, 2, …, N, let U_n denote any of the ${ℱ}_{n-1}^S$ random variables $\frac{1}{n}\left(S_0+S_1+\cdots +S_{n-1}\right)$ max {S₀, S₁, …, S_n−1} min {S₀, S₁, …, S_n−} and set A_n = {S_n > U_n}. Then $A_n\in {ℱ}_n^S$, and the functions in (a), (b) and (c) are of the form

$\tau \left(\omega \right)=\{\begin{array}{ll}\min \left\{n|\omega \in A_n\right\}\hfill & \text{if}\left\{n|\omega \in A_n\right\}\ne \cancel{0},\hfill \\ N\hfill & \text{otherwise}\text{.}\hfill \end{array}$

Therefore,

$\left\{\tau =n\right\}=A_1^{\prime }{A}_2^{\prime }\cdots A_{n-1}^{\prime }{A}_n\in {ℱ}_n^S,n<N,$

and

$\left\{\tau =N\right\}=A_1^{\prime }{A}_2^{\prime }\cdots A_{N-1}^{\prime }\in {ℱ}_{N-1}^S\subseteq {ℱ}_N^S.$

3.  We show by induction on k that

$v_k\left(S_k\left(\omega \right)\right)=f\left(S_k\left(\omega \right)\right)=0$

(†)

for all k > n (= τ₀(ω)). By definition of τ₀, (†) holds for k = n. Suppose (†) holds for arbitrary k ≥ n. Since

$v_k\left(S_k\left(\omega \right)\right)=\max \left(f\left(S_k\left(\omega \right)\right),a{v}_{k+1}\left(S_k\left(\omega \right)u\right)+b{v}_{k+1}\left(S_k\left(\omega \right)d\right)\right)$

and all terms comprising the expression on the right of this equation are nonnegative, (†) implies that

$v_{k+1}\left(S_k\left(\omega \right)u\right)=v_{k+1}\left(S_k\left(\omega \right)d\right)=0,$

that is, v_k+1 (S_k+1(ω)) = ...