Apr 13, 2026
Stop Loss/Start Gain
The SL/SG strategy is to hold one share when the underlying crosses a lower bound and sell it when it crosses an upper bound. We develop a theory and notation where this can be expressed using (X > a)'(X < b).
We use standard mathematical notation with an eye toward computer implemenetation. Given symbols f and x how do we indicate when we want the result of applying the function f\colon X\to Y to the value x\in X?
Right to left parsing with parentheses for grouping.
If f is a function from set X to set Y then f(x), x\in X, is the corresponding element in Y
Evaluation e\colon Y^X \times X\to Y.
An iterable \iota\in I\langle T\rangle over value type T is defined by three primitives: \ast (dereference), + (increment), and \varepsilon (more) where \ast\colon I\langle T\rangle\to T, +\colon I\langle T\rangle\to I\langle T\rangle, and \varepsilon\colon I\langle T\rangle\to\boldsymbol{{B}}, the set of boolean values, indicating more values are available to dereference. If \varepsilon\iota is false then \ast\iota and +\iota are undefined.
Iterables benefit from lazy evaluation. Explicit dereference and increment cause computation.
Many languages implement this under varying guises. In C# iterables
I\langle T\rangle are an interface
IEnumerator<T> requiring methods
T Current() { get; } to dereference and
bool MoveNext() that combines increment and indicates
whether more values are available.
…More examples…
One representation of an iterable on a computer is by an array {\iota = (t_1,\ldots, t_n)} where {\ast\iota = t_1}, {+\iota = (t_2,\ldots,t_n)}, and \varepsilon\iota is n\not=0. Iterables are more flexible than arrays. They can be computed on-the-fly taking only fixed memory and can also be potentially unbounded. Let () be the iterable with \varepsilon() always false.
The iterable primitives allow a large number of algorithms to be defined.
The count of an iterable is \#\iota = 1 + \#+\iota if \varepsilon\iota and 0 otherwise.
If n\in\boldsymbol{{N}} is a non-negative integer and \iota\in I\langle T\rangle define “n take \iota” by n\uparrow \iota is \iota if n = 0 and (n - 1)\uparrow +\iota if n > 0. Note n\uparrow() = () if n = 0, and is undefined for n > 0.
Exercise. Show (\#\iota)\uparrow\iota = ().
If P is a predicate on T define “\iota where P” by \ast(\iota|P) = \ast\iota if P(\ast\iota) and \ast(+U|P) otherwise. This filters an iterable based on the predicate.
*where(i, p) => return p(*i) ? *i : *where(+i, p);
+where(i, p) => while (!p(*+i)); return where(i, p);If \iota_0,\iota_1\in I\langle T\rangle define the concatenation \iota_0,\iota_1 by \ast(\iota_0,\iota_1) = *\iota_0 if \varepsilon\iota_0 otherwise \ast \iota_1. Define +(\iota_0,\iota_1) = +\iota_0,\iota_1 if \varepsilon\iota_0 otherwise +\iota_1. Define \varepsilon(\iota_0,\iota_1) to be \varepsilon\iota_0 or \varepsilon\iota_1.
Define the cyclical comma operator by \iota, = \iota if \varepsilon\iota and (), = \iota.
Exercise. Show \iota,\iota, = \iota, for \iota\in I\langle T\rangle.
Given \iota_0\in I\langle T_0\rangle and \iota_1\in I\langle T_1\rangle define their zip \iota_0 ," \iota_1 by *(\iota_0 ," \iota_1) to be the pair (*\iota_0, *\iota_1), +(\iota_0 ," \iota_1) = (+\iota_0),"(+\iota_1), and \varepsilon(\iota_0 ," \iota_1) to be \varepsilon\iota_0 and \varepsilon\iota_1.
The disjoint union of T_i\subseteq T is \sqcup_{i\in I} T_i = \cup_{i\in I} \{i\}\times T_i. This has injections \nu_i\colon T_i\to \sqcup_i T_i where \nu_i(t) = (i, t).
Exercise. Given q_i\colon T_i\to U, i\in I, show there exists q\colon \sqcup_{j\in I}\to U with q\nu_i = q_i, i\in I.
Hint: Define q((i, t)) = q_i(t) for t\in T_i.
If T_i\subseteq T are indexed by i\in I define the cartesian product \prod_{i\in I} T_i = \{(t_i)_{i\in I}\in I\mid t_i\in T_i\}. It has projections \pi_i\colon \prod_{j\in I} T_j\to T_i defined by \pi_i((t_j)_{j\in I}) = t_i.
Exercise. Given p_i\colon U\to T_i, i\in I, show there exists p\colon U\to\prod_i T_i with \pi_ip = p_i, i\in I.
Hint: Define p(u) = (p_i(u))_{i\in I} for u\in U. If I is totally ordered define (i,t)<(j,u) to be i < j or i = j and t < u. This defines a total order on the disjoint union in terms of the total orders on I and T.
If T is a totally ordered finite set and U\subseteq T we can make U an iterable by defining \ast U = \min\{u\mid u\in U\}, +U = U\setminus\{\ast U\}, and \varepsilon U to be U\not=\emptyset
Note \sqcup_i T_i\subseteq\sqcup_i T so the iterable primatives can be applied. The primatives also come in indexed version.
Define \ast_i\colon\sqcup_{j\in I} T_j by \ast_i = \ast T_i.
Define +_i = \sqcup +T_i \sqcup_{j\not= k} T_i.
Define \varepsilon_i\colon \sqcup_{j\in I}T_j = \varepsilon T_i.
If (X_t)_{t\in T} is a timeseries on \Omega
Consider I\times\cup_{i\in I} T_i. Define \ast(i, t) = \ast T_i. If \ast(i,t) < \ast(j,t) for all j\not=i define …
Define \ast_i\colon \sqcup_{j\in I} T_j\to T_i by \ast_i(t) = \ast T_i.
Concatenate in increasing order. (Zig)
U \subseteq \sqcup_i U_i.
If (M_t) is a (sub)martingale then the number of up crossings of the interval [a,b] then (b - a)N_{a,b}(t) \le E[\max\{M_t - a, 0\}] - \max\{M_0 - a, 0\}.