April 25, 2024
Let \langle T,\le\rangle be a totally ordered set: for all t,u\in T either t \le u or u\le t.
If S is a finite subset of a totally ordered set T define an iterable over T by \begin{aligned} *S &= \min\{s\mid s\in S\} \\ +S &\leftarrow S\setminus\{*S\} \\ ?S &= (S \not= \emptyset) \\ \end{aligned} where left arrow indicates S is updated with the right hand value.
If f\colon T\to U define f(S) as an iterable over U by \begin{aligned} *f(S) &= f(*S) \\ +f(S) &\leftarrow f(+S) \\ ?f(S) &= (S \not= \emptyset) \\ \end{aligned}
Given a predicate P\colon S\to\bm{bool} define S\mid P by \begin{aligned} *(S\mid P) &= P(*S) \text{ if } ?S \text{ else } *(+(S\mid P)) \\ +(S\mid P) &\leftarrow (+S)\mid P \\ ?(S\mid P) &= (S\not=\emptyset) \\ \end{aligned} This will recursively find the next value of S that satisfies the predicate. We call this S given P.
Exercise Show s\in (S\mid P) if and only if s\in S and P(s).
If R\subset T\times T is a relation and t\in T define the predicate P(s) = sRt where sRt means (s,t)\in R. Define SRt to be S\mid P.
Exercise. Show S<t = S\cap\{s\in S\mid s < t\}.
If Q is a boolean valued iterable we use the same notation for the iterable S\mid Q. Define \begin{aligned} *(S\mid Q) &= *S\text{ if }*Q\text{ else } *(+(S\mid Q)) \\ +(S\mid Q) &\leftarrow (+S\mid +Q) \\ ?(S\mid Q) &= ?Q \\ \end{aligned} This will recursively find the next value of S that belongs to the mask. We call this S mask Q.
Exercise Show s\in (S\mid Q) if and only if and s = *+^nS and *+^nQ for some n\ge0.
If \{S_j\}_{0\le j \le n} are finite subsets of T then \sqcup_{0\le j \le n} S_j = \cup_{0\le j \le n} S_j\times\{j\}. The lexicographical order defined by (s,i) \le (t,j) by s < t or s = t and i < j is a total order.
If \{S_j\}_{0\le j < n} are finite subsets of T define ,S_j by *(,S_j) = *(\sqcup_j S_j) = (s, k) and +(,S_j) = \sqcup_j (S_j > s')). where s' = *S_{\mod(k+1,n)}.
Exercise. If (X_t) is right continuouse and T^a = \min\{t\mid X_t \ge a\} then 1_{T\ge t} is left continuous.
Consider a trading strategy where you buy 1 share of stock with the price goes below L and sell when it goes above H. If S_t is the price at time t trade indicators are (S < L),(S > H).