Theorem 3.1 is our first serious theorem. It had an easy but non-trivial proof. It is important because it allows us to know that $F(x)$ satisfies a DP equation (3.7). It holds under the cases of D (discounted), N (negative) or P (positive) programming.
The problem of selling a tulip bulb collection in Section 3.5 is very much like the secretary problem in Section 2.3. The differences are that now (i) we observe values (not just relative ranks), (ii) wish to maximize the expected value of the selected candidate (rather than probability of choosing the best), and (iii) the number of offers is infinite, but with a discount factor $\beta$. We see that one way in which discounting by a factor beta can naturally occur is via a catastrophe, with probability $1-\beta$, bringing the problem to an end.
I asked if you can figure out the answer to the asset selling problem if past offers for the tulip bulb collection remain open (so long as the market has not collapsed). The state $x$ is now the best offer so far received. The DP equation would be
$$
F(x) = \int_0^\infty\max\Bigl[x,y,\beta F(\max\{x,y\})\Bigr] g(y) dy
$$
The validity of this equation is from Theorem 3.1 and that fact that this is a Positive case of dynamic programming.
In the proof of Theorem 3.1 I used that $\lim_{s\to\infty}EF_s(x_1)=E[\lim_{s\to\infty}EF_s(x_1)]$ when $F_s$ is either monotone increasing or decreasing in $s$, as it indeed is in the N and P cases. This is called the Lebesgue monotone convergence theorem. In the D case the interchange of $E$ and $\lim_{s\to\infty}$, it is true because $F_s(x)$ is close to its limit for large $s$, uniformly in $x$.
I remarked today that the D case can be recast as a N or P case. I have place this remark in Section 4.5 of the notes.
The problem of selling a tulip bulb collection in Section 3.5 is very much like the secretary problem in Section 2.3. The differences are that now (i) we observe values (not just relative ranks), (ii) wish to maximize the expected value of the selected candidate (rather than probability of choosing the best), and (iii) the number of offers is infinite, but with a discount factor $\beta$. We see that one way in which discounting by a factor beta can naturally occur is via a catastrophe, with probability $1-\beta$, bringing the problem to an end.
I asked if you can figure out the answer to the asset selling problem if past offers for the tulip bulb collection remain open (so long as the market has not collapsed). The state $x$ is now the best offer so far received. The DP equation would be
$$
F(x) = \int_0^\infty\max\Bigl[x,y,\beta F(\max\{x,y\})\Bigr] g(y) dy
$$
The validity of this equation is from Theorem 3.1 and that fact that this is a Positive case of dynamic programming.
In the proof of Theorem 3.1 I used that $\lim_{s\to\infty}EF_s(x_1)=E[\lim_{s\to\infty}EF_s(x_1)]$ when $F_s$ is either monotone increasing or decreasing in $s$, as it indeed is in the N and P cases. This is called the Lebesgue monotone convergence theorem. In the D case the interchange of $E$ and $\lim_{s\to\infty}$, it is true because $F_s(x)$ is close to its limit for large $s$, uniformly in $x$.
I remarked today that the D case can be recast as a N or P case. I have place this remark in Section 4.5 of the notes.
