#  $p$-Linear schemes for sequences modulo $p^{r}$

Frits Beukers

###### Abstract.

Many interesting combinatorial sequences, such as Apéry numbers and Franel
numbers, enjoy the so-called Lucas property modulo almost all primes $p$.
Modulo prime powers $p^{r}$ such sequences have a more complicated behaviour
which can be described by matrix versions of the Lucas property called
$p$-linear schemes. They are examples of finite $p$-automata. In this paper we
construct such $p$-linear schemes and give upper bounds for the number of
states which, for fixed $r$, do not depend on $p$.

##  1\. Introduction

In this paper we study infinite sequences of integers that have special
properties when they are reduced modulo a prime $p$ or a prime power $p^{r}$.
The most familiar sequences are those sequences $a_{k}$ that have the so-
called Lucas property modulo almost all primes $p$,

$a_{k_{m}p^{m}+\cdots+k_{1}p+k_{0}}\equiv a_{k_{m}}\cdots
a_{k_{1}}a_{k_{0}}({\rm mod}\ p)$

for all $k_{i}$ with $0\leq k_{i}\leq p-1$. Examples are the exponential
sequences like $2^{k}$, the central binomial coefficients $\binom{2k}{k}$, the
Apéry and Franel-type numbers given by

$\sum_{m=0}^{k}\binom{k}{m}^{2}\binom{k+m}{m}^{2},\quad\sum_{m=0}^{k}\binom{k}{m}^{\ell}$

and countless others. Our first result concerns constant term sequences of the
form

$g_{k}=\text{ct}[g({\bf x})^{k}]\quad k\geq 0,$

where $g({\bf x})\in{\mathbb{Z}}[x_{1}^{\pm 1}\cdots x_{n}^{\pm 1}]$ is a
Laurent polynomial in the $n$ variables $x_{1},\ldots,x_{n}$ and ’ct’ is the
operation of taking the constant term. In this paper we denote the Newton
polytope of $g({\bf x})$ by $\Delta$. By $\Delta^{\circ}$ we denote the
topological interior of $\Delta$, i.e $\Delta$ minus its proper faces. In [5,
Thm 1.4] we find the following theorem.

######  Theorem 1.1.

Suppose that $\Delta^{\circ}$ has ${\bf 0}$ as unique lattice point. Then, for
any prime $p$ the constant term sequence $g_{k}$ has the Lucas property.

To my knowledge the first explicit occurrence of this theorem is in the case
$s=1$ of [8, Thm 3.3], which is unfortunately unpublished.

In Section 3 we present a proof, which is basically the same as in [5]. Here
are some examples of interest. In [5, Example 2.3] we find

$\binom{2k}{k}=\text{ct}\left[\left(x+2+\frac{1}{x}\right)^{k}\right].$

In [5, Example 2.4] we find

$\sum_{k_{1},\ldots,k_{n}\geq 0\atop
k_{1}+\cdots+k_{n}=k}\left(\frac{k!}{k_{1}!\cdots
k_{n}!}\right)^{2}=\text{ct}\left[(1+x_{1}+\cdots+x_{n})^{k}\left(1+\frac{1}{x_{1}}+\cdots+\frac{1}{x_{n}}\right)^{k}\right].$

In [5, Example 2.5] we find

$\sum_{m=0}^{k}\binom{m+k}{m}^{2}\binom{k}{m}^{2}=\text{ct}\left[\left(\frac{(x+y)(z+1)(x+y+z)(y+z+1)}{xyz}\right)^{k}\right].$

These are the famous Apéry numbers that occur in the irrationality proof of
$\zeta(3)$. Finally we mention

$\sum_{m=0}^{k}\binom{k}{m}^{n+1}=\text{ct}\left[(1+x_{1})^{k}(1+x_{2})^{k}\cdots(1+x_{n})^{k}\left(1+\frac{1}{x_{1}\cdots
x_{n}}\right)^{k}\right]$

which is not hard to show.

In general one cannot expect the Lucas property to hold modulo $p^{r}$ for
$r>1$, or when $\Delta^{\circ}$ contains another interior lattice point
besides ${\bf 0}$. But one can introduce additional companion sequences, which
result in matrix versions of the Lucas property.

######  Definition 1.2.

A $p$-linear scheme modulo $p^{r}$ for a set of sequences ${\bf
a}_{k}=(a_{1,k},\ldots,a_{s,k})^{t}$, $k=0,1,\ldots$ is a set of $s\times
s$-matrices $M_{\ell}$ with entries in ${\mathbb{Z}}$ such that

(1) ${\bf a}_{kp+\ell}\equiv M_{\ell}{\bf a}_{k}({\rm mod}\ p^{r})$

for every $\ell\in\\{0,1,\ldots,p-1\\}$ and $k\geq 0$.

We say that any constant linear combination of the sequences
$a_{1,k},\ldots,a_{s,k}$ is a sequence that can be generated by a $p$-linear
scheme modulo $p^{r}$. We call $s$ the number of states of this $p$-linear
scheme.

As explained in [10], $p$-linear schemes are special instances of finite
$p$-automata. A simple corollary of Definition 1.2 is

${\bf a}_{k_{m}p^{m}+\cdots+k_{1}p+k_{0}}\equiv M_{k_{0}}M_{k_{1}}\cdots
M_{k_{m}}{\bf a}_{0}({\rm mod}\ p^{r}),$

where $0\leq k_{i}\leq p-1$ for all $i=0,\ldots,m$. So all one needs in order
to compute ${\bf a}_{k}({\rm mod}\ p^{r})$ for any $k$ are the matrices
$M_{\ell}$ for $\ell=0,1,\ldots,p-1$, the vector ${\bf a}_{0}$ and the base
$p$ expansion of $k$. Furthermore, if $r=1$, $s=1$ and $a_{0}=1$ then
Definition 1.2 comes down to the Lucas property modulo $p$.

There is also a formulation of a $p$-linear scheme in terms of generating
power series. Let ${\bf F}(t)=\sum_{k\geq 0}{\bf a}_{k}t^{k}$, a vector of
generating series, and $M(t)=\sum_{\ell=0}^{p-1}M_{\ell}t^{\ell}$. Then (1) is
equivalent to

(2) ${\bf F}(t)\equiv M(t){\bf F}(t^{p})({\rm mod}\ p^{r}).$

As an example we display a generalization of Theorem 1.1. Let
$\Delta_{\mathbb{Z}}^{\circ}$ be the set of lattice points in
$\Delta^{\circ}$. Define the $p$-th _Hasse-Witt matrix_ $H$ associated to
$1-tg({\bf x})$ as the square matrix indexed by $\Delta_{\mathbb{Z}}^{\circ}$,
whose ${\bf u},{\bf v}$-entry is given by

$H_{{\bf u}{\bf v}}(t)=\text{coefficient of ${\bf x}^{p{\bf u}-{\bf v}}$ in
}(1-tg({\bf x}))^{p-1}.$

Note that the entries of $H$ are polynomials of degree $\leq p-1$. For any
${\bf u}\in\Delta_{\mathbb{Z}}^{\circ}$ define

$F_{{\bf u}}(t)=\sum_{k\geq 0}ct[{\bf x}^{{\bf u}}g({\bf x})^{k}]t^{k}.$

We prove in Section 3 the following statement.

######  Theorem 1.3.

For any ${\bf u}\in\Delta_{\mathbb{Z}}^{\circ}$ we have

$F_{{\bf u}}(t)=\sum_{{\bf v}\in\Delta_{\mathbb{Z}}^{\circ}}H_{{\bf u}{\bf
v}}(t)F_{{\bf v}}(t^{p})({\rm mod}\ p).$

A simple example for congruences modulo higher powers $p^{r}$ is the 2-state
$p$-linear scheme modulo $p^{2}$ for the sequences $a=2^{k},b_{k}=k2^{k}$. It
follows from the following congruences, which are easily verified for any odd
prime $p$,

$\displaystyle a_{kp+\ell}$ $\displaystyle\equiv$ $\displaystyle
2^{\ell}a_{k}+p\alpha 2^{\ell}b_{k}({\rm mod}\ p^{2})$ $\displaystyle
b_{kp+\ell}$ $\displaystyle\equiv$ $\displaystyle\ell
2^{\ell}a_{k}+p2^{\ell}(1+\ell\alpha)b_{k}({\rm mod}\ p^{2}),$

where $\alpha\equiv\frac{2^{p-1}-1}{p}({\rm mod}\ p)$. In general $2^{k}({\rm
mod}\ p^{r})$ can be generated by an $r$-state $p$-linear scheme.

An example of historical interest is Ira Gessel’s mod $p^{2}$ congruence for
the Apéry numbers $A_{k}=\sum_{m=0}^{k}\binom{k}{m}^{2}\binom{m+k}{m}^{2}$ in
[4].

######  Theorem 1.4 (Gessel, 1982).

Define for all integers $k\geq 0$,

$A^{\prime}_{k}=\sum_{m=0}^{k}\binom{k}{m}^{2}\binom{m+k}{m}^{2}\left(\frac{1}{m+1}+\cdots+\frac{1}{k}\right).$

Then, for any prime $p$ and any integers $k\geq 0$ and $\ell=0,1,\ldots,p-1$
we have

$A_{kp+\ell}\equiv A_{\ell}A_{k}+pA^{\prime}_{\ell}kA_{k}({\rm mod}\ p^{2}).$

Multiply this result with $kp+\ell$ to get

$(kp+\ell)A_{kp+\ell}\equiv\ell A_{\ell}A_{k}+p(A_{\ell}+\ell
A^{\prime}_{\ell})kA_{k}({\rm mod}\ p^{2}).$

So the sequences $A_{k},kA_{k}$ form the two states of a $p$-linear scheme
modulo $p^{2}$.

Suppose again that $g({\bf x})$ is any Laurent polynomial in
$x_{1},\ldots,x_{n}$ and coefficients in ${\mathbb{Z}}$. The following
statement is an immediate consequence of Theorem 3.1 and Remark 3.2.

######  Corollary 1.5.

Let $r\in{\mathbb{N}}$ and $p$ a prime. Choose an integer $\rho$ such that
$\rho-\lceil\rho/p\rceil\geq r-1$. Then the numbers $g_{k}({\rm mod}\ p^{r})$
can be generated by a $p$-linear scheme with $s$ states, where $s$ is bounded
by $\rho$ times the number of lattice points in $\rho\Delta^{\circ}$.

Note that if $r=1$ we can take $\rho=1$. Since $g_{0}=1$ we recover Theorem
1.1 if ${\bf 0}$ is the unique lattice point in $\Delta^{\circ}$. Notice also
that $\rho$ can always be chosen such that $\rho\leq 2r$.

Our second result concerns coefficients of power series expansions for
rational functions. Let $P({\bf x}),Q({\bf
x})\in{\mathbb{Z}}[x_{1},\ldots,x_{n}]$ and suppose that $P_{0}:=P({\bf
0})\neq 0$. Then we can expand

$\frac{Q({\bf x})}{P({\bf x})}=\sum_{{\bf k}\in{\mathbb{Z}}_{\geq
0}^{n}}a_{{\bf k}}{\bf x}^{{\bf k}}$

with $a_{{\bf k}}\in{\mathbb{Z}}[1/P_{0}]$. Of particular interest in the
literature are the diagonal sequences $a_{k,k,\ldots,k}$. Some typical
examples,

  * -

The diagonal elements of $(1-x_{1}-\cdots-x_{n})^{-1}$ are
$\frac{(kn)!}{(k!)^{n}}$.

  * -

The diagonal elements of
$((1-x_{1}-x_{2})(1-x_{3}-x_{4})-x_{1}x_{2}x_{3}x_{4})^{-1}$ are the Apéry
numbers introduced above. See [9].

In [11] we find a long list of further examples, with further references to
Sloane’s Online Encyclopedia.

In Theorem 4.1 we construct a $p$-linear scheme to compute the coefficients
$a_{{\bf k}}$. This time the coefficients $a_{{\bf k}}$ have
${\mathbb{Z}}_{\geq 0}^{n}$ as domain, rather than ${\mathbb{Z}}_{\geq 0}$.
However, it is straightforward to replace ${\bf a}_{k}$ by ${\bf a}_{{\bf k}}$
in Definition 1.2. Here is a consequence of Corollary 4.2 which also follows
from [6, Theorem 5.2] with $s=1$.

######  Corollary 1.6.

Let notations be as above and suppose that $Q({\bf x})=1$, $P_{0}=1$ and the
degree of $P({\bf x})$ in each variable is $1$. Then, for every prime $p$,
${\bf k}\in{\mathbb{Z}}_{\geq 0}^{n}$ and $\bm{\ell}\in[0,1,\ldots,p-1]^{n}$
we have

$a_{p{\bf k}+\bm{\ell}}\equiv a_{\bm{\ell}}a_{{\bf k}}({\rm mod}\ p).$

To prove this, apply Corollary 4.2 with $r=1$ (hence $\rho=1$) and
$d_{1}=\cdots=d_{n}=1$. Then we get a 1-state $p$-linear scheme to compute the
numbers $a_{{\bf k}}({\rm mod}\ p)$. Since $a_{{\bf 0}}=1$ our corollary
follows. Of course the diagonal sequence $a_{k,k,\ldots,k}$ has now the Lucas
property modulo every prime $p$.

In [6] and [7] the authors propose algorithms to construct finite $p$-automata
and in particular $p$-linear schemes for the computation of sequences
$a_{k}({\rm mod}\ p^{r})$. Unfortunately the algorithms seem rather
inefficient (except in ’small’ cases) and the resulting upper bounds for the
number of states seem exceedingly large with a strong dependence on $p$ (see
[10, Thm 2.4]). These papers were the motivation for writing the present note.
Its method is a simplified version of the techniques in [3], which studies the
action of the so-called Cartier operator on spaces of rational functions.
However, the present paper can be completely understood without having to read
[3].

Acknowledgement: Many thanks to Armin Straub for the stimulating discussions I
had on the subject of this paper and his encouragement.

Many thanks also to the referee for the careful reading of this paper and for
the remarks that very much improved its exposition.

##  2\. Main tool

The starting point in this paper is a Laurent polynomial $f$ in the variables
$x_{1},\ldots,x_{n}$ and coefficients in a ring $R$ which is either
${\mathbb{Z}}$ or ${\mathbb{Z}}[t]$. Let $p$ be any prime and let $\sigma$ be
the Frobius lift on $R$, which is the identity map on ${\mathbb{Z}}$ and which
maps $t$ to $t^{p}$ in the ring ${\mathbb{Z}}[t]$.

Let $\mu$ be a bounded polytope in ${\mathbb{R}}^{n}$ with a finite set of
proper faces removed and such that $\Delta^{\circ}\subset\mu$. The original
polytope is denoted by $\overline{\mu}$, the closure of $\mu$. In [1],[2],[3]
we study the action of the Cartier operator $\mathscr{C}_{p}$ on the
$R$-module $\Omega_{f}(\mu)$ of rational functions of the form $(r-1)!A({\bf
x})/f({\bf x})^{r}$, where $A({\bf x})$ is a Laurent polynomial with
coefficients in $R$ and support in $r\mu$. In Section 3 we use
$\mu=\Delta^{\circ}$ and in Section 4 we use an extension of $\Delta^{\circ}$,
the so-called box closure. Note that in [1] we use only $\mu$ such that
$\overline{\mu}=\Delta$. The main property of these sets is the following.

######  Lemma 2.1.

Let $\mu$ be a bounded polytope with a finite number of proper subfaces
removed. Denote the original polytope by $\overline{\mu}$. Then, for all
positive integers $a,b$ we have

$a\mu+b\overline{\mu}\subset(a+b)\mu.$

Here we use the notation $X+Y=\\{{\bf x}+{\bf y}|{\bf x}\in X,{\bf y}\in Y\\}$
for any two sets $X,Y\subset{\mathbb{R}}^{n}$ and $kX$ the $k$-fold sum of $k$
copies of $X$.

Suppose we are given an infinite Laurent series $\sum_{{\bf k}}a_{{\bf k}}{\bf
x}^{{\bf k}}$, $a_{{\bf k}}\in R$, with support in a positive cone with ${\bf
0}$ as top. Then $\mathscr{C}_{p}$ is defined by

$\mathscr{C}_{p}\left(\sum_{{\bf k}}a_{{\bf k}}{\bf x}^{{\bf
k}}\right)=\sum_{{\bf k}}a_{p{\bf k}}{\bf x}^{{\bf k}}.$

The action of $\mathscr{C}_{p}$ on a rational function $R({\bf
x})\in\Omega_{f}(\mu)$ is obtained by expansion of $R({\bf x})$ in an infinite
Laurent series and then application of $\mathscr{C}_{p}$. Unfortunately the
image is a Laurent series which usually does not come from an element in
$\Omega_{f}(\mu)$. Fortunately, in [1] it is shown that the image does lie in
the $p$-adic completion of $\Omega_{f^{\sigma}}(\mu)$. Here $f^{\sigma}$ is
simply $f$ with $\sigma$ applied to its coefficients.

In the present paper we will only need a mod $p^{r}$ version of this fact and
for its derivation we need the following easy lemma.

######  Lemma 2.2.

Let $K({\bf x}),L({\bf x})$ be two infinite Laurent series in
$x_{1},\ldots,x_{n}$ supported in the same cone with top ${\bf 0}$. Then,

$\mathscr{C}_{p}(K({\bf x}^{p})L({\bf x}))=K({\bf x})\mathscr{C}_{p}(L({\bf
x})).$

Let $r$ be a positive integer and consider the module of rational functions
$\frac{A({\bf x})}{f({\bf x})^{r}}$ with support of $A({\bf x})$ in $r\mu$ by
$M(r)$. By $M^{\sigma}(r)$ we denote module generated by the elements of
$M(r)$ where we have applied $\sigma$ to all coefficients of the rational
functions.

######  Proposition 2.3.

Let $r$ be a positive integer and $p$ a prime. Choose an integer $\rho$ such
that $\rho-\lceil\rho/p\rceil\geq r-1$. Then the Cartier operator
$\mathscr{C}_{p}$ modulo $p^{r}$ maps $M(\rho)$ to $M^{\sigma}(\rho)$.

###### Proof.

Rewrite $A({\bf x})/f({\bf x})^{\rho}$ as $A({\bf x})f({\bf
x})^{p\lceil\rho/p\rceil-\rho}/f({\bf x})^{p\lceil\rho/p\rceil}$. Then note
that $f({\bf x})^{p}=f^{\sigma}({\bf x}^{p})-pG({\bf x})$ for some Laurent
polynomial $G$ with coefficients in $R$ and support in $p\Delta$. Then we use
the $p$-adic expansion

$\frac{A({\bf x})}{f({\bf x})^{\rho}}=\frac{A({\bf x})f({\bf
x})^{p\lceil\rho/p\rceil-\rho}}{(f^{\sigma}({\bf x}^{p})-pG({\bf
x}))^{\lceil\rho/p\rceil}}=\sum_{m\geq 0}p^{m}{\lceil\rho/p\rceil+m-1\choose
m}\frac{G({\bf x})^{m}}{f^{\sigma}({\bf x}^{p})^{m+\lceil\rho/p\rceil}}A({\bf
x})f({\bf x})^{p\lceil\rho/p\rceil-\rho}.$

Apply $\mathscr{C}_{p}$. We find that

$\mathscr{C}_{p}\left(\frac{A({\bf x})}{f({\bf x})^{\rho}}\right)=\sum_{m\geq
0}p^{m}{\lceil\rho/p\rceil+m-1\choose m}\frac{Q_{m}({\bf x})}{f^{\sigma}({\bf
x})^{m+\lceil\rho/p\rceil}},$

where $Q_{m}({\bf x})=\mathscr{C}_{p}(A({\bf x})G({\bf x})^{m}f({\bf
x})^{p\lceil\rho/p\rceil-\rho})$. The support of $A({\bf x})G({\bf
x})^{m}f({\bf x})^{p\lceil\rho/p\rceil-\rho}$ is contained in

$\rho\mu+mp\Delta+(p\lceil\rho/p\rceil-\rho)\Delta\subset\rho\mu+(mp+p\lceil\rho/p\rceil-\rho)\overline{\mu}.$

According to Lemma 2.1 this is contained in $(mp+p\lceil\rho/p\rceil)\mu$,
hence the support of $Q_{m}$ lies in $(m+\lceil\rho/p\rceil)\mu$. Its
coefficients are still in $R$. The terms with $m\geq r$ all vanish modulo
$p^{r}$. Hence

(3) $\mathscr{C}_{p}\left(\frac{A({\bf x})}{f({\bf
x})^{\rho}}\right)\equiv\sum_{m=0}^{r-1}p^{m}{\lceil\rho/p\rceil+m-1\choose
m}\frac{Q_{m}({\bf x})}{f^{\sigma}({\bf x})^{m+\lceil\rho/p\rceil}}({\rm mod}\
p^{r}).$

Since $m+\lceil\rho/p\rceil\leq r-1+\lceil\rho/p\rceil\leq\rho$ we see that
modulo $p^{r}$ the right hand side is in $M^{\sigma}(\rho)$. ∎

##  3\. Constant terms of powers of Laurent polynomials

As a warm up to the proof of Theorem 3.1 we give a simple proof.

###### Proof of Theorem 1.1.

Let notations and assumptions be as in Theorem 1.1. Consider the equality

$\frac{1}{1-tg({\bf x})}=\frac{\sum_{k=0}^{p-1}t^{k}g({\bf
x})^{k}}{1-t^{p}g({\bf x})^{p}}\equiv\frac{\sum_{k=0}^{p-1}t^{k}g({\bf
x})^{k}}{1-t^{p}g({\bf x}^{p})}({\rm mod}\ p).$

Expand on both sides in a power series in $t$ with Laurent polynomials in
$x_{1},\ldots,x_{n}$ as coefficients. Apply the Cartier operator on both
sides. We get

$\mathscr{C}_{p}\left(\frac{1}{1-tg({\bf
x})}\right)\equiv\frac{1}{1-t^{p}g({\bf
x})}\mathscr{C}_{p}\left(\sum_{k=0}^{p-1}t^{k}g({\bf x})^{k}\right)({\rm mod}\
p).$

Take any monomial ${\bf x}^{{\bf w}}$ in $\sum_{k=0}^{p-1}t^{k}g({\bf
x})^{k}$. In order to survive $\mathscr{C}_{p}$ we must have ${\bf w}=p{\bf
v}$ for some lattice point ${\bf v}$. Note that $p{\bf v}\in(p-1)\Delta$,
hence ${\bf v}\in\Delta^{\circ}$. By our assumption on the lattice points in
$\Delta^{\circ}$, ${\bf v}={\bf 0}$. We obtain

$\mathscr{C}_{p}\left(\frac{1}{1-tg({\bf
x})}\right)\equiv\frac{1}{1-t^{p}g({\bf x})}\sum_{k=0}^{p-1}g_{k}t^{k}({\rm
mod}\ p).$

Take the constant term in $x_{1},\ldots,x_{n}$ on both sides. We get

$\sum_{\ell\geq
0}g_{\ell}t^{\ell}\equiv\left(\sum_{k=0}^{p-1}g_{k}t^{k}\right)\left(\sum_{m\geq
0}g_{m}t^{pm}\right)({\rm mod}\ p).$

By comparison of the coefficients of $t^{\ell}=t^{pm+k}$ on both sides we get
$g_{pm+k}\equiv g_{k}g_{m}({\rm mod}\ p)$, which proves our theorem. ∎

We now switch to the general situation. We apply Proposition 2.3 to construct
$p$-linear schemes for sequences of the form

$\text{ct}[q({\bf x})g({\bf x})^{k}]({\rm mod}\ p^{r}),$

where ’ct’ is the operation of selecting the constant term and $q({\bf
x}),g({\bf x})\in{\mathbb{Z}}[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ are Laurent
polynomials. Let $f({\bf x})=1-tg({\bf x})$ and $\mu=\Delta^{\circ}$, the
euclidean interior of the Newton polytope $\Delta$. Let $p$ be a prime, $r$ a
positive integer and $\rho$ an integer such that
$r-1\leq\rho-\lceil\rho/p\rceil$. For any monomial ${\bf x}^{{\bf
u}}\in\rho\Delta^{\circ}$ expand ${\bf x}^{{\bf u}}/f({\bf x})^{\rho}$ as
power series in $t$ and take the constant term of each coefficient. We get a
power series

$F_{{\bf u}}(t):=\text{ct}\left[\frac{{\bf x}^{{\bf u}}}{(1-tg({\bf
x}))^{\rho}}\right]=\sum_{k\geq 0}{\rho+k-1\choose\rho}\text{ct}[{\bf x}^{{\bf
u}}g({\bf x})^{k}]t^{k},$

which is the generating series of the numbers
${\rho+k-1\choose\rho}\text{ct}[{\bf x}^{{\bf u}}g({\bf x})^{k}]$. The
functions $t^{\ell}F_{{\bf u}}(t)$ with $\ell=0,1,\ldots,\rho-1$ and ${\bf
u}\in\rho\Delta^{\circ}$ will be the states of our $p$-linear scheme. Usually
we are interested in sequences of the form $\text{ct}[g({\bf x})^{k}]$, which
correspond to the constant term of $1/f({\bf x})$. However, we can use the
observation

$\frac{1}{f({\bf x})}=\frac{f({\bf x})^{\rho-1}}{f({\bf x})^{\rho}}.$

So the constant term of $1/f({\bf x})({\rm mod}\ p^{r})$ is a
${\mathbb{Z}}$-linear combination of the states $t^{\ell}F_{{\bf v}}(t)$ with
${\bf v}\in\rho\Delta^{\circ}$ and $\ell=0,1,\ldots,\rho-1$.

######  Theorem 3.1.

Let $p,r,\rho$ and $F_{{\bf u}}{t}$ be as before. Then there exist polynomials
$\lambda_{{\bf u},\ell,{\bf v},m}(t)$ in $t$ of degree $<p$ such that

$t^{\ell}F_{{\bf u}}(t)\equiv\sum_{{\bf
v}\in\rho\Delta^{\circ}}\sum_{m=0}^{\rho-1}\lambda_{{\bf u},\ell,{\bf
v},m}(t)(t^{p})^{m}F_{{\bf v}}(t^{p})({\rm mod}\ p^{r}).$

for all ${\bf u}\in(\rho\Delta^{\circ})_{\mathbb{Z}}$ and
$\ell=0,1,\ldots,\rho-1$.

######  Remark 3.2.

Note that we have found a $p$-linear scheme with states $t^{\ell}F_{{\bf
u}}({\rm mod}\ p^{r})$, where $\ell=0,1,\ldots,\rho-1$ and ${\bf
u}\in\rho\Delta^{\circ}$. The cardinality of
$(\rho\Delta^{\circ})_{\mathbb{Z}}$ is bounded polynomially in $\rho$ of
degree $n$. For every prime $p$ the choice $\rho=2r$ suffices so the number of
such states has order $O(r^{n+1})$, independent of $p$.

###### Proof of Theorem 3.1.

Consider (3) with $A({\bf x})={\bf x}^{{\bf u}}$ and expand as

(4) $\mathscr{C}_{p}\left(\frac{{\bf x}^{{\bf u}}}{f({\bf
x})^{\rho}}\right)\equiv\frac{1}{f^{\sigma}({\bf x})^{\rho}}\sum_{{\bf
v}\in\rho\mu}q_{{\bf u},{\bf v}}(t){\bf x}^{{\bf v}}({\rm mod}\ p^{r}).$

The coefficients $q_{{\bf u},{\bf v}}(t)$ are polynomials in $t$ and one
easily verifies that their degrees are at most
$p(r-1)+p\lceil\rho/p\rceil-\rho$. Let us choose polynomials $\lambda_{{\bf
u},\ell,{\bf v},m}$ of degree $<p$ such that

$t^{\ell}q_{{\bf u},{\bf v}}=\sum_{m=0}^{\rho-1}\lambda_{{\bf u},\ell,{\bf
v},m}(t)t^{pm}.$

This is possible because the degree of the left hand side is at most

$p(r-1)+p\lceil\rho/p\rceil-\rho+\rho-1=p(r-1+\lceil\rho/p\rceil)-1\leq
p\rho-1.$

The constant terms with respect to $x_{1},\ldots,x_{n}$ on both sides of (4)
are equal modulo $p^{r}$ and by a straightforward computation our assertion
follows. ∎

###### Proof of Theorem 1.3.

Set $r=\rho=1$. Inspection of (3) with $A({\bf x})={\bf x}^{{\bf u}}$ yields

(5) $\mathscr{C}_{p}\left(\frac{{\bf x}^{{\bf u}}}{1-tg({\bf
x})}\right)\equiv\frac{1}{1-t^{p}g({\bf x})}\mathscr{C}_{p}\left({\bf x}^{{\bf
u}}(1-tg({\bf x}))^{p-1}\right)({\rm mod}\ p).$

The support of ${\bf x}^{{\bf u}}(1-tg({\bf x}))^{p-1}$ is in
$p\Delta^{\circ}$ and we get

$\mathscr{C}_{p}\left({\bf x}^{{\bf u}}(1-tg({\bf x}))^{p-1}\right)=\sum_{{\bf
v}\in\Delta^{\circ}_{\mathbb{Z}}}H_{{\bf u}{\bf v}}(t){\bf x}^{{\bf v}}.$

We complete the proof of the theorem by expansion of (5) as power series in
$t$ and taking the constant term with respect to $x_{1},\ldots,x_{n}$ on both
sides. ∎

##  4\. Power series coefficients of rational functions

Our second main application of Proposition 2.3 is to the coefficients of power
series expansions of rational functions. Let $P$ be a polynomial in
$x_{1},\ldots,x_{n}$ with integer coefficients and a constant term which is
not divisible by the prime $p$. Let us call this constant $P_{0}$. Let
$\Delta$ be the Newton polytope of $P$. For any polynomial $Q$ with integer
coefficients we can expand $Q/P$ as power series. Let us write

$\frac{Q({\bf x})}{P({\bf x})}=\sum_{{\bf k}}a_{{\bf k}}{\bf x}^{{\bf k}},$

and note that $a_{{\bf k}}\in{\mathbb{Z}}[1/P_{0}]$ for all ${\bf k}$. We will
show that for every $r\geq 1$ there is a $p$-linear scheme which generates the
numbers $a_{{\bf k}}({\rm mod}\ p^{r})$. In particular this holds for
coefficients $a_{k{\bf v}}$ with ${\bf v}=(1,1,\ldots,1)$ and
$k\in{\mathbb{Z}}_{\geq 0}$. These are the diagonal elements of the power
series.

In Theorem 4.1 we use the concept of _box closure_. For any ${\bf
x}\in{\mathbb{R}}^{n}$ with non-negative entries we denote $B({\bf
x})=[0,x_{1}]\times\cdots\times[0,x_{n}]$ where ${\bf
x}=(x_{1},\ldots,x_{n})$. We take $B({\bf x})$ empty if $x_{i}<0$ for at least
one index $i$. For any subset $S\subset{\mathbb{R}}^{n}$ we define its box
closure by $B(S)=\cup_{{\bf x}\in S}B({\bf x})$.

For us, the main property of box closures is that for any
$\bm{\ell}\in{\mathbb{R}}_{\geq 0}^{n}$ we have $B(S-\bm{\ell})\subset B(S)$,
where $S-\bm{\ell}$ is the translate of $S$ over the vector $-\bm{\ell}$.

In the following theorem we heavily use $B(\Delta^{\circ})$, where $\Delta$ is
the Newton polytope of $P$.

######  Theorem 4.1.

Let $p$ be a prime not dividing $P_{0}$. Choose $\rho$ such that
$r-1\leq\rho-\lceil\rho/p\rceil$. For every ${\bf u}\in(\rho
B(\Delta^{\circ}))_{\mathbb{Z}}$ consider the power series

$\frac{{\bf x}^{{\bf u}}}{P({\bf x})^{\rho}}=\sum_{{\bf k}}a_{{\bf u},{\bf
k}}{\bf x}^{{\bf k}},$

where $a_{{\bf u},{\bf k}}\in{\mathbb{Z}}[1/P_{0}]$. Then there exist
$\lambda_{{\bf u},{\bf v},\bm{\ell}}\in{\mathbb{Z}}$ with ${\bf u},{\bf
v}\in(\rho B(\Delta^{\circ}))_{\mathbb{Z}}$ and
$\bm{\ell}\in[0,1,\ldots,p-1]^{n}$ such that for every ${\bf k}$ we have

$a_{{\bf u},p{\bf k}+\bm{\ell}}\equiv\sum_{{\bf v}\in(\rho
B(\Delta^{\circ}))_{\mathbb{Z}}}\lambda_{{\bf u},{\bf v},\bm{\ell}}a_{{\bf
v},{\bf k}}({\rm mod}\ p^{r}).$

###### Proof.

In this proof we shall use the convention that for any two sets
$X,Y\subset{\mathbb{R}}^{n}$ we denote $X+Y=\\{{\bf x}+{\bf y}|{\bf x}\in
X,{\bf y}\in Y\\}$. Furthermore $kX$ with $k$ a positive integer denotes the
repeated sum of $k$ copies of $X$.

We apply (3) in Proposition 2.3 with $f=P$ and $\mu=B(\Delta^{\circ})$. For
any $\bm{\ell}\in[0,1,\ldots,p-1]^{n}$ and any ${\bf u}\in\rho
B(\Delta^{\circ})$ we have

$\mathscr{C}_{p}\left(\frac{{\bf x}^{{\bf u}-\bm{\ell}}}{P({\bf
x})^{\rho}}\right)\equiv\sum_{m=0}^{r-1}p^{m}\binom{\lceil\rho/p\rceil+m-1}{\rho}\frac{Q_{m}({\bf
x})}{P({\bf x})^{m+\lceil\rho/p\rceil}}({\rm mod}\ p^{r}).$

with

$Q_{m}({\bf x})=\mathscr{C}_{p}\left({\bf x}^{{\bf u}-\bm{\ell}}G({\bf
x})^{m}P({\bf x})^{p\lceil\rho/p\rceil-\rho}\right).$

In the present context we have that $G({\bf x})$ is $P({\bf x}^{p})-P({\bf
x})^{p}$ divided by $p$. The terms in ${\bf x}^{{\bf u}-\bm{\ell}}G({\bf
x})^{m}P({\bf x})^{p\lceil\rho/p\rceil-\rho}$ have their support in

(6) $\\{{\bf
u}\\}+\\{-\bm{\ell}\\}+pm\Delta+(p\lceil\rho/p\rceil-\rho)\Delta\subset\\{-\bm{\ell}\\}+p(m+\lceil\rho/p\rceil)B(\Delta^{\circ}).$

Here we applied Lemma 2.1 with $\mu=B(\Delta^{\circ}),{\bf u}\in\rho\mu$ and
$\Delta\subset\overline{\mu}$. The operator $\mathscr{C}_{p}$ selects the
terms whose support belongs to the set (6) and which are divisible by $p$.
Since the components of $\bm{\ell}$ are all $\leq p-1$ such support vectors
have non-negative coordinates. By the main property of box closures the points
in $\\{-\bm{\ell}\\}+p(m+\lceil\rho/p\rceil)B(\Delta^{\circ})$ with non-
negative coordinates is contained in
$p(m+\lceil\rho/p\rceil)B(\Delta^{\circ})$.

Hence we conclude that the support of $Q_{m}$ is in
$(m+\lceil\rho/p\rceil)B(\Delta^{\circ})$. Note that by our choice of $\rho$
we have $m+\lceil\rho/p\rceil\leq r-1+\lceil\rho/p\rceil\leq\rho$.
Consequently there exist coefficients $\lambda_{{\bf u},{\bf
v},\bm{\ell}}\in{\mathbb{Z}}$ such that

$\mathscr{C}_{p}\left(\frac{{\bf x}^{{\bf u}-\bm{\ell}}}{P({\bf
x})^{\rho}}\right)\equiv\sum_{{\bf v}\in(\rho
B(\Delta^{\circ}))_{\mathbb{Z}}}\lambda_{{\bf u},{\bf v},\bm{\ell}}\frac{{\bf
x}^{{\bf v}}}{P({\bf x})^{\rho}}({\rm mod}\ p^{r}).$

The theorem follows by taking the power series expansion of both sides and
then selection of the coefficient of ${\bf x}^{{\bf k}}$ on both sides. ∎

######  Corollary 4.2.

Let $a_{{\bf k}}$ be as in the beginning of this section and suppose that the
numerator $Q$ has its support in $B(\Delta^{\circ})$. Let $d_{i}$ be the
degree of $P$ in the variable $x_{i}$ for $i=1,\ldots,n$. Let $r$ be a
positive integer and $\rho$ an integer such that
$r-1\leq\rho-\lceil\rho/p\rceil$. Then there is a $p$-linear scheme, with at
most $\rho^{n}d_{1}\cdots d_{n}$ states, which generates the numbers $a_{{\bf
k}}({\rm mod}\ p^{r})$.

###### Proof.

We apply Theorem 4.1. Note that the lattice points in $\rho B(\Delta^{\circ})$
are contained in the box $[0,\rho d_{1}-1]\times\cdots\times[0,\rho d_{n}-1]$.
Hence the number of lattice points is bounded above by $\rho^{n}d_{1}\cdots
d_{n}$. One easily checks that $Q/P$ can be written as ${\mathbb{Z}}$-linear
combination of ${\bf x}^{{\bf u}}/P^{\rho}$ with ${\bf u}\in(\rho
B(\Delta^{\circ}))_{\mathbb{Z}}$. ∎

## References

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  * [2] F. Beukers, M. Vlasenko, _Dwork crystals II_ , Int. Math. Res. Notices, 2021 (2021), 4427–4444
  * [3] F. Beukers, M. Vlasenko, _Dwork crystals III: from excellent Frobenius lifts towards supercongruences_ , arXiv:math/2105.14841(2021), online: https://doi.org/10.1093/imrn/rnad101
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  * [6] E. Rowland and R. Yassawi, _Automatic congruences for diagonals of rational functions._ Journal de Théorie des Nombres de Bordeaux, 27(1), 2015, 245–288.
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  * [8] K. Samol, D. van Straten, _Dwork congruences and reflexive polytopes_ , arXiv: 0911.0797.
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http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/


# A Stock Market Model Based on CAPM and Market Size

Andrey Sarantsev, Blessing Ofori-Atta, Brandon Flores  Department of
Mathematics and Statistics, University of Nevada, Reno<EMAIL_ADDRESS>
###### Abstract.

We introduce a new system of stochastic differential equations which models
dependence of market beta and unsystematic risk upon size, measured by market
capitalization. We fit our model using size deciles data from Kenneth French’s
data library. This model is somewhat similar to generalized volatility-
stabilized models in (Pal, 2011; Pickova, 2013). The novelty of our work is
twofold. First, we take into account the difference between price and total
returns (in other words, between market size and wealth processes). Second, we
work with actual market data. We study the long-term properties of this system
of equations, and reproduce observed linearity of the capital distribution
curve. In the Appendix, we analyze size-based real-world index funds.

###### Key words and phrases:

JEL Classification: C58, G17. Capital Asset Pricing Model; Stochastic
differential equations; Capital distribution curve; Stochastic stability;
Market weight

##  1\. Introduction

###  1.1. Size effect and the Capital Asset Pricing Model

The size of a stock is measured by its market capitalization, or market cap:
current stock price multiplied by the number of shares. For a stock portfolio,
its market cap is defined as weighted sum of market caps of constituent
stocks, with weights equal to the portfolio weights. The size is a very
important fundamental characteristic of a stock or a portfolio.

It is observed that small stocks have higher returns but higher risk than
large stocks. An explanation is that small companies are in their dynamic
growth phase and they have higher growth potential relative to large mature
companies, but small companies are more vulnerable to failure and bankruptcy.
Some researchers claim that even after adjusting for risk, small stocks have
higher returns than large stocks. This adjustment can be made rigorous within
the framework of the Capital Asset Pricing Model (CAPM). Take a portfolio of
stocks with total returns (including dividends, not inflation-adjusted) $Q(t)$
during time $t$. Here, we operate in a discrete-time setting. Compare it with
risk-free returns from short-term Treasury bills $R(t)$. An investor deserves
premium reward for taking risk and investing in stocks rather than in safe
Treasury bills. We calculate this equity premium $P(t)$ by subtracting
$P(t)=Q(t)-R(t)$. Next, we compute this equity premium $P_{0}(t)$ for a market
portfolio, used as a benchmark. An example of such benchmark is the Standard &
Poor (S&P) 500, a widely used benchmark for large U.S. stocks. Run a linear
regression:

(1) $P(t)=\alpha+\beta P_{0}(t)+\varepsilon(t).$

The parameter $\beta$ shows market exposure, how much risk the portfolio is
exposed to because of fluctuations in the benchmark. The parameter $\alpha$
shows excess return, how much can one earn from this portfolio on top of this
market return. They are often called by their Greek names: beta and alpha. The
residual $\varepsilon(t)$ is called unsystematic risk which can be eliminated
by diversification. According to the CAPM, $\alpha=0$ and the only risk which
deserves rewards is the systematic risk (due to market exposure) since other
risk can be diversified away. The CAPM was proposed in the classic article
[25]. Subsequent research cast doubt on the consistency of CAPM with actual
market data. In particular, [4] found that taking a portfolio of small stocks
generates positive $\alpha$. That is, small stocks have higher returns than
large stocks even after adjusting for market exposure (which is greater than 1
for small stocks). Subsequent classic article [7] confirmed this. Further
research on the size effect can be found in [24] and [26] and references
therein. See also critique of CAPM in [8].

###  1.2. Our model

We study dependence of $\alpha,\beta,\sigma$ (the standard deviation of
$\varepsilon(t)$) on the size, measured by the market cap $S(t)$, or, more
precisely, by relative size to $S_{0}(t)$:

(2) $C(t)=\ln\frac{S_{0}(t)}{S(t)}.$

We would like to find functions $\alpha,\beta,\sigma$ of $C$ such that for
standardized white noise terms $Z(t)$, with $\mathbb{E}[Z(t)]=0$ and
$\mathbb{E}[Z^{2}(t)]=1$:

(3) $P(t)=\alpha(C(t))+\beta(C(t))P_{0}(t)+\sigma(C(t))Z(t).$

This allows us to quantify how exactly $\alpha$ and $\beta$ (and $\sigma$ the
standard deviation of unsystematic risk) depend on the relative size measure.
We then consider a version of the equation (3) in which equity premia are
replaced by price returns, that is, returns due to price changes (or,
equivalently, market cap changes). That is, we replace $P(t)$ and $P_{0}(t)$
with $\ln(S(t+1)/S(t))$ and $\ln(S_{0}(t+1)/S_{0}(t))$, ignoring for now risk-
free rate and dividends. This gives us:

(4)
$\ln\frac{S(t+1)}{S(t)}=\alpha(C(t))+\beta(C(t))\ln\frac{S_{0}(t+1)}{S_{0}(t)}+\sigma(C(t))Z(t)$

with $C(t)$ from (2). This equation (4) includes only market caps of benchmark
$S_{0}(t)$ and the portfolio $S(t)$. This time series equation or its
continuous-time version, a stochastic differential equation, allows us to
model $S(t)$ and $S_{0}(t)$ separately from dividends and risk-free returns.
On top of this, we add the equation (3). Surprisingly, market data gives us
almost the same functions $\alpha,\beta,\sigma$ in (3) and (4), but with
differing coefficients. Moreover, the white noise terms in (3) and (4) are
almost perfectly (more than 99%) correlated. This does not follow from any
theoretical considerations, and seems simply a piece of good luck which
simplifies analysis.

We also adapt (3) and (4) for continuous time: Equity premium $P(t)$ becomes
then $\mathrm{d}\ln V(t)$, where $V(t)$ is the wealth process adjusted for
risk-free returns. More precisely, $V(t)=U(t)/U^{*}(t)$, where $U(t)$ is the
wealth accumulated from investing $U(0)=1$ in stock portfolio and reinvesting
dividends, while $U_{*}$ is a similar wealth process from investing in
Treasury bills. Then (3) takes the form

(5) $\mathrm{d}\ln V(t)=\alpha(C(t))\,\mathrm{d}t+\beta(C(t))\,\mathrm{d}\ln
V_{0}(t)+\sigma(C(t))\,\mathrm{d}W(t),$

where $V_{0}$ is the adjusted wealth process for the benchmark, and $W$ is a
Brownian motion: A real-valued continuous process with
$W(t)-W(s)\sim\mathcal{N}(0,t-s)$ independent of $W(u),\,0\leq u\leq s$, for
all $0\leq s<t$. This Brownian motion can be viewed as a zoomed out random
walk with very small but very frequent steps. For simplicity, we assume that
$\ln V_{0}(t)$ is also a Brownian motion with positive drift (which captures
the tendency of long-term stock returns to be greater than risk-free returns).
Although equity premia have heavy tails and thus are not well-described by the
Gaussian distribution, the Brownian motion provides a simple first
approximation. Similarly, equation (4) becomes

(6) $\mathrm{d}\ln S(t)=\alpha(C(t))\,\mathrm{d}t+\beta(C(t))\,\mathrm{d}\ln
S_{0}(t)+\sigma(C(t))\,\mathrm{d}W(t),$

with $C(t)$ from (2). By the above remark, we can assume that the driving
Brownian motion $W$ from (5) and (6) is the same.

###  1.3. Dependence upon the size measure

What is the dependence of market exposure $\beta$, excess return $\alpha$, and
standard deviation of unsystematic risk $\sigma$ upon relative size?

Our statistical analysis does not yield conclusive results. The white noise
tests for $Z$ unfortunately fail. Thus we cannot claim that a model (3), (4)
passes goodness of fit tests. The most resonable guess for
$\alpha,\beta,\sigma$ seems to be:

(7) $\displaystyle\alpha(c),\ \beta(c),\
\sigma(c)=\begin{cases}\alpha_{+}c,\hskip 32.72049pt1+\beta_{+}c,\
\sigma_{+}c,\,&c\geq c_{+}\\\ -\alpha_{-}|c|^{1/2},\ 1+\beta_{-}c,\
\sigma_{-}|c|^{1/2},\,&c\leq-c_{-}.\end{cases}$

Here, $\alpha_{\pm},\beta_{\pm},\sigma_{\pm},c_{\pm}$ are positive constants.
Our data analysis does not allow us to suggest these functions in a
neighborhood of zero. This is due to the fact that observed $C(t)$ in our data
do not come very close to zero.

###  1.4. Stochastic Portfolio Theory

The continuous-time version is useful because we can use stochastic calculus
and immerse these models in Stochastic Portfolio Theory (SPT). This is a
framework for stock market modeling which does not depend on particular
models. It suggests overweighing small stocks and continuous rebalancing. This
means investing in small stocks in proportion greater than their market cap
would dictate. One example of this is taking the equal-weighted portfolio. The
central result of SPT: under mild conditions, diversity (no stock dominates
the entire market) and sufficient intrinsic volatility, such portfolios
outperform the market portfolio, which invests in each stock in proportion to
its market cap; see [1], [10], [11]. This theory has solid theoretical basis,
and is consistent with observed data. For references, see the book [9] and a
more recent survey [12]. SPT is based on the same observation as above: small
stocks have higher return and risk than large stocks.

Although we mentioned above that SPT is model-independent, there are some SPT
models which attempt to capture this observation: competing Brownian
particles, where logarithms of market caps evolve as Brownian motions with
drift and diffusion coefficients dependent on their current ranks relative to
other particles, [2], [3], [13]; their generalizations with jumps, or with
dependence on both name and rank (so-called second-order models): see articles
[3], [5], [21]; volatility-stabilized models, where $\ln S(t)$ are modeled by
stochastic differential equations (SDE) with volatility inversely proportional
to $S(t)$, [18], and their generalizations, [20]. As the number of stocks
tends to infinity, the limiting behavior of competing Brownian particles and
volatility-stabilized models is studied in [6], [22], [23].

In this article, we recognize the difference between price and total returns,
and model them separately as (3) and (4) for discrete time or (5) and (6) for
continuous time. SPT is much more developed for continuous-time diffusive
models based on SDE than for discrete-time, see [19]. Thus it is reasonable to
switch to continuous time in (5) and (6).

Stochastic Portfolio Theory deals with diversification benefits in the form of
excess growth rate, and functionally generated portfolios. The standard
assumption in SPT is that there are no dividends; that is, price and total
returns are the same. But here this is no longer true. To model separately
price returns (which drive the market capitalization processes) and total
returns (which drive the wealth processes) requires an extension of the SPT,
in particular the concept of functionally generated portfolios. This is left
for future research.

###  1.5. Data analysis

We take real U.S. market data from Kenneth French’s Data Library online:
market cap, price and total returns for equal-weighted portfolios made from
size deciles (stocks split into top 10%, next 10%, etc. according to their
size). This library contains data processed from original raw data from the
Center for Research in Securities Prices (CRSP) at the University of Chicago,
July 1926–June 2020 (84 years), monthly data.

We study models of $n$ portfolios and the benchmark with market caps
$S_{0},\ldots,S_{n}$, relative size measures $C_{0},\ldots,C_{n}$, and wealth
processes $V_{0},\ldots,V_{n}$. The corresponding Brownian motions
$W_{1},\ldots,W_{n}$ from (6) and (5) (recall that these two equations have
the same Brownian motions) are assumed to be i.i.d. for simplicity (although
our analysis shows this to be inconsistent with actual data). Consider market
weights:

$\mu_{i}(t)=\frac{S_{i}(t)}{S_{0}(t)+\ldots+S_{n}(t)},\,i=0,\ldots,n,$

representing the proportion of the $i$th stock in the overall market. Small
stocks have smaller market weights. The market weight vector
$\mathbf{\mu}=(\mu_{0},\ldots,\mu_{n})$ is a Markov process on the
$n$-dimensional simplex $\triangle_{n}$. If this market weight vector
converges to a unique stationary distribution as $t\to\infty$ in the total
variation norm (see definitions in Section 4), then we call the system stable.
In this article, we state and prove that our market model is stable under
certain conditions on $\alpha,\beta,\sigma$. We also investigate whether
collisions: $S_{i}(t)=S_{j}(t)$ happen (when small stocks grow and overtake
large stocks). Finally, at each time rank weights from top to bottom:

$\mu_{(0)}(t)\geq\ldots\geq\mu_{(n)}(t).$

The capital distribution curve is the plot of ranked weights vs their ranks on
the double logarithmic scale:

$\left(\ln k,\ln\mu_{(k)}(t)\right),\,k=0,\ldots,n.$

For the actual market data, this plot is linear, except at the endpoint. See
[9, Figure 5.1] for the capital distribution curve for the CRSP stock universe
for 8 days, chosen to be the last days (December 31) of the eight decades –
December 31, 1929; December 31, 1939; …December 31, 1999. Strikingly, these 8
curves almost completely coincide. In other words, this curve is stable in
time. Previously mentioned competing Brownian particles and volatility-
stabilized models reproduce this feature, see [6] and [18] respectively. In
this article, we establish this property for our model using both theoretical
analysis and simulations.

###  1.6. Our contributions

We propose a new model within SPT which is consistent with CAPM and captures
the size effect. This model is consistent with long-term market data (although
not fully consistent) and captures its important features. We study its
properties: long-term stability, collisions of particles, market weights, and
the capital distribution curve.

###  1.7. Organization of the article

In Section 2, we describe our data in detail and analyze it. This provides a
motivation for continuous-time modeling. In Section 3, we introduce a system
of SDE as in (6) and (5), and prove existence and uniqueness of the solution.
We also discuss applicability of SPT, since now size and wealth processes are
different. In Section 4, we show stability under certain conditions and see
how they are compared with data analysis in Section 2. In Section 5, we
replicate the linear capital distribution curve. Finally, Section 6 is devoted
to conclusions and suggestions for future research. The Appendix contains some
data analysis for existing exchange-traded funds based on size. The code and
data can be found on GitHub: asarantsev/CAPM-SPT.

###  1.8. Acknowledgements

We thank the Department of Mathematics and Statistics at our University of
Nevada, Reno, for welcoming and supportive atmosphere and for fostering
research collaboration between faculty and students (undergraduate and
graduate). We thank the referees for useful remarks and positive responses.

##  2\. Data Analysis

###  2.1. Data description

Our main data source, as mentioned in the Introduction, is the CRSP database
at the University of Chicago, taken from Kenneth French’s online data library.
We take from this library price and total monthly returns for equally-weighted
portfolios made of stock in each decile from July 1926 to June 2020. We
analyze only the top 8 deciles, corresponding to large-cap, mid-cap, and
small-cap stocks. The two bottom deciles are micro-cap stocks which we
exclude. Deciles are created based on the market cap at the end of each June.
After a year, these deciles are reconstituted. The data library also has
average market capitalizations for the end of each month for each decile.
Risk-free monthly returns are computed as $R(t)=\ln(1+r(t)/12)$, where $r(t)$
are monthly data for short-term Treasury bills taken from the Federal Reserve
Economic Data (FRED) website: January 1934–June 2020 series TB3MS and July
1926–December 1933 (discontinued) series M1329AUSM193NNBR.

Stock and portfolio returns can be computed in two ways: arithmetic $A$ and
geometric $G$, which are related as follows: $G=\ln(1+A)$. Arithmetic returns
are quoted regularly for their practicality in computing portfolio returns
such as in the data library which we used as our data source. The arithmetic
return of a portfolio is equal to the weighted average of arithmetic returns
of constituent stocks, but here we convert arithmetic returns to their
geometric versions according to the above formula.

The advantage of geometric over arithmetic returns for our research is
apparent when combining compound interest rates. For example, 20% and 30%
arithmetic returns combined gives 56% returns, whereas geometric returns
combine for 50% as expected.

Our time unit is equal to a month, of total $T=12\cdot(2020-1926)=1128$
months, $t=0$ corresponding to June 1926, $t=T$ corresponding to June 2020.

(a) $\beta_{k}(n)-1$ vs $C_{k}(n)$

(b) $(\beta_{k}(n)-1)/C_{k}(n)$ vs $C_{k}(n)$

Figure 1. Beta $\beta_{k}(n)$ vs $C_{k}(n)$ for top-decile benchmark

For a decile $k$ (with $k=1$ for the top decile, $k=8$ for the bottom decile),
its average market cap at end of month $t$ is denoted by $S_{k}(t)$, the price
returns are $Q_{k}(t)$, and the equity premium (total returns including
dividends minus risk-free returns) is $P_{k}(t)$.

###  2.2. Beta analysis for price returns

Split $T=1128$ months into $N=47$ two-year, $K=24$-month time periods. For
each period $n=1,\ldots,N$ and each decile $k=1,\ldots,8$, except the top one,
which we use as the benchmark (this top decile roughly corresponds to Standard
& Poor 500 constituent stocks), regress

(8)
$Q_{k}(t)=\alpha_{k}(n)+\beta_{k}(n)Q_{1}(t)+\delta_{k}(t),\,k=2,\ldots,8,$

where $t$ is in this period, and $\alpha_{k}(n),\beta_{k}(n)$ are intercept
and slope (excess return and market exposure), found using ordinary least
squares; and $\delta_{k}(t)$ are residuals. Thus we compute beta
$\beta_{k}(n)$ for each decile from 2nd to 8th and each of 47 two-year
periods. For size measure of the $k$th decile vs top decile, take

$C_{k}(n)=\ln\frac{S_{1}(Kn)}{S_{k}(Kn)},$

where $S_{1}(Kn)$ and $S_{k}(Kn)$ are average market capitalizations for the
beginning of $n$th period. Then we plot $\beta_{k}(n)-1$ vs $C_{k}(n)$ and
$(\beta_{k}(n)-1)/C_{k}(n)$ in Figure 1. We plot $\beta_{k}(n)-1$ instead of
$\beta_{k}(n)$ since our benchmark for beta is $1$: The beta for the top
decile (which coincides with the benchmark) is $1$.

###  2.3. Model suggestions for price returns

From Figure 1 (A), we get the suggestion that $\beta_{k}(n)-1$ is proportional
to $C_{k}(n)$. This is confirmed by Figure 1 (B). This transformation helps
make variance constant. We still cannot claim that these normalized quantities
are i.i.d. white noise, since they fail white noise tests. Still, let us
extract the trend in $\beta_{k}(n)$ by replacing it with $1+\gamma C_{k}(n)$
for some coefficient. Next, comparing with (8), consider the residual
$Q_{k}(t)-(1+\gamma C_{k}(n))Q_{1}(t)$. We make it dependent on $n$ (the
overall two-year period), not individual months $t$ in this period, by taking
the sum of geometric price returns $Q_{k}(t)$ over $t$ in this two-year
period. This sum is equal to overall price returns $\overline{Q}_{k}(t)$ in
this two-year period. Then we let:

(9) $\varepsilon_{k}(n):=\overline{Q}_{k}(n)-(1+\gamma
C_{k}(n))\overline{Q}_{1}(n),\ k=2,\ldots,8;\ n=1,\ldots,N.$

(a) $\varepsilon_{k}(n)$ vs $C_{k}(n)$

(b) $\varepsilon_{k}(n)/C_{k}(n)$ vs $C_{k}(n)$

Figure 2. Residual $\varepsilon_{k}(n)$ vs $C_{k}(n)$ for top decile benchmark

We then plot $\varepsilon_{k}(n)$ vs $C_{k}(n)$ in Figure 2 (A), together with
$\varepsilon_{k}(n)/C_{k}(n)$ vs $C_{k}(n)$ in Figure 2 (B). We see again in
Figure 2 (A) that $\varepsilon_{k}(n)$ depends on $C_{k}(n)$ linearly, and in
Figure 2 (B) that the variance becomes constant. Again, we cannot claim that
$\varepsilon_{k}(n)/C_{k}(n)$ are i.i.d. since white noise tests fail. But if
we model this as i.i.d. $\mathcal{N}(\mu,\rho^{2})$, we get:

(10) $\frac{\varepsilon_{k}(n)}{C_{k}(n)}=\mu+\rho Z_{k}(n),$

where $Z_{k}(n)$ are i.i.d. standard normal random variables. A generalization
of this could be:

(11) $(Z_{2}(n),\ldots,Z_{8}(n))\sim\mathcal{N}_{7}(\mathbf{0},\Sigma)$

i.i.d. multivariate normal with mean zero vector and (not identity) covariance
matrix with units on the main diagonal (that is, $\mathbb{E}[Z_{k}(n)]=0$ and
$\mathbb{E}[Z_{k}^{2}(n)]=1$). Combining (9) and (10), we get:

(12) $Q_{k}(n)=(1+\gamma C_{k}(n))Q_{1}(n)+\mu C_{k}(n)+\rho
C_{k}(n)Z_{k}(n).$

We can estimate $\gamma=0.0045,\mu=0.0069,\rho=0.052$. This gives us (7), the
top row.

(a) $\beta_{k}(n)-1$ vs $C_{k}(n)$

(b) $\varepsilon_{k}(n)/\sqrt{|C_{k}(n)}$ vs $C_{k}(n)$

Figure 3. 8th decile (bottom) benchmark

###  2.4. Bottom decile as benchmark

If we repeat this analysis in previous subsection with benchmark 8th decile
(which in our research is the bottom decile since we ignore the 9th and 10th
deciles), we get: For $\beta_{k}(n)$ in (8), we plot $\beta_{k}(n)-1$ vs
$C_{k}(n)$ in Figure 3 (A). In this case, dependence is also linear. Next,
take $\gamma$, the mean of these quantities. Create residuals similarly to
(9):

(13)
$\varepsilon_{k}(n)=Q_{k}(n)-(1+\gamma\sqrt{|C_{k}(n)|})Q_{8}(n),\,k=1,\ldots,7,\,n=1,\ldots,N.$

The plot of these residuals vs $C_{k}(n)$, normalized by dividing by
$\sqrt{|C_{k}(n)|}$, is shown in Figure 3 (B). Making white noise test for
$\varepsilon_{k}(n)$, we fail to reject this hypothesis. Assume
$\varepsilon_{k}(n)\sim\mathcal{N}(\mu,\rho^{2})$. Combining this with (13),
we get:

(14)
$\displaystyle\begin{split}Q_{k}(n)&=(1+\gamma\sqrt{|C_{k}(n)|})Q_{8}(n)+\mu\sqrt{|C_{k}(n)|}+\rho\sqrt{|C_{k}(n)|}Z_{k}(n),\\\
k&=1,\ldots,7,\quad n=1,\ldots,N;\\\
Z_{k}(n)&=(\varepsilon_{k}(n)-\mu)/\rho\sim\mathcal{N}(0,1)\quad\mbox{i.i.d.}\end{split}$

This verifies (7), the bottom row. From (14), we see that linear functions
from (12) are not the only possible and reasonable functions for $\beta$. In
Section 4, we state and prove results for general $\alpha,\beta,\rho$. Our
coefficients are $\gamma=0.12,\mu=0.0055,\rho=0.090$.

###  2.5. Data analysis for equity premia

Repeating similar analysis for equity premia instead of price returns, we get
similar functions for both cases: top and bottom deciles as benchmarks.
Moreover, point estimates for price returns and equity premia are close;
$\gamma=0.045,\mu=0.0017,\rho=0.052$ for the top decile benchmark, and
$\gamma=0.12,\mu=0.0024,\rho=0.088$ for the 8th decile benchmark. Thus
functions $\beta$ and $\sigma$ for price returns and equity premia are the
same. Unfortunately, for $\alpha$ this is not true. We will have separate
functions for price returns and equity premia. As noted in Section 1, white
noise terms $Z_{k}(n)$ for price returns and equity premia are almost
perfectly correlated, since the Pearson correlation coefficient is greater
than 99%. However, price returns and equity premia for the benchmark are not
perfectly correlated. Thus noise terms for size and wealth processes are
different.

###  2.6. Conclusions

Data analysis in previous subsections implies that excess return for small
stocks is positive, and market exposure for small stocks is greater than $1$.
Large stocks have negative excess return and market exposure less than $1$. In
other words, small stocks are riskier than larger stocks, but they have higher
returns even after adjusting for market exposure. We will accommodate this for
continuous time in the next section.

##  3\. A Continuous-Time Model

###  3.1. Model description

Take a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ with the
filtration satisfying the usual conditions (each $\mathcal{F}_{t}$ contains
all $\mathbb{P}$-null sets; that is, $A\in\mathcal{F}_{t}$ with
$\mathbb{P}(A)=0$ and $B\subseteq A$ implies $B\in\mathcal{F}_{t}$; and the
filtration is right-continuous, that is,
$\mathcal{F}_{t}=\cap_{s>t}\mathcal{F}_{s}$ for all $t\geq 0$). All stochastic
processes $X=(X(t),\,t\geq 0)$ below are adapted, that is, $X(t)$ is
$\mathcal{F}_{t}$-measurable for every $t\geq 0$. We say that a stochastic
process $B=(B(t),\,t\geq 0)$ is a standard Brownian motion if
$B(t)-B(s)\sim\mathcal{N}(0,t-s)$ is independent of $\mathcal{F}_{s}$ for
every $0\leq s<t$.

Replace white noise terms $Z_{k}(n)$ in (12) or (14) by standard Brownian
motions $W_{k}(t)$, or, more exactly, its differential $\mathrm{d}W_{k}(t)$.
Replace price returns $Q_{k}(n)$ by $\mathrm{d}\ln S_{k}(t)$. Recall that
$S_{k}(t)$ is the average market capitalization of the $k$th portfolio at time
$t$. Replace equity premia $P_{k}(n)$ with $\mathrm{d}\ln V_{k}(t)$, where
$V_{k}(t)$ is the wealth (including reinvested dividends) of the $k$th decile,
starting from $V_{k}(0)=1$, divided by wealth similarly computed from risk-
free returns. Set three functions: $\alpha,\beta,\sigma$ of $C_{k}(t)$ (alpha,
beta, and standard deviation of unsystematic risk). Set a separate function
$\alpha_{*}$ of $C_{k}(t)$ for equity premia instead of price returns. We
shall not set separate functions for $\beta$ and $\sigma$ since from our data
analysis we found that these functions are the same for price returns and
equity premia. We have $n$ portfolios indexed by $1,\ldots,n$, and the
benchmark indexed with $0$ (that is, total $n+1$ portfolios). We write
separate functions for price returns and equity premia (which we denote by
asterisks). Consider the following system of SDE:

(15) $\mathrm{d}\ln
S_{k}(t)=\alpha(C_{k}(t))\,\mathrm{d}t+\beta(C_{k}(t))\,\mathrm{d}\ln
S_{0}(t)+\sigma(C_{k}(t))\,\mathrm{d}W_{k}(t),$ (16) $\mathrm{d}\ln
V_{k}(t)=\alpha_{*}(C_{k}(t))\,\mathrm{d}t+\beta(C_{k}(t))\,\mathrm{d}\ln
V_{0}(t)+\sigma(C_{k}(t))\,\mathrm{d}W_{k}(t),$

where $C_{k}(t)$ is the relative size measure defined in the Introduction:

(17) $C_{k}(t)=\ln\frac{S_{0}(t)}{S_{k}(t)}.$

We also model $(S_{0},V_{0})$ using two-dimensional geometric Brownian motion:
$(\ln S_{0},\ln V_{0})$ has drift vector $(g_{S},g_{V})$ and covariance matrix

$\mathbf{\Sigma}_{(S,V)}=\begin{bmatrix}\sigma_{S}^{2}&\rho_{0}\sigma_{S}\sigma_{V}\\\
\rho_{0}\sigma_{S}\sigma_{V}&\sigma_{V}^{2}\end{bmatrix}$

That is, for $t>s$ we have: $\ln S_{0}(t)-\ln
S_{0}(s)\sim\mathcal{N}(g_{S}(t-s),\sigma_{S}^{2}(t-s))$ and $\ln V_{0}(t)-\ln
V_{0}(s)\sim\mathcal{N}(g_{V}(t-s),\sigma^{2}_{V}(t-s))$; here, $\rho_{0}$ is
the correlation between these two increments. Thus

(18) $\displaystyle\begin{split}\ln S_{0}(t)&=\ln
S_{0}(0)+g_{S}t+\sigma_{S}W_{S}(t),\\\ \ln V_{0}(t)&=\ln
V_{0}(0)+g_{V}t+\sigma_{V}W_{V}(t),\end{split}$

where $W_{S}$ and $W_{V}$ are (correlated) standard Brownian motions. We
assume $W_{1},\ldots,W_{n}$ are independent of $(B_{S},B_{V})$. Note that
$W_{1},\ldots,W_{n}$ can be dependent of each other standard Brownian motions.
We assume that $W=(W_{1},\ldots,W_{n})$ is an $n$-dimensional Brownian motion
with zero drift vector and covariance matrix $\mathbf{\Sigma}_{W}$ with units
on the main diagonal.

######  Definition 1.

For functions $\alpha,\beta,\sigma:\mathbb{R}\to\mathbb{R}$, real numbers
$g_{S},g_{V}$, $2\times 2$ covariance matrix $\mathbf{\Sigma}_{(S,V)}$, and
$n\times n$ correlation matrix $\mathbf{\Sigma}_{W}$, the system of equations
(15), (16), (17), (18) is called a CAPM-size market model of $N+1$ portfolios,
indexed by $0,\ldots,N$. The process $S_{k}$ is called market size, or market
cap, of the $k$th portfolio; and the process $V_{k}$ is called the wealth
process for this $k$th portfolio. The portfolio indexed by $k=0$ is called the
benchmark. The functions $\alpha$ and $\beta$ are called by their Greek names.
The function $\sigma$ is called the standard error (of the diversifiable
risk).

###  3.2. Existence and uniqueness

We can rewrite (15) using (17):

(19)
$\mathrm{d}C_{k}(t)=-\alpha(C_{k}(t))\,\mathrm{d}t+(1-\beta(C_{k}(t))\,\mathrm{d}\ln
S_{0}(t)+\sigma(C_{k}(t))\,\mathrm{d}W_{k}(t).$

######  Remark 1.

Each $k$th equation (15) is independent of other equations: Does not contain
$S_{l}$ for other $l=1,\ldots,n$. It depends only on $S_{k}$ and $S_{0}$.

######  Definition 2.

Define the explosion time as follows:

(20) $\mathcal{T}:=\mathcal{T}_{1}\wedge\ldots\wedge\mathcal{T}_{n},$

where $\mathcal{T}_{k}$ is the explosion time for (15), or, equivalently, for
(19).

On $[0,\mathcal{T})$, equations (15) and (16) have a unique solution.

######  Theorem 1.

(a) If $\alpha,\beta,\sigma,\alpha_{*}:\mathbb{R}\to\mathbb{R}$ are measurable
and locally bounded, there exists a unique strong solution to the system of
SDE (15), (16), (17), until the explosion time $\mathcal{T}$.

(b) If, in addition, we have the following linear bound:

$|\alpha(c)|+|\beta(c)|+|\sigma(c)|+|\alpha_{*}(c)|\leq K(1+|c|),$

then the explosion time is infinite: $\mathcal{T}=\infty$.

###### Proof.

(a) First, let us show existence and uniqueness in (19). The diffusion part in
this equation is given by
$\tilde{\sigma}(C_{k}(t))\,\mathrm{d}\tilde{W}_{k}(t)$, where

$\tilde{\sigma}^{2}(c):=\sigma^{2}_{S}(1-\beta(c))^{2}+\sigma^{2}(c),$

and $\tilde{W}_{k}$ is a standard Brownian motion which is a combination of
$W_{S}$ and $W_{k}$. The drift part is given by
$\tilde{\gamma}(C_{k}(t))\,\mathrm{d}t$, where

(21) $\tilde{\gamma}(c):=-\alpha(c)+g_{S}(1-\beta(c)).$

Thus we can rewrite (19) as follows:

(22)
$\mathrm{d}C_{k}(t)=\tilde{\gamma}(C_{k}(t))\,\mathrm{d}t+\tilde{\sigma}(C_{k}(t))\,\mathrm{d}\tilde{W}_{k}(t).$

These functions $\tilde{\gamma}$ and $\tilde{\sigma}$ are measurable, and we
apply [27] to prove strong existence and pathwise uniqueness of the solution
$C_{k}$ to this SDE, at least until the explosion time $\mathcal{T}_{k}$. Here
we use Remark 1: the system of $n$ equations $C_{1},\ldots,C_{n}$ consists of
$n$ independent one-dimensional SDE. Next, we can reconstruct
$S_{1},\ldots,S_{n}$ from $C_{1},\ldots,C_{n}$ and $S_{0}$:
$S_{k}(t)=S_{0}(t)e^{-C_{k}(t)}$. Since $S_{0}$ is well-defined for infinite
time horizon (as geometric Brownian motion), the strong solution $S_{k}$ for
(15) exists and is pathwise unique, too, at least until the explosion time
$\mathcal{T}_{k}$. Finally, strong existence and pathwise uniqueness for (16)
can be shown as follows. Let

$\mathcal{T}_{k,m}:=\inf\\{t\geq 0\mid|C_{k}(t)|=m\\},\,k=1,\ldots,n;\
m=1,2,\ldots$

Since $\alpha_{*},\beta,\sigma$ are locally bounded, they are bounded on
$[-m,m]$. We can rewrite (16) as

(23) $\mathrm{d}\ln
V_{k}(t)=\overline{\gamma}(C_{k}(t))\,\mathrm{d}t+\overline{\sigma}(C_{k}(t))\,\mathrm{d}\overline{W}_{k}(t),$

with $\overline{W}_{k}$ a standard Brownian motion, and

(24)
$\overline{\gamma}(c):=\alpha_{*}(c)+g_{V}\beta(c),\quad\overline{\sigma}^{2}(c):=\sigma^{2}_{V}\beta^{2}(c)+\sigma^{2}(c).$

These two functions from (24) are bounded on $[-m,m]$. Rewriting (23) as

$\ln
V_{k}(t)=\int_{0}^{t}\overline{\gamma}(C_{k}(s))\,\mathrm{d}s+\int_{0}^{t}\overline{\sigma}(C_{k}(s))\,\mathrm{d}\overline{W}_{k}(s),$

we see that (strong) solution until $t<\mathcal{T}_{k,m}$ is well-defined and
pathwise unique, because the integrals (both usual and stochastic) are finite.
As $m\to\infty$, we have: $\mathcal{T}_{k,m}\uparrow\mathcal{T}_{k}$ almost
surely. Thus (16) have a pathwise unique strong solution until
$\mathcal{T}_{k}$. Combining this with (20), we complete the proof of (a).

Next, (b) follows from (19) and (23) and estimates of linear growth for
$\tilde{\gamma}$ and $\tilde{\sigma}$:

$|\tilde{\gamma}(c)|+|\tilde{\sigma}(c)|\leq K_{*}(1+|c|),$

following from similar estimates for $\alpha,\alpha_{*},\beta,\sigma$. ∎

##  4\. Main Results

###  4.1. Long-term stability results

Recall the definition of market weights.

######  Definition 3.

For the market model in (15), we define market weights as follows:

$\mu_{i}(t)=\frac{S_{i}(t)}{S_{0}(t)+\ldots+S_{n}(t)},\,i=0,\ldots,n,\,t\geq
0.$

These market weights sum up to $1$ for every $t\geq 0$, and are in one-to-one
correspondence with $C_{1}(t),\ldots,C_{n}(t)$: There exists a bijection
$\Phi:\triangle_{n}\to\mathbb{R}^{n}$ such that

$\Phi:(m_{0},\ldots,m_{n})\mapsto\left(\ln\frac{m_{0}}{m_{1}},\ldots,\ln\frac{m_{0}}{m_{n}}\right).$

The process $\mathbf{C}=(C_{1},\ldots,C_{n})$, as its each component, is a
Markov process. The same is true for the market weight vector
$\mu=(\mu_{0},\ldots,\mu_{n})$.

######  Definition 4.

We call a probability measure $\pi_{\mu}$ on $\triangle_{n}$ a stationary
distribution if $\mu(0)\sim\pi_{\mu}$ implies $\mu(t)\sim\pi_{\mu}$ for all
$t\geq 0$. The market is called stable if this market weight vector has a
unique stationary distribution $\pi_{\mu}$, and when we start from another
initial distribution $\mu(0)$, then the distribution of $\mu(t)$ converges to
$\pi_{\mu}$ as $t\to\infty$ in the total variation (TV) norm:

$\sup\limits_{D\subseteq\triangle_{n}}|\mathbb{P}(\mu(t)\in
D\mid\mu(0)=x)-\pi_{\mu}(D)|\to 0\quad\mbox{as}\quad t\to\infty.$

We can similarly (and equivalently) define stability for the process
$\mathbf{C}=(C_{1},\ldots,C_{n})$, which is $\mathbb{R}^{n}$-valued. These
definitions of stability are equivalent, since there is a one-to-one
continuous mapping between $C$ and $\mu$.

######  Theorem 2.

Under assumptions of Theorem 1, suppose

(25)
$\varliminf\limits_{c\to\infty}\left[\alpha(c)+g_{S}(\beta(c)-1)\right]>0\quad\mbox{and}\quad\varlimsup\limits_{c\to-\infty}\left[\alpha(c)+g_{S}(\beta(c)-1)\right]<0.$

Then the market system is stable.

###### Proof.

We apply the method of Lyapunov functions. As a Lyapunov function, take the
following infinitely differentiable function $V:\mathbb{R}\to[0,\infty)$:

$V(c):=\begin{cases}|c|,\,c\geq 2;\\\ 0,\,c\leq 1.\end{cases}$

This function can be constructed by smoothing kernel and convolution. The
generator for each $C_{k}$ in (15) is given by

$\mathcal{L}f(c):=\tilde{\gamma}(c)f^{\prime}(c)+\frac{1}{2}\tilde{\sigma}^{2}(c)f^{\prime\prime}(c).$

Recall (21): $\varlimsup\limits_{c\to\infty}\tilde{\gamma}(c)<0$ and
$\varliminf\limits_{c\to-\infty}\tilde{\gamma}(c)>0$. Combining this with
(25), we get:

$\varlimsup\limits_{|c|\to\infty}\mathcal{L}V(c)<0.$

From the articles [16] and [17], we get tightness of each $C_{k}$. That is,
$\sup_{t\geq 0}\mathbb{P}(|C_{k}(t)|\geq c)\to 0$ as $c\to\infty$. The same is
true for the vector $\mathbf{C}=(C_{1},\ldots,C_{n})$. Combining this
observation with the following property: for each $i$,
$\mathbb{P}(a<C_{i}(t)<b\mid C_{i}(0)=x)>0,\quad x,a,b\in\mathbb{R}$, we
complete the proof. ∎

######  Remark 2.

We also have convergence

$\displaystyle\sup\limits_{g\in\mathcal{G}}$
$\displaystyle|\mathbb{E}[g(C_{k}(t))]-(\pi_{C},g)|\to 0,\,t\to\infty,$
$\displaystyle\mathcal{G}$
$\displaystyle:=\\{g:\mathbb{R}\to\mathbb{R}\mid\sup\limits_{z\in\mathbb{R}}\frac{|g(z)|}{1+|z|}<\infty\\}.$

In particular, we have $\mathbb{E}[C_{k}(t)]\to m_{C}$, where $m_{C}$ is the
mean of the distribution $\pi_{C}$.

######  Example 1.

If $\beta(c)=1+\gamma c$, $\alpha(c)=\mu c$, then
$\alpha(c)+g_{S}(\beta(c)-1)=(\mu+g_{S}\gamma)c$, and (25) is equivalent to

(26) $\Gamma:=\mu+g_{S}\gamma>0.$

######  Example 2.

From the data analysis in Section 2, let

$\alpha(c):=\begin{cases}\alpha_{+}|c|^{\gamma_{+}},\ c\geq 0;\\\
-\alpha_{-}|c|^{\gamma_{-}},\,c\leq 0;\end{cases}\
\beta(c):=1+\begin{cases}\beta_{+}c,\,c\geq 0;\\\ \beta_{-}c,\,c\leq
0;\end{cases}\ \sigma(c):=\begin{cases}\sigma_{+}|c|^{\gamma_{+}},\,c\geq
0;\\\ \sigma_{-}|c|^{\gamma_{-}},\,c\leq 0.\end{cases}$

Here, $\alpha_{\pm},\beta_{\pm},\gamma_{\pm},\sigma_{\pm}>0$. Let us find
conditions for (25): For the first condition, if $\gamma_{+}<1$, we have
$g_{S}\beta_{+}>0$; if $\gamma_{+}=1$, we have $\alpha_{+}+g_{S}\beta_{+}>0$;
if $\gamma_{+}>1$, we have $\alpha_{+}>0$. Similarly for the second condition
in (25). The actual estimates from Section 2 satisfy these conditions.

###  4.2. Hitting times

We wish to allow $S_{i}$ and $S_{0}$ to exchange ranks. That is, we want to
allow for $S_{i}(0)<S_{0}(0)$ but $S_{i}(t)>S_{0}(t)$ for some $t>0$, or vice
versa. This is consistent with real world market behavior, when portfolios
exchange ranks based on size. In terms of relative size measures $C_{i}$, we
wish that $C_{i}$ can move from positive half-line to negative half-line. In
particular, it must hit zero with positive probability.

This is not true for Example 26 if $\sigma(c)=\rho c$ for $\rho>0$. Indeed,
then $C_{i}$ is a geometric Brownian motion with drift $-\Gamma$ and thus
converges to $0$ almost surely as $t\to\infty$. Thus the limiting stationary
distribution is the delta measure at the origin, $\delta_{(0,\ldots,0)}$. The
corresponding limiting distribution for $\mu$, the market weight vector is
$\delta_{(1/(n+1),\ldots,1/(n+1))}$. If $\sigma(c)$ is bounded away from zero,
this changes the behavior of $\mathbf{C}$.

######  Theorem 3.

Under conditions of Theorem 2, assume

(27) $\sigma(c)\geq\sigma_{*}>0\quad\mbox{for all}\quad c\in\mathbb{R}.$

(a) For every $c\in\mathbb{R}$ with positive probability there exists a $t>0$
such that $C_{i}(t)=c$.

(b) The stationary distribution $\pi_{C}$ for
$\mathbf{C}=(C_{1},\ldots,C_{n})$ has support on $\mathbb{R}^{n}$. The
stationary distribution $\pi_{\mu}$ for $\mu$ has support on $\Delta_{n}$.

###### Proof.

Return again to the equation (22), which is a simplified equation (19).

(a) Compute the scale function $s$ for the diffusion $C_{k}$: Its derivative
is

$s^{\prime}(c)=\exp\left[-2\int_{0}^{c}\frac{\tilde{\gamma}(u)}{\tilde{\sigma}^{2}(u)}\,\mathrm{d}u\right].$

There exist $\gamma_{*}>0$ and $c_{*}>0$ such that

$\tilde{\gamma}(u)\leq-\gamma_{*},\,c\geq
c_{*};\quad\tilde{\gamma}(u)\geq\gamma_{*},\,c\leq-c_{*}.$

Moreover, $\tilde{\sigma}(u)\geq\sigma(u)\geq\sigma_{*}$ for all
$u\in\mathbb{R}$. Thus for $c\geq c_{*}$,

(28) $s^{\prime}(c)\geq
s^{\prime}(c_{*})\exp\left[2\int_{c_{*}}^{c}\frac{\gamma_{*}}{\sigma_{*}^{2}}\,\mathrm{d}c\right]=s^{\prime}(c_{*})\exp\left[2(c-c_{*})\frac{\gamma_{*}}{\sigma_{*}^{2}}\right].$

A similar estimate is true for $c\leq-c_{*}$:

(29) $s^{\prime}(c)\leq
s^{\prime}(-c_{*})\exp\left[2(|c|-c_{*})\frac{\gamma_{*}}{\sigma_{*}^{2}}\right].$

The speed measure has bounded Lebesgue density
$1/(s^{\prime}(c)\sigma^{2}(c))$, as shown in (28), (29), (27). Apply Feller’s
test from [14, Chapter 5, Section 5] to complete the proof.

(b) The statement for $\pi_{C}$ follows from ellipticity of the elliptic
partial differential equation governing the Lebesgue density of this
stationary distribution. The statement for $\pi_{\mu}$ follows from one-to-one
mapping between $\mathbf{C}$ and $\mu$. ∎

It seems to us that (27) is a reasonable assumption, since we would want to
allow for exchange of ranks of portfolios. This does not contradict our
statistical analysis, since we can observe only $C_{i}(t)>c_{+}$ or
$C_{i}(t)<-c_{-}$. Our suggested functions $\sigma(c)$ do give $\sigma(0)=0$
if we extend them to $[-c_{-},c_{+}]$ as is. But we could not observe
$C_{i}(t)$ in a neighborhood of zero, thus we can extend it as a piecewise
function.

Under assumptions of Theorem 3, the stationary distribution for each $C_{i}$
has (after normalization) density as above: $1/(s^{\prime}\sigma^{2})$, which
is supported on the whole real line but is bounded. This is different from the
case $\sigma(c)=\sigma_{0}c$ discussed above in Example 26, when the
stationary distribution is concentrated at one point, but the components of
the stationary distribution for the overall vector $C$ are not independent
since the SDE for individual relative size measures are dependent.

###  4.3. Lack of propagation of chaos

For interacting particle systems, sometimes dependence (as a process itself,
or stationary distribution) vanishes as the number of particles tends to
infinity. This phenomenon is called propagation of chaos, since the system
becomes less interdependent and more chaotic. Such results were shown for
competing Brownian particles and volatility-stabilized models (see citations
in the Introduction). But it is unreasonable to expect this for the current
system since all particles are dependent. The limiting density, if it exists,
will likely be a solution to a stochastic partial differential equation. To
derive this large system limit is left for future research.

##  5\. Capital Distribution Curve

###  5.1. Modified plots

Let us study the capital distribution curve $(\ln k,\ln\mu_{(k)}(t))$ in this
model. We solve the system of stochastic differential equations explicitly. We
use the values of $C_{k}(t)$ to plot the capital distribution curve:

$C_{k}(t)=\ln\frac{S_{0}(t)}{S_{k}(t)}=\ln\frac{S_{0}(t)}{S(t)}-\ln\mu_{k}(t).$

Thus the ranking of $C_{k}(t)$ reverts the ranking of ranked market weights:

$0\leq C_{(1)}(t)\leq\ldots\leq C_{(n)}(t).$

Thus we can plot the modified curve $(\ln k,C_{(k)}(t))$. If this curve is
linear, the same can be said for the original capital distribution curve. Now
we shall study the system of stochastic differential equations (19) and plot
$(\ln k,C_{(k)}(t))$ for fixed $t$.

###  5.2. Degenerate case

Even if the system is stable, under conditions of Theorem 2, capital
distribution curve can be degenerate, equal to one point:
$(1/(n+1),\ldots,1/(n+1))$. This is the case when $C_{k}(t)\to 0$ a.s. as
$t\to\infty$ for each $k=1,\ldots,n$. Indeed, in this case $\mu_{k}(t)\to
1/(n+1)$ a.s. as $t\to\infty$. In particular, this is true in Example 1 with
$\sigma(c)=\rho c$ for $\rho>0$. Below, we consider the case when the capital
distribution curve is not trivial.

###  5.3. Linear case

This is the case when

(30) $\alpha(c)=\mu c,\quad\beta(c)=1+\gamma c,\quad\sigma(c)=\rho.$

Then the system (19) is linear:

$\mathrm{d}C_{k}(t)=-\mu C_{k}(t)\,\mathrm{d}t-\gamma
C_{k}(t)(g_{S}\,\mathrm{d}t+\sigma_{S}\,\mathrm{d}W_{S}(t))+\rho\,\mathrm{d}W_{k}(t).$

As shown in [14, Chapter 5, Section 6.C], we can solve this system explicitly:

$\displaystyle C_{k}(t)$
$\displaystyle=Z(t)\left[C_{k}(0)-\int_{0}^{t}Z^{-1}(u)(\rho\,\mathrm{d}W_{k}(u)-\gamma\sigma_{S}\rho\,\mathrm{d}u)\right];$
$\displaystyle Z(t)$ $\displaystyle=\exp\left[-(\mu+\gamma
g_{S})t-\gamma\sigma_{S}\,W_{S}(t)-\frac{1}{2}\gamma^{2}\sigma_{S}^{2}t\right].$

Below, we show the simulation results for $n=100$, with initial conditions
$C_{k}(0)=0$, $k=1,\ldots,n$. We take estimated values $\mu=0.0069$,
$\gamma=0.0045$. For $\rho$, we take the value $0.1$, which is consistent with
estimates. Finally, estimates for mean $g_{S}$ and standard deviation
$\sigma_{S}$ of monthly price returns for the benchmark (top decile) are given
by $g_{S}=0.0044$ and $\sigma_{S}=0.0541$. We simulate until $t=100$, assuming
all $W_{1},\ldots,W_{n},W_{S}$ are independent Brownian motions.

Figure 4. Capital Distribution Curve $(\ln k,C_{(k)})$

##  6\. Conclusions

We developed a model in this article which can be viewed as an enhancement of
the Capital Asset Pricing Model, which stresses dependence of stock portfolios
upon the overall market. The portfolios are based on size (market cap), and
the quantities $\alpha$, $\beta$, and standard error $\sigma$ (of
diversifiable idiosyncratic risk) depends on size (more precisely, size of
portfolio relative to the size of benchmark). Thus we wrote a system of
stochastic differential equations.

We write separately systems of equations for equity premia (total returns,
including dividends, minus risk-free returns), and for market size (that is,
price returns). They are very similar and the idiosyncratic risk can be taken
the same. The equity premium and price returns of the benchmark are driven by
different random processes (although correlated).

Using CRSP 1926–2020 monthly data of size deciles, we find reasonable guesses
for $\alpha$, $\beta$, $\sigma$ as functions of relative size. Our statistical
analysis is not fully rigorous, because it fails white noise tests. However,
we do find some reasonable results.

On the theoretical side, we prove long-term stability results: under some
conditions on $\alpha,\beta,\sigma$, the vector of relative size measures
converges as $t\to\infty$ to a stationary distribution.

Finally, an important feature of real-world markets: stability and linearity
of the capital distribution curve, is reproduced in our model by numerical
simulation.

Future research can include making more sophisticated time series models which
take into account autocorrelations, or non-Gaussian fluctuations of the
market. It seems important to develop the SPT for the case of dividends, when
price and total returns (and by extension market capitalizations and wealth
processes) are different. Finally, we would like to derive a large system
limit.

## Appendix. Statistical Analysis of Size-Based Index Funds

These size deciles of the CRSP universe are not directly investable. But there
exist size-based funds available for individual investors. Among many of them,
let us take JKJ, JKG, JKD: iShares Morningstar Small-Cap, Mid-Cap, and Large-
Cap exchange-traded funds. These are based on largest 70%, next 20%, and next
7% of the total universe of stocks. In other words, Large-Cap corresponds to
Deciles 1–7 weighted by their market capitalizations, Mid-Cap corresponds to
Deciles 8–9 weighted by their market capitalizations, and Small-Cap
corresponds to the top 7% of the bottom Decile 10.

Monthly total arithmetic returns for these funds are taken from BlackRock web
site, July 2004 – August 2020. For risk-free returns, we take 1-month Treasury
Constant Maturity Rate from Federal Reserve Economic Data web site, observed
at the last day of each month June 2004 – July 2020. We compute geometric
versions of these returns. From each such rate $r$, we obtain geometric total
monthly returns for the next month $\ln(1+r/1200)$. Then we compute equity
premia $P_{S},P_{M},P_{L}$ for these funds. Regress the first two upon the
third:

(31)
$\displaystyle\begin{cases}P_{S}(t)=\alpha_{S}+\beta_{S}P_{L}(t)+\varepsilon_{S}(t);\\\
P_{M}(t)=\alpha_{M}+\beta_{M}P_{L}(t)+\varepsilon_{M}(t);\end{cases}\quad\begin{bmatrix}\varepsilon_{S}(t)\\\
\varepsilon_{M}(t)\end{bmatrix}\sim\mathcal{N}_{2}\left(\begin{bmatrix}0\\\
0\end{bmatrix},\Sigma\right).$

The quantile-quantile plots, Shapiro-Wilk and Jarque-Bera normality tests, and
autocorrelation function plots allow us to assume that each series of
residuals can be modeled by i.i.d. normal distribution. Thus we can apply
standard Student tests for regression coefficients. The 95% confidence
intervals for each of $\alpha_{S}$ and $\alpha_{M}$ contain zero. Thus we can
assume that $\alpha_{S}=\alpha_{M}=0$, but the confidence intervals for
$\beta_{S}$ and $\beta_{M}$ do not contain $1$. Point estimates of these
coefficients are: $\beta_{S}=1.27$, $\beta_{M}=1.15$. Estimates for standard
errors for residuals, and cross-correlation between residuals are:
$\sigma_{S}=0.026$, $\sigma_{M}=0.019$, $\rho=0.83$. The $R^{2}$ values for
each regression are 87% and 80%. Thus we see that the CAPM works for actual
traded size-based funds.

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