# Schubert coefficients of sparse paving matroids

Jon Pål Hamre

###### Abstract.

The Chow class of the closure of the torus orbit of a point in a Grassmannian
only depends on the matroid associated to the point. The Chow class can be
extended to a matroid invariant of arbitrary matroids. We call the
coefficients appearing in the expansion of the Chow class in the Schubert
basis the Schubert coefficients of the matroid. These Schubert coefficients
are conjectured by Berget and Fink to be non-negative. We compute the Schubert
coefficients for all sparse paving matroids, and confirm their non-negativity.

##  1\. Introduction

To a point $x$ in a complex Grassmannian $G(r,n)$ we associate two objects.
The matroid $M_{x}$ whose bases are given by the non-zero Plücker coordinates
of $x$ and the projective, toric subvariety of $G(r,n)$ given by the closure
of the orbit of $x$ under the action of the algebraic torus
$T=({\mathbb{C}}^{*})^{n}$, denoted ${\overline{Tx}}$. The class of
${\overline{Tx}}$ in the Chow ring of $G(r,n)$ is an additive, valuative
matroid invariant. Using results of Fink, Speyer and Derksen in [FS12] and
[DF10] we consider the unique valuative extension of this invariant to all
matroids, not just those representable over ${\mathbb{C}}$. We denote it by
$\operatorname{\text{Sc}}(M)\in A^{\bullet}(G(r,n))$. The Chow ring
$A^{\bullet}(G(r,n))$ is generated as a free abelian group by the Schubert
cycles $\sigma_{\lambda}$ where $\lambda$ ranges over all partitions whose
Young diagram is contained in the $r\times(n-r)$ rectangle. Hence
$\operatorname{\text{Sc}}(M)$ can be expressed as the sum
$\operatorname{\text{Sc}}(M)=\sum_{\lambda}d_{\lambda}(M)\sigma_{\lambda}$. We
call the integers $d_{\lambda}(M)$ the Schubert coefficients of $M$.

The non-negativity of the Schubert coefficients was conjectured by Berget and
Fink in [BF22]. We compute the Schubert coefficients of sparse paving matroids
and confirm their non-negativity.

######  Theorem 1.1.

Let $M$ be a connected sparse paving matroid of rank $r$ on $[n]$ then
$d_{\lambda}(M)=d_{\lambda}(U_{r,n})$ for all partitions $\lambda$ except for
the complement of the hook partition $h^{c}$ for which
$d_{h^{c}}(M)=\beta(M)$.

######  Corollary 1.2.

Let $M$ be a connected sparse paving matroid. The Schubert coefficients
$d_{\lambda}(M)$ are non-negative.

We suspect that sparse paving matroids are the only matroids such that
$d_{\lambda}(M)=d_{\lambda}(U_{r,n})$ for all partitions $\lambda\neq h^{c}$,
and make the following conjecture.

######  Conjecture 1.3.

Let $M$ be a connected matroid of rank $r$ on $[n]$. $M$ is sparse paving if
and only if $d_{\lambda}(M)=d_{\lambda}(U_{r,n})$ for all partitions
$\lambda\neq h^{c}$.

Section 2 consists of preliminaries on Grassmannians, matroids and valuations.
Section 3 introduces the concept of Schubert coefficients, and recaps what is
previously known. Section 4 contains the proof of 1.1 and some evidence for
1.3.

##  2\. Preliminaries

###  2.1. Grassmannians

We denote the complex Grassmannian of sub-vector spaces of ${\mathbb{C}}^{n}$
of dimension $r$ by $G(r,n)$. It is a smooth projective variety embedded into
${\mathbb{P}}^{\binom{n}{r}-1}$ by the well known Plücker embedding. The
algebraic torus $T=({\mathbb{C}}^{*})^{n}$ acts on $G(k,n)$ by dilating the
coordinates of ${\mathbb{C}}^{n}$. We will be interested in the closure in
$G(r,n)$ of the $T$ orbit of $x\in G(r,n)$ denoted ${\overline{Tx}}$.

###  2.2. Partitions

######  Definition 2.1.

A partition $\lambda$ is a tuple $\lambda=(\lambda_{1},\dots,\lambda_{k})$ of
positive integers with $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{k}$.

We think of partitions in terms of their Young diagrams (sometimes called
Ferrer diagrams), i.e. a collection of boxes with $\lambda_{i}$ boxes in row
$i$. We write $|\lambda|=\sum_{i}\lambda_{i}$ for the number of boxes. There
is a partial ordering of partitions defined by $\lambda\subseteq\mu$ if the
Young diagram of $\lambda$ is contained in the Young diagram of $\mu$. We will
be interested in the partitions $\lambda\subseteq r\times(n-r)$ where
$r\times(n-r)$ is the partition whose Young diagram is the $r\times(n-r)$
rectangle. For each $\lambda\subseteq r\times(n-r)$ let $\lambda^{c}$ be the
partition given by the complement of the Young diagram of $\lambda$ in the
$r\times(n-r)$ rectangle, rotated by 180 degrees. We denote the so called hook
partition by $h$, i.e. $h=(n-r,1,\dots,1)$ with $r-1$ ones. Notice that
$h^{c}=(r-1)\times(n-r-1)$. The Young diagrams of $h$ and $h^{c}$ for $r=3$
and $n=7$ are shown below.

######  Definition 2.2.

A Young tableaux of shape $\lambda$ is a filling of the boxes of the Young
diagram of $\lambda$ with the integers $1,2,\dots,|\lambda|$. A Young tableaux
is called standard if it is weakly increasing in rows and columns.

The combinatorics of partitions and Young tableaux appears in many areas of
matematics. Such as in the description of the Chow ring of Grassmannians.

###  2.3. Chow ring of Grassmannians

The Chow ring $A^{\bullet}(G(r,n))$ is generated as a free abelian group by
the Schubert cycles $\sigma_{\lambda}$ where $\lambda$ ranges over all
partitions $\lambda\subseteq r\times(n-r)$. Here $\sigma_{\lambda}$ is in the
degree $|\lambda|$ piece of $A^{\bullet}(G(r,n))$ i.e. $\sigma_{\lambda}\in
A^{|\lambda|}(G(r,n))$. The top piece $A^{r(n-r)}(G(r,n))$ is generated by the
Schubert cycle $\sigma_{r\times(n-r)}$ and the isomorphism with ${\mathbb{Z}}$
sending $\sigma_{r\times(n-r)}$ to $1$ is called the degree map $\deg\colon
A^{r(n-r)}(G(r,n))\to{\mathbb{Z}}$.

The product in $A^{\bullet}(G(r,n))$ can be described purely by partition
combinatorics known as Schubert calculus. There are two important formulas to
do computations. First the complimentary dimension formula which states that
if $|\lambda|+|\mu|=r(n-r)$ then $\deg(\sigma_{\lambda}\sigma_{\mu})=1$ if
$\mu=\lambda^{c}$ and $0$ otherwise. The other formula is called Pieri’s
formula. For a partition $\lambda\subseteq r\times(n-r)$ and a positive
integer $b$ we have $\sigma_{\lambda}\sigma_{(b)}=\sum_{\mu\in\lambda\otimes
b}\sigma_{\mu}$, where $\lambda\otimes b$ is the set of partitions in
$r\times(n-r)$ that can be obtained from the Young diagram of $\lambda$ by
adding $b$ boxes, at most one per column. For a comprehensive reference see
[EH16].

###  2.4. Matroids

We mainly use the basis definition of a matroid, and identify a matroid $M$ of
rank $r$ on $[n]$ with its set of bases $\mathcal{B}\subseteq\binom{[n]}{r}$.
For references on matroid theory see [Oxl11], [Kat16] and [Ard21]. We denote
the number of connected components of $M$ by $\kappa(M)$.

To a point $x\in G(r,n)$ we associate a matroid $M_{x}$. The bases of $M_{x}$
are the sets $I\in\binom{[n]}{r}$ with non-zero Plücker coordinates
$p_{I}(x)\neq 0$. A matroid is called representable over ${\mathbb{C}}$ if it
is obtained from some point in $G(r,n)$. We will be interested in a subclass
of matroids called sparse paving matroids. A matroid $M$ of rank $r$ is called
paving if all its circuits have cardinally $r$ or $r+1$. A matroid $M$ is
called sparse paving if both $M$ and its dual $M^{*}$ are paving. It is
conjectured in [DW11] and [Oxl11] that asymptotically almost all matroids are
sparse paving.

We will also describe another class of matroids called lattice path matroids
in order to make several other definitions easier, see [BdN03] for
comprehensive definitions. Let $P$ and $Q$ be lattice paths from $(0,0)$ to
$(n-r,r)$ such that $P$ never goes below $Q$. The lattice path matroid
$M[P,Q]$ is a rank $r$ matroid on $[n]$ with bases in bijection with the
lattice paths from $(0,0)$ to $(n-r,r)$ bounded by $P$ and $Q$. To get a basis
$B$ of $M[P,Q]$ from a lattice path $S$, label the steps of $S$ by the
integers $1,\dots,n$ from $(0,0)$ to $(n-r,r)$. Then the basis $B$ is the set
of labels of vertical steps.

One special case of lattice path matroids are those where $P$ is the path
starting with $r$ vertical steps followed by $n-r$ horizontal steps. In this
case the matroids $M[P,Q]$ are called Schubert matroids. If
$I\in\binom{[n]}{r}$ is the set indicating the vertical steps of $Q$ we denote
the Schubert matroid by $SM_{I}=M[P,Q]$. We introduce the matroids that will
be most important to us in the following examples.

######  Example 2.3.

The Schubert matroid given by $I=\\{n-r+1,n-r+2,\dots,n\\}$ is the uniform
matroid $SM_{I}=U_{r,n}$. To see this consider the example $r=3$ and $n=7$.
Since the paths $P$ and $Q$ forms the $3$ by $4$ rectangle, the bases of
$SM_{I}$ are in bijection with the lattice paths from $(0,0)$ to $(4,3)$. We
see that any set $B\in\binom{[7]}{3}$ can be obtained as labels of vertical
steps of some path.

######  Example 2.4.

The Schubert matroid given by $I=\\{2,3,\dots,r-1,n\\}$ is called the minimal
matroid, and is denoted $T_{r,n}=SM_{I}$. It is, up to isomorphism, the unique
connected matroid of rank $r$ on $[n]$ with minimal number of bases. In the
example where $r=3$ and $n=7$ it is given by the diagram

and is represented by the columns of the matrix

$\left(\begin{array}[]{ccccccc}1&0&0&1&1&1&1\\\ 0&1&0&1&1&1&1\\\
0&0&1&1&1&1&1\\\ \end{array}\right).$

In general $T_{r,n}$ is obtained from $U_{r,r+1}$ by adding $n-r-1$ elements
parallel to one of the elements in $U_{r,r+1}$.

######  Example 2.5.

The Panhandle matroids were introduced in [Han+22]. They are given by three
positive integers $r\leq s<n$. Here we define them as the Schubert matroid
$\operatorname{\text{Pan}}_{r,s,n}=SM_{I}$ where
$I=\\{s-r+2,s-r+3,\dots,s,n\\}$. The diagram of the panhandle matroid
$\operatorname{\text{Pan}}_{4,6,10}$ is

which looks like a panhandle, hence the name. Notice that
$\operatorname{\text{Pan}}_{r,r,n}=T_{r,n}$ and
$\operatorname{\text{Pan}}_{r,n-1,n}=U_{r,n}$.

###  2.5. Matroid polytopes and invariants

In this section we follow the conventions of [DF10] and [FS12]. Throughout let
$S(r,[n])$ denote the set of matroids of rank $r$ on $[n]$. We are interested
in functions $f\colon S(r,[n])\to A$ with values in an abelian group $A$. Such
functions are called matroid invariants if $f(M_{1})=f(M_{2})$ whenever
$M_{1}$ and $M_{2}$ are isomorphic. Also $f$ is called additive if $f(M)=0$
whenever $M$ is not connected. Another important property of functions of
matroids is called valuativity, which we will explain in what follows.

To any matroid $M$ on $[n]$ the matroid (base) polytope
$P(M)\subset{\mathbb{R}}^{n}$ is defined as the convex hull of the indicator
vectors $e_{B}=\sum_{i\in B}e_{i}$ for all bases $B$ of $M$. In [Gel+87] it
was shown that an equivalent definition of matroids can be given in terms of
its matroid polytope, so the matroid polytope $P(M)$ contains all the
information of $M$. The dimension of $P(M)$ is $n-\kappa(M)$. We will be
interested in how we can chop up matroid polytopes into smaller matroid
polytopes.

######  Definition 2.6.

A matroid subdivision of a matroid $M$ is a polyhedral complex consisting of
matroid polytopes with facets $P_{1},P_{2},\dots P_{k}$ such that

$P(M)=P_{1}\cup P_{2}\cup\dots\cup P_{k}.$

Given such a subdivision, for any set $J\subseteq[k]$ we denote
$P_{J}=\cap_{j\in J}P_{j}$ and $P_{\emptyset}=P(M)$. We may identify the set
$S(r,[n])$ with the set of matroid polytopes of matroids in $S(r,[n])$.

######  Definition 2.7.

A function $f\colon S(r,[n])\to A$ is called valuative if for any $M\in
S(r,[n])$ and any matroid subdivision of $M$ as above, the relation

$\sum_{J\subseteq[k]}(-1)^{|J|}f(P_{J})=0$

holds.

That is, $f$ is valuative if it satisfies an inclusion-exclusion property with
respect to matroid subdivisions. Notice that if $f$ is both valuative and
additive, since the matroid of $P_{J}$ is disconnected for all $J$ except
$1,2,\dots,k$ and $\emptyset$, the formula simplifies to

$f(M)=\sum_{i=1}^{k}f(P_{i}).$

In [DF10], Derksen and Fink proved that a function $f\colon S(r,[n])\to A$ is
valuative if and only if it factors through the abelian group generated by
indicator functions of matroid polytopes of $S(r,[n])$, denoted $P(r,[n])$.
They also show that the quotient $P^{sym}(r,[n])$ of $P(r,[n])$ given by
identifying isomorphic matroids is generated as an abelian group by the
classes of the indicator functions of Schubert matroids. This means that any
valuative matroid invariant is determined by its values on the Schubert
matroids.

The most obvious example of a valuative matroid invariant is the volume of a
matroid polytope.

######  Example 2.8.

Let $M$ be a matroid on $[n]$. We denote the normalized $s$ dimensional volume
of $P(M)$ by $\operatorname{\text{Vol}}_{s}(P(M))$111We may drop the subscript
if it does not cause too much confusion.. Then

$\operatorname{\text{Vol}}_{s}(P(-))\colon S(r,[n])\to{\mathbb{Z}}$

is a valuative matroid invariant. Moreover, if $s=n-1$ it is an additive
matroid invariant since $M$ is connected if and only if $n-\kappa(M)=s$.

The next important example was introduced by Crapo in [Cra67].

######  Example 2.9.

The beta invariant is a function

$\beta\colon S(r,[n])\to{\mathbb{Z}}.$

It is completely determined by the properties that if $M$ contains a loop or
coloop then $\beta(M)=0$, if $i$ is neither a loop or coloop of $M$ then

$\beta(M)=\beta(M/i)+\beta(M\operatorname{\smallsetminus}i),$

and $\beta(U_{1,2})=1$. It is well known that $\beta(M)$ is non-negative for
all $M$ and $\beta(M)=0$ if and only if $M$ is disconnected (excluding the
special cases $U_{0,1}$ and $U_{1,1}$). The beta invariant is in fact also a
valuative and additive matroid invariant.

The next example calculates the beta invariant of minimal matroids.

######  Example 2.10.

As we have seen, the minimal matroid $T_{r,n}$ is obtained from $U_{r,r+1}$ by
adding $n-r-1$ elements parallel to one of the elements in $U_{r,r+1}$. Let
$i$ be one of these parallel elements. Then
$T_{r,n}\operatorname{\smallsetminus}i=T_{r,n-1}$ and $T_{r,n}/i$ contains
$n-r-1$ loops, so

$\beta(T_{r,n})=\beta(T_{r,n}/i)+\beta(T_{r,n}\operatorname{\smallsetminus}i)=\beta(T_{r,n-1}).$

We may iterate this recursion until $T_{r,m}$ does not contain any parallel
elements to choose, i.e. when $m-r-1=0$. Hence

$\beta(T_{r,n})=\beta(T_{r,r+1})=\beta(U_{r,r+1}).$

The beta invariant of a uniform matroid is known to be

$\beta(U_{r,n})=\binom{n-2}{r-1}$

so

$\beta(T_{r,n})=\beta(U_{r,r+1})=\binom{r-1}{r-1}=1.$

The following theorem characterises sparse paving matroids in terms of matroid
subdivisions, and can be found for example in [Fer20] or in [JS17].

######  Theorem 2.11.

A matroid $M\in S(r,[n])$ is sparse paving if and only if there exists a
matroid subdivision of $U_{r,n}$ with facets $P(M)$ and
$P(M_{1}),\dots,P(M_{k})$ where all matroids $M_{i}$ are isomorphic to
$T_{r,n}$.

Hence for any additive and valuative matroid invariant $f$ and sparse paving
matroid $M$ of rank $r$ on $[n]$ we have

$\displaystyle f(U_{r,n})$ $\displaystyle=f(M)+k\cdot f(T_{r,n})$
$\displaystyle f(M)$ $\displaystyle=f(U_{r,n})-k\cdot f(T_{r,n}).$

##  3\. Schubert coefficients

In this section we define Schubert coefficients of matroids and recap what is
previously known. For each point $x\in G(r,n)$ we may consider the class of
the torus orbit closure ${\overline{Tx}}$ in $A^{\bullet}(G(r,n))$. We express
this class in the basis of Schubert cycles as

$[{\overline{Tx}}]=\sum_{\lambda}d_{\lambda}\sigma_{\lambda},$ (3.1)

where the sum is over all partitions $\lambda\subseteq r\times(n-r)$ with
$|\lambda|=\operatorname{Codim}({\overline{Tx}})$ and
$d_{\lambda}\in{\mathbb{Z}}$. More precisely the coefficient
$d_{\lambda}=\deg([{\overline{Tx}}]\sigma_{\lambda^{c}})$ is the number of
points in the intersection of ${\overline{Tx}}$ and the Schubert variety
$X_{\lambda^{c}}(V_{\bullet})$ for a generic flag $V_{\bullet}$. Hence the
coefficients are non-negative.

In [Gel+87] it is shown that ${\overline{Tx}}$ is isomorphic to the projective
toric variety of the matroid polytope $P(M_{x})$. Hence many properties of
${\overline{Tx}}$ can be read off from the matroid $M_{x}$. For example

$\dim({\overline{Tx}})=\dim(P(M_{x}))=n-\kappa(M_{x})$ (3.2)

and

$\deg({\overline{Tx}})=\operatorname{\text{Vol}}(P(M_{x})).$ (3.3)

Here $\deg({\overline{Tx}})$ denotes the degree of ${\overline{Tx}}$ embedded
in ${\mathbb{P}}^{\binom{n}{r}-1}$ and $\operatorname{\text{Vol}}(P(M_{x}))$
is the normalized $n-\kappa(M_{x})$ dimensional volume of $P(M_{x})$.

As we will see, the class of ${\overline{Tx}}$ in $A^{\bullet}(G(r,n))$ only
depends on the matroid $M_{x}$, so we may denote the coefficients in Equation
3.1 by $d_{\lambda}(M_{x})$. The coefficient $d_{\lambda}(M_{x})$ only makes
sense if $|\lambda|=r(n-r)-(n-\kappa(M_{x}))$. For all other cases we define
$d_{\lambda}(M_{x})=0$. This way we can consider $d_{\lambda}$ as functions
from the set of matroids representable over ${\mathbb{C}}$ of rank $r$ on
$[n]$ to ${\mathbb{Z}}$. We call the integers $d_{\lambda}(M_{x})$ the
Schubert coefficients of the matroid $M_{x}$.

In [Kly85], Klyachko gives a formula for $d_{\lambda}(U_{r,n})$ in terms of
evaluations of Schur polynomials.

######  Theorem 3.1.

[Kly85, Theorem 6] Let $\lambda\subseteq r\times(n-r)$ be a partition with
$|\lambda|=(r-1)(n-r-1)$. The associated Schubert coefficient of $U_{r,n}$ is

$d_{\lambda}(U_{r,n})=\sum_{i=0}^{r}(-1)^{i}\binom{n}{i}s_{\lambda^{c}}(1^{r-i}).$

In [Spe09], Speyer proved the following important theorem describing the
Schubert coefficient associated to the hook complement.

######  Theorem 3.2.

[Spe09, Theorem 5.1] For $x\in G(r,n)$

$d_{h^{c}}(M_{x})=\beta(M_{x}).$

Notice that this is true even if $M_{x}$ is disconnected, since in that case
both $d_{h^{c}}(M_{x})$ and $\beta(M_{x})$ are zero.

One issue with the Schubert coefficients of matroids is that points in the
Grassmannian only give rise to matroids representable over ${\mathbb{C}}$.
This is rectified by Berget and Fink in [BF22] where they define a class of a
matroid in $A^{\bullet}(G(r,n))$ for any matroid $M$ of rank $r$ on $[n]$,
which we denote $\operatorname{\text{Sc}}(M)$, coinciding with
$[{\overline{Tx}}]$ whenever $M=M_{x}$ for $x\in G(r,n)$. Note that
$\operatorname{\text{Sc}}(M)\in A^{r(n-r)-(n-\kappa(M))}(G(r,n))$. From the
class $\operatorname{\text{Sc}}(M)$ we get Schubert coefficients
$d_{\lambda}(M)$ for arbitrary matroids. We summarize the most relevant
takeaways from [BF22] in a theorem.

######  Theorem 3.3.

For each partition $\lambda\subseteq r\times(n-r)$ there are functions
$d_{\lambda}\colon S(r,[n])\to{\mathbb{Z}}$ coinciding with the coefficients
of Equation 3.1 whenever $M=M_{x}$ for $x\in G(r,n)$. These functions are
valuative matroid invariants and whenever $|\lambda|=r(n-r)-(n-1)$ then
$d_{\lambda}$ is additive.

As mentioned in [BF22] there is another way of defining Schubert coefficients
of arbitrary matroids by the theory of tautological classes of matroids from
[Ber+23]. In this setting
$d_{\lambda}(M)=\deg_{X_{n}}(s_{\lambda^{c}}(\mathcal{S}_{M}^{\vee}))$ where
$X_{n}$ is the $(n-1)$-dimensional permutahedral variety, $\mathcal{S}_{M}$ is
the tautological sub-bundle of $M$ and
$s_{\lambda^{c}}(\mathcal{S}_{M}^{\vee})$ is the evaluation of the Schur
polynomial $s_{\lambda^{c}}$ at the Chern roots of $\mathcal{S}_{M}^{\vee}$.

For representable matroids $M_{x}$ we could conclude that $d_{\lambda}(M_{x})$
is non-negative since it is the number of points in an intersection of
varieties. This argument only works for representable matroids, but the non-
negativity of $d_{\lambda}(M)$ for arbitrary matroids $M$ is conjectured in
[BF22].

######  Conjecture 3.4.

[BF22, Conjecture 9.13] The Schubert coefficients $d_{\lambda}(M)$ are non-
negative for all matroids $M$.

##  4\. Schubert coefficients of sparse paving matroids

In this section we compute the Schubert coefficients of sparse paving
matroids. The observation that allows us to do so is the following.

######  Proposition 4.1.

For $x\in G(r,n)$ let $s=\dim({\overline{Tx}})=n-\kappa(M_{x})$. The following
equality holds

$\deg([{\overline{Tx}}]\sigma_{(1)}^{s})=\operatorname{\text{Vol}}_{s}(P(M_{x})).$
(4.1)

This gives the linear relation among the Schubert coefficients of $M_{x}$

$\sum_{\lambda}c_{\lambda^{c}}d_{\lambda}(M_{x})=\operatorname{\text{Vol}}_{s}(P(M_{x})),$
(4.2)

where $c_{\lambda^{c}}$ are the integer coefficients found in the expansion of
$\sigma_{(1)}^{s}$. That is,
$c_{\lambda^{c}}=\deg(\sigma_{(1)}^{s}\sigma_{\lambda}).$ Here the sum is
taken over all partitions $\lambda\subseteq r\times(n-r)$ with
$|\lambda|=r(n-r)-s$.

###### Proof.

There are two ways of computing the degree $D$ of ${\overline{Tx}}$ in
${\mathbb{P}}^{\binom{n}{r}-1}$. First, since ${\overline{Tx}}$ is a
subvariety of $G(r,n)$, $D=\deg([{\overline{Tx}}]\sigma_{(1)}^{s})$. Second,
since ${\overline{Tx}}$ is isomorphic to the projective toric variety obtained
from $P(M_{x})$, $D=Vol_{s}(P(M_{x}))$. Expanding $[{\overline{Tx}}]$ and
$\sigma_{(1)}^{s}$ gives

$\displaystyle[{\overline{Tx}}]$
$\displaystyle=\sum_{\lambda}d_{\lambda}(M_{x})\sigma_{\lambda}\text{ and }$
$\displaystyle\sigma_{(1)}^{s}$
$\displaystyle=\sum_{\lambda}c_{\lambda^{c}}\sigma_{\lambda^{c}}.$

Now by the complimentary dimension formula the left hand side of Equation 4.1
becomes the left hand side of Equation 4.2. ∎

The proof above only works for matroids representable over ${\mathbb{C}}$, but
we can extend the result to arbitrary matroids using the fact that valuative
matroid invariants are determined by their values on Schubert matroids.

######  Corollary 4.2.

Let $M$ be a rank $r$ matroid on $[n]$ and $s=n-\kappa(M)$ then

$\deg(\operatorname{\text{Sc}}(M)\sigma_{(1)}^{s})=Vol_{s}(P(M)).$

This gives the linear relation

$\sum_{\lambda}c_{\lambda^{c}}d_{\lambda}(M)=\operatorname{\text{Vol}}_{s}(P(M)),$

where $c_{\lambda^{c}}$ are as in Proposition 4.1.

###### Proof.

For a fixed positive integer $s\leq r(n-r)$, notice that both

$\displaystyle\operatorname{\text{Vol}}_{s}(P(-))\colon$ $\displaystyle
S(r,[n])\to{\mathbb{Z}}\text{ and }$
$\displaystyle\deg\big{(}(\sum_{\lambda}d_{\lambda}(-)\sigma_{\lambda})\sigma_{(1)}^{s}\big{)}\colon$
$\displaystyle S(r,[n])\to{\mathbb{Z}},$

where the sum is over all partitions $\lambda\subseteq r\times(n-r)$ with
$|\lambda|=r(n-r)-s$, are valuative matroid invariants. By Theorem 6.3 of
[DF10] all valuative matroid invariants are determined by their values on the
Schubert matroids $SM_{I}$. Since the Schubert matroids are representable over
${\mathbb{C}}$ we know that
$\deg((\sum_{\lambda}d_{\lambda}(SM_{I})\sigma_{\lambda})\sigma_{(1)}^{s})=\operatorname{\text{Vol}}_{s}(P(SM_{I}))$
by Proposition 4.1. Hence these invariants are equal for any matroid. The
corollary follows from setting $s=n-\kappa(M)$. ∎

We will use this to compute the Schubert coefficients of sparse paving
matroids. First we compute those of the minimal matroids.

######  Lemma 4.3.

The Schubert coefficients of the minimal matroid $T_{r,n}$ are
$d_{\lambda}(T_{r,n})=0$ for $\lambda\neq h^{c}$ and
$d_{h^{c}}(T_{r,n})=\beta(T_{r,n})=1$, i.e.
$\operatorname{\text{Sc}}(T_{r,n})=\sigma_{h^{c}}$.

###### Proof.

By Proposition 4.1

$\sum_{\lambda}c_{\lambda^{c}}d_{\lambda}(T_{r,n})=\operatorname{\text{Vol}}(P(T_{r,n})).$
(4.3)

The matroid polytope $P(T_{r,n})$ is a pyramid of lattice height $1$ over the
product of simplicies $\Delta_{r-1}\times\Delta_{n-r-1}$ and hence has volume
$\operatorname{\text{Vol}}(P(T_{r,n}))=\binom{n-2}{r-1}$, see also example 7.4
of [FS22]. Next we compute the coefficient of the hook partition $c_{h}$. By
using Pieri’s formula and considering the expansion of $\sigma_{(1)}^{n-1}$ in
$A^{\bullet}(G(r,n))$ we see that $c_{\lambda^{c}}=|SYT(\lambda^{c})|$ is the
number of standard Young tableaux with shape $\lambda^{c}$. The following
formula for $|SYT(\lambda^{c})|$ is given in Theorem $1$ of [FRT54]222This
paper gives the formula for the degree of the corresponding irreducible
representation of the symmetric group on $|\lambda|$ elements. There is a well
known basis of this representation in bijection with standard Young tableaux
of shape $\lambda$.,

$|SYT(\lambda^{c})|=\frac{|\lambda^{c}|!}{\prod_{b\in\lambda^{c}}hl(b)}$

where the product is over all boxes in the Young diagram of $\lambda^{c}$ and
$hl(b)$ is the hook length of $b$. Applying this formula to the hook partition
$h$ gives

$\displaystyle c_{h}=|SYT(h)|$
$\displaystyle=\frac{(n-1)!}{(n-1)(r-1)!(n-r-1)!}$
$\displaystyle=\frac{(n-2)!}{(r-1)!(n-2-(r-1))!}$
$\displaystyle=\binom{n-2}{r-1}$

hence $\operatorname{\text{Vol}}(P(T_{k,n}))=c_{h}$. Since
$d_{h^{c}}(T_{r,n})=\beta(T_{r,n})=1$ Equation 4.3 becomes

$c_{h}+\sum_{\lambda\neq
h^{c}}c_{\lambda^{c}}d_{\lambda}(T_{r,n})=\operatorname{\text{Vol}}(P(T_{r,n})).$

The integers $c_{\lambda^{c}}$ are positive since it is the degree of the
Schubert variety $X_{\lambda}(V_{\bullet})$ and the Schubert coefficients
$d_{\lambda}(T_{r,n})$ are non-negative since $T_{r,n}$ is representable over
${\mathbb{C}}$. Hence $d_{\lambda}(T_{r,n})=0$ for all partitions $\lambda\neq
h^{c}$. ∎

By using Lemma 4.3 we are now ready to prove 1.1.

###### Proof of 1.1.

Since $M$ is sparse paving there is a matroid subdivision of $P(U_{r,n})$
containing as facets $P(M)$ and $k$ polytopes of matroids isomorphic to
$T_{r,n}$. Since $d_{\lambda}$ are additive and valuative matroid invariants
this subdivision yields

$d_{\lambda}(U_{r,n})=d_{\lambda}(M)+k\cdot d_{\lambda}(T_{r,n})\iff
d_{\lambda}(M)=d_{\lambda}(U_{k,n})-k\cdot d_{\lambda}(T_{r,n}).$

By Lemma 4.3 if $\lambda\neq h^{c}$ then $d_{\lambda}(M)=d_{\lambda}(U_{r,n})$
and $d_{h^{c}}(M)=\beta(M)=\beta(U_{r,n})-k$. ∎

Now Corollary 1.2 follows since $U_{r,n}$ is representable over ${\mathbb{C}}$
so $d_{\lambda}(M)=d_{\lambda}(U_{r,n})$ and $\beta(M)$ are non-negative.

The proof of 1.1 applies to the following more general case.

######  Corollary 4.4.

Let $M$ and $M^{\prime}$ be connected matroids of rank $r$ on $[n]$ and assume
there is a matroid subdivision of $P(M^{\prime})$ containing as facets $P(M)$
and $k$ polytopes of matroids isomorphic to $T_{r,n}$. Then
$d_{\lambda}(M)=d_{\lambda}(M^{\prime})$ for all $\lambda\neq h^{c}$.

######  Example 4.5.

Some well known examples of matroids that are not representable over
${\mathbb{C}}$ are the Fano matroid $F$, the non-Pappus matroid $NP$ and the
Vamos matroid $V$. These are sparse paving matroids and the classes of $F$,
$NP$ and $V$ in $A^{\bullet}(G(r,n))$ are shown below.

$\displaystyle\operatorname{\text{Sc}}(F)=$ $\displaystyle
6\sigma_{(4,2,0)}+3\sigma_{(4,1,1)}+3\sigma_{(3,3,0)}+8\sigma_{(3,2,1)}+\sigma_{(2,2,2)}\in
A^{\bullet}(G(3,7))$ $\displaystyle\operatorname{\text{Sc}}(NP)=$
$\displaystyle
15\sigma_{(6,4,0)}+15\sigma_{(6,3,1)}+6\sigma_{(6,2,2)}+13\sigma_{(5,5,0)}$
$\displaystyle+24\sigma_{(5,4,1)}+15\sigma_{(5,3,2)}.+6\sigma_{(4,4,2)}+3\sigma_{(4,3,3)}\in
A^{\bullet}(G(3,9))$ $\displaystyle\operatorname{\text{Sc}}(V)=$
$\displaystyle
4\sigma_{(4,4,1,0)}+20\sigma_{(4,3,2,0)}+12\sigma_{(4,3,1,1)}+12\sigma_{(4,2,2,1)}$
$\displaystyle+15\sigma_{(3,3,3,0)}+20\sigma_{(3,3,2,1)}+4\sigma_{(3,2,2,2)}\in
A^{\bullet}(G(4,8))$

One might try to expand this argument to prove non-negativity of the Schubert
coefficients for a class of matroids larger than the sparse paving matroids.
In what follows we give an example showing this does not work for paving
matroids that are not sparse paving.

As we have seen, sparse paving matroids can be described as those matroids
obtained from uniform matroids by cutting of minimal matroid polytopes from
the hypersimplex. In a similar way, as described in [Han+22], one can describe
paving matroids as those obtained from $U_{r,n}$ by cutting of panhandle
matroids. Recall that the panhandle matroid
$\operatorname{\text{Pan}}_{r,r,n}$ is the minimal matroid $T_{r,n}$ and
$\operatorname{\text{Pan}}_{r,n-1,n}$ is the uniform matroid $U_{r,n}$. The
next example shows that 1.1 does not hold for paving matroids that are not
sparse paving.

######  Example 4.6.

The smallest non-minimal, non-uniform example of a panhandle matroid is
$\operatorname{\text{Pan}}_{2,3,5}$. There is a matroid subdivision of
$U_{2,5}$ of the form

$P(U_{2,5})=P(\operatorname{\text{Pan}}_{2,3,5})\cup P(T_{2,5}).$ (4.4)

That is $\operatorname{\text{Pan}}_{2,3,5}$ is a sparse paving matroid, and
$T_{2,5}$ is a paving matroid that is not sparse paving. By Lemma 4.3 and 3.1
we have

$d_{(1,1)}(T_{2,5})=0\neq 1=d_{(1,1)}(U_{2,5})$

so 1.1 does not hold for paving matroids in general. For completeness by 3.1,
Lemma 4.3 and Equation 4.4 we have

$\displaystyle\operatorname{\text{Sc}}(U_{2,5})$
$\displaystyle=\sigma_{(1,1)}+3\sigma_{(2)},$
$\displaystyle\operatorname{\text{Sc}}(T_{2,5})$ $\displaystyle=\hskip
42.67912pt\sigma_{(2)}\text{ and }$
$\displaystyle\operatorname{\text{Sc}}(\operatorname{\text{Pan}}_{2,3,5})$
$\displaystyle=\sigma_{(1,1)}+2\sigma_{(2)}.$

This leads us to state the following conjecture.

######  Conjecture 4.7.

Let $M$ be a connected matroid of rank $r$ on $[n]$ then $M$ is sparse paving
if and only if $d_{\lambda}(M)=d_{\lambda}(U_{r,n})$ for all partitions
$\lambda\neq h^{c}$.

##  5\. Acknowledgments

This research is supported in part by the Trond Mohn Foundation project
"Algebraic and topological cycles in complex and tropical geometry" and the
Center for Advanced Study Young Fellows Project "Real Structures in Discrete,
Algebraic, Symplectic, and Tropical Geometries". We would like to thank Kris
Shaw, Alex Fink, Andrew Berget, Lorenzo Vecchi and Benjamin Schröter for all
the help with and interest in this project.

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Lipschitz continuous. The next proposition shows (under certain assumptions)
that the value function $\alpha$ belongs to $C^{0,1}(\mathcal{D})$, where
$\mathcal{D}=\mathbb{R}^{d}\times(0,T)\times(\varphi_{min},\infty)$. The
following proposition and its proof are contained in our recent paper ([66,
Theorem 3]).

######  Proposition 9.

[66, Proposition 1] Let $\triangle\subset\mathbb{R}^{n}$ be a given compact
decision set. Assume that the functions $\mu(x,t,{\bm{\theta}})$ and
$\sigma(x,t,{\bm{\theta}})^{2}$ are globally Lipschitz continuous in
$x\in\mathbb{R}^{d},t\in[0,T]$ and ${\bm{\theta}}\in\triangle$ variables and
there exist positive constants $\omega,L>0$ such that
$\omega\leq\frac{1}{2}\sigma(x,t,{\bm{\theta}})^{2}\leq L$ for any
$x\in\mathbb{R}^{d},t\in[0,T]$, and ${\bm{\theta}}\in\triangle$.

Then, $\alpha\in C^{0,1}(\mathcal{D})$. Moreover, the function $\alpha$ is
strictly increasing, and

$0<\omega\leq\frac{\alpha(x,\tau,\varphi_{2})-\alpha(x,\tau,\varphi_{1})}{\varphi_{2}-\varphi_{1}}\leq
L,\quad\text{for any}\ (x,\tau,\varphi_{i})\in\mathcal{D},$ (76)

i.e., $\omega\leq\alpha^{\prime}_{\varphi}(x,\tau,\varphi)\leq L$, and

$|\nabla_{x}\alpha(x,\tau,\varphi)|\leq p(x,\tau)+L_{0}|\varphi|,$ (77)

for a.e., $(x,\tau,\varphi)\in\mathcal{D}$, where
$p(x,\tau):=\max_{{\bm{\theta}}\in\triangle}|\nabla_{x}\mu(x,t,{\bm{\theta}})|$
and
$L_{0}:=\max_{{\bm{\theta}}\in\triangle,t\in[0,T],x\in\mathbb{R}^{d}}|\nabla_{x}\sigma^{2}(x,t,\theta)|$,
where $t=T-\tau$.

###### Proof.

First, let
$\alpha^{\bm{\theta}}(x,\tau,\varphi):=-\mu(x,t,{\bm{\theta}})+\frac{\varphi}{2}\sigma(x,t,{\bm{\theta}})^{2}$,
where $t=T-\tau$. Then,

$\alpha(x,\tau,\varphi)=\min_{{\bm{\theta}}\in\triangle}\alpha^{\bm{\theta}}(x,\tau,\varphi)\,.$

For any given ${\bm{\theta}}\in\triangle$, the function
$\alpha^{\bm{\theta}}(x,\tau,\varphi)$ is globally Lipschitz continuous in all
variables. Therefore, the minimal function $\alpha$ is globally Lipschitz
continuous. Moreover, the function $\alpha^{\bm{\theta}}(x,\tau,\varphi)$
satisfies the inequality (76) for any ${\bm{\theta}}\in\triangle$, and so does
the minimal function $\alpha$.

Next, we prove inequality (77). Let $x_{1},x_{2}\in\mathbb{R}^{d}$ such that
$x_{2}=x_{1}+he^{i}$, where $e^{i},i=1,\cdots,d,$ is the standard normal
vector, i.e., $e^{i}=(0,0,...,0,1,0,...,0)^{T}$. Then, we have

$\displaystyle\alpha^{\bm{\theta}}(x_{1},\tau,\varphi)-\alpha^{\bm{\theta}}(x_{2},\tau,\varphi)=-(\mu(x_{1},\tau,{\bm{\theta}})-\mu(x_{2},\tau,{\bm{\theta}}))+\frac{\varphi}{2}(\sigma(x_{1},\tau,{\bm{\theta}})^{2}-\sigma(x_{2},\tau,{\bm{\theta}})^{2})$
$\displaystyle=\int_{0}^{h}(-\partial_{x_{i}}\mu(x_{1}+\xi
e^{i},\tau,{\bm{\theta}}))d\xi+\int_{0}^{h}\frac{\varphi}{2}\partial_{x_{i}}\sigma^{2}(x_{1}+\xi
e^{i},\tau,{\bm{\theta}})d\xi$
$\displaystyle\leq\int_{0}^{h}|\partial_{x_{i}}\mu(x_{1}+\xi
e^{i},\tau,{\bm{\theta}}))|d\xi+\int_{0}^{h}\frac{|\varphi|}{2}|\partial_{x_{i}}\sigma^{2}(x_{1}+\xi
e^{i},\tau,{\bm{\theta}})|d\xi$
$\displaystyle\leq\max_{\bm{\theta}\in\triangle,0\leq\xi\leq
h}|\partial_{x_{i}}\mu(x_{1}+\xi
e^{i},\tau,{\bm{\theta}})|\,h+\max_{{\bm{\theta}}\in\triangle,x\in\mathbb{R}^{d}}|\partial_{x_{i}}\sigma^{2}(x,\tau,\theta)|\frac{|\varphi|}{2}h.$

Hence,

$\alpha^{\bm{\theta}}(x_{1},\tau,\varphi)\leq\alpha^{\bm{\theta}}(x_{2},\tau,\varphi)+\max_{{\bm{\theta}}\in\triangle,0\leq\xi\leq
h}|\partial_{x_{i}}\mu(x_{1}+\xi
e^{i},\tau,{\bm{\theta}})|\,h+\max_{{\bm{\theta}}\in\triangle,x\in\mathbb{R}^{d}}|\partial_{x_{i}}\sigma^{2}(x,\tau,\theta)||\varphi|h.$

We note that $x_{2}-x_{1}=he^{i}$ so that $|x_{2}-x_{1}|=h$. Taking minimum
over ${\bm{\theta}}\in\triangle$, we obtain

$\alpha(x_{1},\tau,\varphi)\leq\alpha(x_{2},\tau,\varphi)+\max_{\bm{\theta}\in\triangle,0\leq\xi\leq
h}|\partial_{x_{i}}\mu(x_{1}+\xi
e^{i},\tau,{\bm{\theta}})|\,h+\max_{{\bm{\theta}}\in\triangle,x\in\mathbb{R}^{d}}|\partial_{x_{i}}\sigma^{2}(x,\tau,\theta)||\varphi|h.$

By exchanging the role of $x_{1}$ and $x_{2}$ and taking the limit as
$x_{2}\to x_{1}$, i.e., $h\to 0$, we obtain inequality (77), as stated.

∎

According to Proposition 9, the value function $\alpha$ given in (73)
satisfies the assumptions of Theorem 4 provided that the functions

$p(x,\tau)=\max_{{\bm{\theta}}\in\triangle}|\nabla_{x}\mu(x,\tau,{\bm{\theta}})|,\quad\text{and}\
h(x,\tau)=\alpha(x,\tau,0)=-\max_{{\bm{\theta}}\in\triangle}\mu(x,\tau,{\bm{\theta}})$

belong to the Banach space $L^{\infty}((0,T);H)$.

The next result was proved in [36]. It gives sufficient conditions imposed on
the decision set $\triangle$ and functions $\mu$ and $\sigma$ that guarantee
higher smoothness of the value function $\alpha$. Its proof is based on the
classical envelope theorem due to Milgrom and Segal [48] and the result on the
Lipschitz continuity of the minimizer
$\hat{\bm{\theta}}=\hat{\bm{\theta}}(x,\tau,\varphi)$ belonging to a convex
compact set $\triangle$ according to Klatte [39].

######  Theorem 8.

[36, Theorem 1] Suppose that $\triangle\subset\mathbb{R}^{n}$ is a convex
compact set and the functions $\mu(x,t,{\bm{\theta}})$ and
$\sigma(x,t,{\bm{\theta}})^{2}$ are $C^{1,1}$ smooth such that the objective
function
$f(x,t,\varphi,{\bm{\theta}}):=-\mu(x,t,{\bm{\theta}})+\frac{\varphi}{2}\sigma(x,t,{\bm{\theta}})^{2}$
is strictly convex in the variable ${\bm{\theta}}\in\triangle$ for any
$\varphi\in(\varphi_{min},\infty)$. Then, the function $\alpha$ belongs to the
space $C^{1,1}(\mathcal{D})$.

####  7.2.3. Point-wise a-priori estimates of solution with their existence
and uniqueness

In this section, we present a-priori estimates on a solution $\varphi$ to the
Cauchy problem (74). We will assume that the function $\mu$ is independent of
time $t\in[0,T]$, and $\sigma$ is independent of $x\in\mathbb{R}$ and
$t\in[0,T]$, i.e.,

$\mu=\mu(x,\bm{\theta}),\qquad\sigma=\sigma(\bm{\theta}).$

Then, the value function $\alpha=\alpha(x,\varphi)$ is also independent of the
$\tau=T-t$ variable. Next, we prove a-priori estimates for the transformed
function $\psi=\psi(x,\tau)$ defined as
$\psi(x,\tau)=\alpha(x,\varphi(x,\tau))$. Since the function $\alpha$ is
strictly increasing in the $\varphi$ variable, there exists an inverse
function $\beta(x,\psi)$ such that $\alpha(x,\beta(x,\psi))=\psi$. Therefore,
the function $\varphi(x,\tau)$ is a solution to (74) if and only if the
function $\psi(x,\tau)$ is a solution to the following linear parabolic PDE:

$-\partial_{\tau}\psi+a(x,\tau)\partial_{x}^{2}\psi+b(x,\tau)\partial_{x}\psi+c(x,\tau)\psi=0,$

where

$\displaystyle a(x,\tau)$ $\displaystyle=$
$\displaystyle\alpha^{\prime}_{\varphi}(x,\varphi(x,\tau)),\quad
b(x,\tau)=-\alpha^{\prime}_{\varphi}(x,\varphi(x,\tau))\varphi(x,\tau)-\alpha(x,\varphi(x,\tau)),$
$\displaystyle c(x,\tau)$ $\displaystyle=$
$\displaystyle\alpha^{\prime}_{x}(x,\varphi(x,\tau)).$

Note that $0<\omega\leq a(x,\tau)\leq L$. Next, suppose that the function
$c(x,\tau)$ is bounded from above by a constant $\lambda\geq 0$. Then, the
function $\psi_{\lambda}(x,\tau)=\psi(x,\tau)e^{-\lambda\tau}$ is a solution
to the linear PDE:

${\mathcal{L}}[\psi_{\lambda}]=0,\qquad\text{where}\ \
{\mathcal{L}}[\psi_{\lambda}]\equiv\partial_{\tau}\psi_{\lambda}-a(x,\tau)\partial_{x}^{2}\psi_{\lambda}-b(x,\tau)\partial_{x}\psi_{\lambda}-c_{\lambda}(x,\tau)\psi_{\lambda},$

where $c_{\lambda}(x,\tau)=c(x,\tau)-\lambda$ is nonpositive, i.e.,
$c_{\lambda}(x,\tau)\leq 0$ for all $x,\tau$. Let $\underline{\psi}\leq 0$ be
a constant. Then,
${\mathcal{L}}[\psi_{\lambda}-\underline{\psi}]=c_{\lambda}(x,\tau)\underline{\psi}\geq
0$. We employ the maximum principle for parabolic equations on unbounded
domains according to Meyer and Needham [46, Theorem 3.4] to obtain
$\psi_{\lambda}(x,\tau)-\underline{\psi}\geq 0$ for all $x,\tau$, provided
that $\psi_{\lambda}(x,0)-\underline{\psi}=\psi(x,0)-\underline{\psi}\geq 0$
for all $x$. In other words, $\underline{\psi}$ is a subsolution. Similarly,
if $\overline{\psi}\geq 0$ is a given constant, then
${\mathcal{L}}[\psi_{\lambda}-\overline{\psi}]=c_{\lambda}(x,\tau)\overline{\psi}\leq
0$ and $\psi_{\lambda}(x,\tau)-\overline{\psi}\leq 0$ for all $x,\tau$,
provided that $\psi(x,0)-\overline{\psi}\leq 0$ for all $x$, i.e.,
$\overline{\psi}$ is a supersolution. Consequently, we have the following
implication:

$\underline{\psi}\leq\psi(x,0)\leq\overline{\psi}\quad\Longrightarrow\quad\underline{\psi}e^{\lambda\tau}\leq\psi(x,\tau)\leq\overline{\psi}e^{\lambda\tau}\quad\text{for
all}\ x\in\mathbb{R},\tau\in[0,T].$

In terms of the solution $\varphi$ to the Cauchy problem (74)–(75), we obtain
the following a-priori estimate:

$\underline{\psi}e^{\lambda\tau}\leq\alpha(x,\varphi(x,\tau))\leq\overline{\psi}e^{\lambda\tau}\quad\text{for
all}\ x\in\mathbb{R},\tau\in[0,T],$ (78)

where

$\underline{\psi}=\min\\{0,\inf_{x\in\mathbb{R}}\alpha(x,\varphi(x,0))\\},\qquad\overline{\psi}=\max\\{0,\sup_{x\in\mathbb{R}}\alpha(x,\varphi(x,0))\\}.$
(79)

Now, we can apply the general Theorem 4 on existence and uniqueness of a
solution.

######  Theorem 9.

[66, Theorem 5] Let the decision set $\triangle\subset\mathbb{R}^{n}$ be
compact and $u:\mathbb{R}\to\mathbb{R}$ be an increasing utility function such
that $\varphi_{0}(x)=-u^{\prime\prime}(x)/u^{\prime}(x)$ belongs to the space
$L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Suppose that the drift
$\mu(x,\bm{\theta})$ and volatility function $\sigma^{2}(\bm{\theta})>0$ are
$C^{1}$ continuous in the $x$ and $\bm{\theta}$ variables and the value
function $\alpha(x,\varphi)$ given in (73) satisfies $p\in
L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R}),h\in
L^{\infty}(\mathbb{R}),\;\text{and}\;\partial^{2}_{x}h\in L^{2}(\mathbb{R})$,
where

$\displaystyle
p(x)=\max_{{\bm{\theta}}\in\triangle}|\partial_{x}\mu(x,{\bm{\theta}})|,\quad
h(x)=-\max_{{\bm{\theta}}\in\triangle}\mu(x,{\bm{\theta}}).$

Then, for any $T>0$, there exists a unique solution $\varphi$ to the Cauchy
problem

$\partial_{\tau}\varphi-\partial^{2}_{x}\alpha(\cdot,\varphi)=-\partial_{x}\left(\alpha(\cdot,\varphi)\varphi\right),\quad\varphi(x,0)=\varphi_{0}(x),\quad(x,\tau)\in\mathbb{R}\times(0,T),$
(80)

satisfying $\varphi\in C([0,T];H)\cap L^{2}((0,T);V)\cap
L^{\infty}((0,T)\times\mathbb{R})$.

###### Proof.

Since $\sigma^{2}(\bm{\theta})>0$ and $\triangle$ is a compact set, there
exist constants $0<\omega\leq L$ such that
$0<\omega\leq\sigma^{2}(\bm{\theta})\leq L$ for all $\bm{\theta}\in\triangle$.
It follows from Proposition 9 that

$\omega|\varphi|\leq|\alpha(x,\varphi)-\alpha(x,0)|\leq L|\varphi|.$ (81)

Since $\varphi_{0},h\in L^{\infty}(\mathbb{R})$ and $h(x)=\alpha(x,0)$, we
obtain $M:=\sup_{x\in\mathbb{R}}|\alpha(x,\varphi_{0}(x))|<\infty$.

Now, let us define the shift diffusion function by
$\tilde{\alpha}(x,\varphi)=\alpha(x,\varphi)-\alpha(x,0)$. Note that
$\alpha(x,0)=\min_{{\bm{\theta}}\in\triangle}-\mu(x,{\bm{\theta}})=h(x)$.
Then, equation (80) is equivalent to

$\partial_{\tau}\varphi+A\tilde{\alpha}(\cdot,\varphi)=\tilde{\alpha}(\cdot,\varphi)+\partial^{2}_{x}h-\partial_{x}\left(\alpha(\cdot,\varphi)\varphi\right),$

where $A=I-\partial^{2}_{x}$.

Next, let $g_{0}(\varphi)=\tilde{\alpha}(\cdot,\varphi)+\partial^{2}_{x}h$ and
$g_{1}(\varphi)=-w(\alpha(\cdot,\varphi))\varphi$. Here,
$w:\mathbb{R}\to\mathbb{R}$ is a suitable cutoff function

$w(\alpha)=\left\\{\begin{array}[]{ll}\underline{\psi}e^{\lambda
T},&\text{if}\ \ \alpha\leq\underline{\psi}e^{\lambda T},\\\
\alpha,&\text{if}\ \ \underline{\psi}e^{\lambda
T}<\alpha<\overline{\psi}e^{\lambda T},\\\ \overline{\psi}e^{\lambda
T},&\text{if}\ \ \alpha\geq\overline{\psi}e^{\lambda T},\\\
\end{array}\right.$

where $\overline{\psi}=M,\underline{\psi}=-M$. Then, the functions
$g_{0},g_{1}:H\to H$ are globally Lipschitz continuous.

Note that the diffusion function $\tilde{\alpha}$ satisfies the assumptions of
Theorem 4 with $\tilde{h}(x)=\tilde{\alpha}(x,0)\equiv 0$. Now, applying
Theorems 4 and 5, we obtain the existence and uniqueness of a solution
$\varphi\in C([0,T];H)\cap L^{2}((0,T);V)$ to the Cauchy problem (27). The
solution $\varphi$ satisfies the point-wise estimate (78). Thus,
$w(\alpha(x,\varphi(x,\tau)))=\alpha(x,\varphi(x,\tau))$, and $\varphi$ is a
solution to the Cauchy problem (80).

Finally, from (81), we deduce the $L^{\infty}((0,T)\times\mathbb{R})$ estimate
for the solution $\varphi$ since
$\sup_{x\in\mathbb{R}}|\alpha(x,\varphi(x,\tau))|\leq Me^{\lambda\tau}$, where
$\lambda=\sup_{x\in\mathbb{R}}p(x)$. Moreover, $\varphi\in
L^{\infty}((0,T)\times\mathbb{R})$, and

$\sup_{x\in\mathbb{R},\tau\in[0,T]}|\varphi(x,\tau)|\leq\omega^{-1}(Me^{\lambda
T}+\max_{x\in\mathbb{R}}|h(x)|).$

∎

####  7.2.4. Application to stochastic dynamic optimal portfolio selection
problem

This subsection presents examples of the stochastic process (68). We also
present examples of the corresponding utility function. For the the stochastic
process (68), we can consider a portfolio optimization problem with regular
cash inflow ($\varepsilon>0$)/outflow ($\varepsilon<0$) to a portfolio
representing pension funds savings. Here, we consider Slovak pension fund
savings according to Kilianová and Ševčovič [35]). It is well-known that in a
stylized financial market, the stochastic process $\\{y_{t}\\}_{t\geq 0}$
driven by the stochastic differential equation

$dy_{t}=(\varepsilon(y_{t})+{\bm{\mu}}^{T}{\bm{\theta}}y_{t})dt+\sigma(\bm{\theta})y_{t}dW_{t},$
(82)

represents a stochastic evolution of the value of a synthetized portfolio
$y_{t}$ consisting of $n$-assets with weights
${\bm{\theta}}=(\theta_{1},\cdots,\theta_{n})^{T}$, mean returns
${\bm{\mu}}=(\mu_{1},\cdots,\mu_{n})^{T}$, and an $n\times n$ positive
definite covariance matrix ${\bm{\Sigma}}$, i.e.,
$\sigma({\bm{\theta}})^{2}={\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}}$.

In this study, we assume that the value of the inflow/outflow rate
$\varepsilon=\varepsilon(y)$ also depends on the $y$ such that it is
characterized by the following

$\varepsilon(y)=\left\\{\begin{array}[]{ll}0,&\hbox{if}\ 0<y<y_{-},\\\
C,&\hbox{if}\ y_{-}<y,\end{array}\right.$

where $C$ is a constant. It represents a realistic pension saving model in
which there is no inflow/outflow provided that the value $y$ of the portfolio
is very small. Based on the logarithmic transformation $x=\ln y$ and Itô’s
lemma, the stochastic process $\\{x_{t}\\}$ satisfies (68), where
$\mu(x,{\bm{\theta}})={\bm{\mu}}^{T}{\bm{\theta}}-\frac{1}{2}\sigma({\bm{\theta}})^{2}+\varepsilon(e^{x})e^{-x}$.

Recently, Kilianová and Trnovská [37] established a further generalization of
the drift and volatility functions arising from the so-called worst-case
portfolio optimization problem. In such case, the volatility function is given
by

$\sigma({\bm{\theta}})^{2}=\max_{{\bm{\Sigma}}\in{\mathcal{K}}}{\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}},$

where $\mathcal{K}$ is a bounded uncertainty convex set of positive definite
covariance matrices. In general, only a part of the covariance matrix can be
calculated precisely, whereas entries are not precisely determined. For
instance, if only the diagonal $d$ is known, we have
${\mathcal{K}}=\\{{\bm{\Sigma}}\succ 0,\ diag({\bm{\Sigma}})=d\\}$.
Furthermore, the drift function is given by

$\mu(x,{\bm{\theta}})=\min_{{\bm{\mu}}\in{\mathcal{E}}}{\bm{\mu}}^{T}{\bm{\theta}}-\frac{1}{2}\sigma({\bm{\theta}})^{2}+\varepsilon(e^{x})e^{-x},$

where $\mathcal{E}$ is a given bounded uncertainty convex set of mean returns.

######  Remark 8.

Consider a class of utility function characterized by a pair of exponential
functions:

$u(x)=\begin{cases}-e^{-a_{0}x}-c^{*},&x\leq x^{\ast},\\\
-(a_{0}/a_{1})e^{-a_{1}x+(a_{1}-a_{0})x^{\ast}},&x>x^{\ast},\end{cases}$ (83)

where $c^{*}=e^{-a_{0}x^{*}}(a_{0}-a_{1})/a_{1}$ and
$a_{0},a_{1}\in\mathbb{R}>0$ are given constants. Here,
$x^{\ast}\in\mathbb{R}$ is a point at which the risk aversion changes. We
observe that $u$ is an increasing $C^{1}$ function with a jump in the second
derivative at the point $x^{\ast}$.

If $a_{0}>a_{1}>0$, then the utility function $u$ is called the decreasing
absolute risk aversion (DARA) function, representing an investor with a non-
constant decreasing risk aversion. It means that the higher the wealth of
investors, the lower their risk aversion, and hence the higher exposition of
the portfolio to more risky assets. According to Post, Fang and Kopa [53], the
piece-wise exponential DARA utility function plays an essential role in the
analysis of decreasing absolute risk aversion stochastic dominance introduced
by Vickson [67] (see also [34]). It is worth noting that the coefficients of
absolute risk aversion of the above utility functions
$-u^{\prime\prime}(x)/u^{\prime}(x)$ is equal to $a_{0}$ if $x\leq x^{*}$ or
to $a_{1}$ if $x>x^{*}$.

The piece-wise constant function
$\varphi_{0}(x)=-u^{\prime\prime}(x)/u^{\prime}(x)$ should be truncated
outside of some interval $(-\gamma,\gamma)$, where $\gamma$ is large enough.
Then, $\varphi_{0}\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$.
Therefore, the underlying utility function is modified by linear functions for
$x<-\gamma$ and $x>\gamma$.

Another simple example of a convex-concave utility function is the function
$u(x)=\arctan(x)$. Then,
$\varphi_{0}(x)=-u^{\prime\prime}(x)/u^{\prime}(x)=2x/(1+x^{2})$. Clearly,
$\varphi_{0}\in H=L^{2}(\mathbb{R})$. It is worth noting that the individual’s
reduction in marginal utility arising from a loss is absolutely greater than
the marginal utility from a financial gain. In the domain of gain ($x>x^{*}$),
the utility function is concave, indicating that investors show risk aversion
in this domain. In contrast, investors become risk-seeker when dealing with
losses, i.e., the utility function is convex for $x\leq x^{*}$.

####  7.2.5. Numerical examples

This subsection presents examples of the value function
$\alpha=\alpha(\varphi)$ to the parametric optimization problem with different
decision sets. First, we consider a simple decision set
$\triangle=\\{{\bm{\theta}}\in\mathbb{R}^{2},\,{\bm{\theta}}\geq
0,\bm{1}^{T}{\bm{\theta}}=1\\},n=2,\mu({\bm{\theta}})={\bm{\mu}}^{T}{\bm{\theta}},\sigma^{2}({\bm{\theta}})={\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}}$,
where ${\bm{\Sigma}}$ is a positive definite covariance matrix, and
${\bm{\mu}}$ is a positive vector of mean return. The value function
$\alpha=\alpha(\varphi)$ can be explicitly expressed as follows:

$\alpha(\varphi)=\left\\{\begin{array}[]{ll}E^{-}\varphi+D^{-},&\hbox{if}\
0<\varphi\leq\varphi_{*}^{-},\\\ A-\frac{B}{\varphi}+C\varphi,&\hbox{if}\
\varphi_{*}^{-}<\varphi<\varphi_{*}^{+},\\\ E^{+}\varphi+D^{+},&\hbox{if}\
\varphi_{*}^{+}\leq\varphi.\end{array}\right.$

Here, $(\varphi_{*}^{-},\varphi_{*}^{+})$ is the maximal interval where the
optimal value $\hat{\bm{\theta}}(\varphi)\in\triangle$ of the function
${\bm{\theta}}\mapsto-{\bm{\mu}}^{T}{\bm{\theta}}+\frac{\varphi}{2}{\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}}$
is strictly positive ($\hat{\bm{\theta}}(\varphi)>0$) for
$\varphi\in(\varphi_{*}^{-},\varphi_{*}^{+})$, and $C,E^{\pm}>0$, $B\geq 0$,
$A,D^{\pm}$ are constants explicitly depending on the covariance matrix
${\bm{\Sigma}}$ and the vector of mean return ${\bm{\mu}}$ such that the
function $\alpha$ is $C^{1}$ continuous at $\varphi_{*}^{\pm}$, i.e.,
$E^{\pm}=B/(\varphi_{*}^{\pm})^{2}+C$ and
$D^{\pm}=A-B/\varphi_{*}^{\pm}+C\varphi^{\pm}-E^{\pm}\varphi_{*}^{\pm}$. It is
clear that $\alpha$ is only $C^{1,1}$ continuous function with two points
$\varphi_{*}^{\pm}$ of discontinuity of the second derivative
$\alpha^{\prime\prime}$.

Furthermore, consider a decision set consisting of finite number of points.
Then, the value function $\alpha(\varphi)$ corresponding to such decision set
is only piece-wise linear. In other words, if
$\hat{\triangle}=\\{{\bm{\theta}}^{1},\cdots,{\bm{\theta}}^{k}\\}\subset\\{{\bm{\theta}}\in\mathbb{R}^{2},\,{\bm{\theta}}\geq
0,\bm{1}^{T}{\bm{\theta}}=1\\}$, then
$\alpha(\varphi)=\min_{i=1,\cdots,k}\alpha^{i}(\varphi)$, where
$\alpha^{i}(\varphi)=E^{i}\varphi+D^{i}$ is a linear function with the slope
$E^{i}=(1/2)({\bm{\theta}}^{i})^{T}{\bm{\Sigma}}{\bm{\theta}}^{i}>0$ and
intercept $D^{i}=-{\bm{\mu}}^{T}{\bm{\theta}}^{i}$.

Figure 7.2.5. a) shows the graph of the value function $\alpha$ corresponding
to the Slovak pension fund system with the two types of decision sets.
According to the data-set obtained from [38], the portfolio consists of the
stock index (with a high mean return $\mu_{s}=0.10$ and high volatility
$\sigma_{s}=0.3$) and bonds (with mean return $\mu_{b}=0.03$ and very low
volatility $\sigma_{s}=0.01$). The returns on stocks index and bonds are
negatively correlated $\varrho=-0.15$.
${\bm{\mu}}=(\mu_{s},\mu_{b})^{T}=(0.1,0.05)^{T}$. Then
${\bm{\Sigma}}_{11}=\sigma_{2}^{2},{\bm{\Sigma}}_{22}=\sigma_{b}^{2},{\bm{\Sigma}}_{12}={\bm{\Sigma}}_{21}=\varrho\sigma_{s}\sigma_{b}$.
As shown in Figure 7.2.5. a), the solid blue line corresponds to the convex
compact decision set
$\triangle=\\{{\bm{\theta}}\in\mathbb{R}^{2},\,{\bm{\theta}}\geq
0,\bm{1}^{T}{\bm{\theta}}=1\\}$. The piece-wise linear value function $\alpha$
(dotted red line) corresponds to the discrete decision set
$\hat{\triangle}=\\{{\bm{\theta}}^{1},{\bm{\theta}}^{2},{\bm{\theta}}^{3}\\}\subset\triangle$.
It represents the Slovak pension fund system consisting of three funds: growth
funds with ${\bm{\theta}}^{1}=(0.8,0.2)^{T}$ (80% of stocks and 20% of bonds),
balanced funds with ${\bm{\theta}}^{2}=(0.5,0.5)^{T}$ (equal proportion of
stocks and bonds), and conservative funds with ${\bm{\theta}}^{3}=(0,1)^{T}$
(only bonds) (c.f. [38]). Figure 7.2.5. b) shows the graph of the second
derivative $\alpha^{\prime\prime}_{\varphi}(\varphi)$ of the value function
$\alpha(\varphi)$ corresponding to the convex compact decision set
$\triangle$. It has the first point of discontinuity $\varphi_{*}^{-}$ close
to the value 2. For $n>2$, the number of discontinuities of
$\alpha_{\varphi}^{\prime\prime}$ increases (c.f. [35]). Figure 7.2.5. shows
another example of the value function and its second derivative for the
portfolio consisting of five stocks (BASF, Bayer, Degussa-Huls, FMC, and
Schering) entering DAX30 German stocks index. The covariance matrix
${\bm{\Sigma}}$ is taken from [21]. We set the vector of yields
${\bm{\mu}}=(0.03,0.02,0.04,0.01,0.01)^{T}$.

Figure 3.: The path $\hat{\bm{\theta}}(\varphi)$ as a function of $\varphi$.

The minimizer $\hat{\bm{\theta}}(\varphi)$ of the convex optimization problem

$\alpha(\varphi)=\min_{\bm{\theta}\in\triangle}\\{-\bm{\mu}^{T}\bm{\theta}+\frac{\varphi}{2}\bm{\theta}^{T}\Sigma\bm{\theta}\\}$

when considered as a function of the risk aversion parameter $\varphi$ is only
Lipschitz continuous in $\varphi$. According to Millgrom–Segal envelope
theorem, the derivative $\alpha^{\prime}(\varphi)$ is given by
$\alpha^{\prime}(\varphi)=\frac{1}{2}\hat{\bm{\theta}}(\varphi)^{T}\Sigma\hat{\bm{\theta}}(\varphi)$.
Figure 7.2.5. shows the path $\hat{\bm{\theta}}(\varphi)$ as a function of
$\varphi$ when increasing $\varphi$ from $\varphi=0$ to $\varphi\to\infty$.
For small values of $\varphi$, only one asset with maximal mean return is
active, i.e., $\theta_{1}>0,\theta_{2}=\theta_{3}=0$. For intermediate values
of $\varphi$, two assets are active
${\theta_{1}>0,\theta_{2}>0},\theta_{3}=0$. Moreover, for larger values of
$\varphi$, all three assets are active, i.e.,
${\theta_{1}>0,\theta_{2}>0,\theta_{3}>0}$. The path
$\varphi\mapsto\hat{\bm{\theta}}(\varphi)$ has a discontinuity in the first
derivative when it leaves lower dimensional object (vertex, edge) enter a
higher-dimensional object volume.

a) b)

Figure 4.: a) A graph of the value function $\alpha$, b) its second derivative
$\alpha^{\prime\prime}(\varphi)$ for the portfolio consisting of the stocks
index and bonds (c.f. [38]) for the convex compact decision set $\triangle$.
The dotted line in a) corresponds to the discrete decision set
$\hat{\triangle}=\\{{\bm{\theta}}^{1},{\bm{\theta}}^{2},{\bm{\theta}}^{3}\\}\subset\triangle$.
Source: our computation is based on the method from [66].

a) b)

Figure 5.: a) A graph of the value function $\alpha(\varphi)$, and b) the
second derivative $\alpha^{\prime\prime}_{\varphi}(\varphi)$ corresponding to
five stocks (BASF, Bayer, Degussa–Huls, FMC Scheringfrom) entering DAX30
index. Source: our computation is based on the method from [66].

The advantage of the Riccati transformation of the original HJB is twofold.
First, the diffusion function $\alpha$ can be computed in advance as a result
of quadratic optimization problem when the vector $\bm{\mu}$ and the
covariance matrix $\bm{\Sigma}$ are given or semidefinite programming problem
when they belong to a uncertainity set of returns and covariance matrices
(c.f. [37]). Figure 7.2.5. shows the vector of optimal weights $\bm{\theta}$,
as a function of the parameter $\varphi$, obtained as the optimal solution to
the quadratic optimization problem with the covariance matrix from [21]
corresponding to the five assets (BASF, Bayer, Degussa–Huls, FMC,
Scheringfrom) entering DAX30 index from 2008. There are more nontrivial
weights $\theta_{i}$ when the parameter $\varphi$ increases.

Figure 6.: The optimal vector $\bm{\theta}=(\theta_{1},\cdots,\theta_{n})^{T}$
as a function of $\varphi$ for the German DAX30 index. Source: our computation
is based on the method from [66], [35].

In contrast to the fully nonlinear character of the original HJB equation
(65), the transformed equation (74) represents a quasilinear parabolic
equation in the divergence form. Thus, efficient numerical schemes can be
constructed for this class of equation. In our computational experiments, we
employ the finite volume discretization scheme proposed and investigated by
Kilianová and Ševčovič [34, 35, 36]). Figure 7.2.5. a) shows the results of
time dependent sequence of profiles $\varphi(x,\tau)$ for a constant initial
condition $\varphi_{0}\equiv 9$. Figure 7.2.5. b) shows the solution profiles
for the initial condition $\varphi_{0}$ attaining four decreasing values
$\\{9,8\\}$. It represents DARA utility function. The function
$\varphi(x,\tau)$ is increasing in the $x$ variable and decreasing in the
$\tau=T-t$ variable. Therefore, the optimal vector $\bm{\theta}(x,\tau)$
contains more diversified portfolio of assets when $x$ increases and the time
$t\to T$ (see Figure 7.2.5.). Furthermore, it is reasonable to invest in an
asset with the highest expected return when the account value $x$ is low,
whereas an investor has to diversify the portfolio when $x$ is large and time
$t$ is approaching terminal maturity $T$.

Figure 7.: A solution $\varphi(x,\tau)$ for the DARA utility function with
$a_{0}=9$, $a_{1}=8$, $x^{\ast}=2$. Source: our computations based on the
numerical method from [66].

##  8\. Conclusions

This review paper presents the analysis of solution to nonlinear and nonlocal
partial integro-differential equation arising from financial market.
Specifically, we studied and analyzed linear and nonlinear PIDEs arising from
option pricing and portfolio selection problem. For the option pricing, we
investigated the systematic relationships between the corresponding PIDEs and
Black–Scholes models that study market’s illiquidity when the underlying asset
price follows a Lévy stochastic process with jumps. We employ the theory of
abstract semilinear parabolic equation in the Bessel potential spaces to
establish the qualitative properties of solutions to nonlocal linear and
nonlinear PIDE for a general class of the so-called admissible Lévy measures
satisfying suitable growth conditions at infinity and origin are also
established in the multidimensional space. We considered a general shift
function arising from nonlinear option pricing models, which takes into
account a large trader stock-trading strategy with the underlying asset price
following the Lévy process. Then, for the portfolio allocation problem, we
presented the qualitative properties results to the fully nonlinear HBJ
equation arising from stochastic dynamic optimization problem in Sobolev
spaces using the theory of monotone operator technique. Such HJB equation,
presented in abstract setting, arises from portfolio management problem, where
the goal of an investor is to maximize the condition expected terminal utility
of a portfolio. We also presented a stable, convergent, and consistent
numerical scheme that approximates solution of such PIDE. Several numerical
simulations are conducted to demonstrate the influence of a large trader and
intensity of jumps on the option price.

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