---
abstract: 'Using methods from the theory of total positivity, we provide a full classification of attainable term structure shapes in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The mathematical tools from total positivity can likely be applied to higher-dimensional generalizations of the Vasicek model and to other interest rate models as well.'
address: 'Institute for Mathematical Stochastics, TU Dresden'
author:
- 'Martin Keller-Ressel'
bibliography:
- 'references.bib'
title: 'Total positivity and the classification of term structure shapes in the two-factor Vasicek model'
---

Introduction
============

The term structure of interest rates – summarized in the form of the yield or forward curve – is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question – to be asked of any mathematical model of the term structure – which shapes of yield and forward curves the model is able to (re-)produce. Already in [@vasicek1977equilibrium] a paragraph is dedicated to this question, with Vasicek concluding that [`normal`]{} (increasing), [`inverse`]{} (decreasing) and [`humped`]{} (endowed with a single maximum) shapes can be attained in his single-factor model. The same classification of shapes has been shown to hold in the Cox-Ingersoll-Ross model and furthermore in all one-dimensional affine term structure models (including short-rate models with jumps), see [@cox1985theory Eq. (26)f], [@keller-ressel2008yield; @keller-ressel2018correction].\
It is also well-known, that in the Hull-White extended Vasicek model [@hull1990pricing] *any* initial term structure can be perfectly fitted and therefore that any shape of the term structure can be reproduced at the time of calibration. However, as time progresses, this initial shape will disappear and – due to ergodicity effects – the model will behave closer and closer to a Vasicek model with time-homogeneous coefficients. Therefore, even in view of Hull-White-extended models, the classification of attainable term structure shapes in time-homogeneous short-rate models is a reasonable and important question.\
Here, we provide for the first time a systematic classification of term structure shapes beyond the one-dimensional case. In our main result, Theorem \[thm:main\], we classify all attainable shapes for both the yield and forward curve in the two-dimensional Vasicek model. As expected, several additional shapes, such as a [`dipped`]{} curve, which are not attainable in the one-dimensional case become attainable in the two-factor model. We also give some stronger attainment results, showing for instance that also the locations of humps and dips can typically be chosen without restrictions.\
Our main mathematical tool is the theory of total positivity (see e.g. [@karlin1968total]), a theory linked to the variation-diminishing properties of certain matrices, function systems and integral kernels. Total positivity has broad applications in numerical interpolation, differential equations and stochastic processes. Within mathematical finance, it has been applied to study monotonicity and convexity of options prices [@kijima2002monotonicity] and to the principal-component-analysis of the term structure of interest rates [@salinelli2006correlation; @lord2007level]. Our application to the shape analysis of the term structure is new and fundamentally different from the results in [@salinelli2006correlation; @lord2007level]. While the results in this paper are limited to the two-dimensional Vasicek model, we are confident that the underlying theory can be applied to other multi-factor interest rate models as well.

Preliminaries
=============

Shapes of the term structure
----------------------------

In our terminology *term structure* refers to either the yield curve or the forward curve. The *shape* $\mathsf{S}$ of the term structure is defined by the number and sequence of local maxima or minima of the term structure curve. In common financial market terminology a local maximum is called a ‘hump’ and a local minimum a ‘dip’. As the term structure curves produced by the Vasicek model (or most other models) are smooth, it is clear that the shape of the term structure curve can be conveniently analyzed by considering its derivative: Any sign change of the derivative (from strictly positive to strictly negative or vice versa) corresponds to a local extremum of the term structure; the type of sign change (${\textup{\texttt{+}}}$ to ${\textup{\texttt{-}}}$ or ${\textup{\texttt{-}}}$ to ${\textup{\texttt{+}}}$) determines the type of the extremum (hump or dip). The basic shapes and their conventional names are listed in Table \[tab:shape\]. For ‘higher order’ shapes we use the letters `H` for a hump and `D` for a dip, e.g., the shape ${\texttt{HDH}}{}$ corresponds to a term structure with two local maxima, interlaced by a single local minimum.

\[tab:shape\]

  Shape $\mathsf{S}$ of the term structure   Description                                              Sign sequence of derivative
  ------------------------------------------ -------------------------------------------------------- ---------------------------------------------------------------------------------
  [`normal`]{}                               strictly increasing                                      ${\bm{[}{\textup{\texttt{+}}}\bm{]}}$
  [`inverse`]{}                              strictly descreasing                                     ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$
  [`humped`]{}                               single local maximum                                     ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$
  [`dipped`]{}                               single local minimum                                     ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
  [`HD`]{}                                   hump-dip, i.e. local maximum followed by local minimum   ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$
  [`DH`]{}, [`HDH`]{}, etc.                  further sequences of multiple ‘dips’ and ‘humps’         ${\bm{[}\dotsc\bm{]}}$

  : Shapes of the term structure

Sign sequences
--------------

We introduce several notions associated to a *sign sequence*. The primary purpose of a sign sequence will be to keep track of the number and the directions of sign changes of a numeric sequence or of a continuous function. The notion of a sign sequence appears implicitly in many of the results related to total positivity, however, the terminology introduced here is new.

(i) A **sign sequence** is a non-empty sequence of the symbols ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$. Only finite sign sequences will be considered here. Also zeroes can be allowed; we comment on this later. We include sign sequences in square brackets and write e.g. $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$$ for some valid sign sequences.

(ii) A sign sequence is called **pure** if the signs ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$ alternate, e.g., the sequences $${\bm{[}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$$ are pure. Any sign sequence can be reduced to a **pure sign sequence** by replacing blocks of ${\textup{\texttt{+}}}$’s by a single ${\textup{\texttt{+}}}$ and blocks of ${\textup{\texttt{-}}}$’s by a single ${\textup{\texttt{-}}}$, e.g. $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \quad \text{reduces to} \quad {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$ Note that this reduction preserves the number and direction of sign changes, which is our primary object of interest.

(iii) Two sign sequences are called **equivalent**, if they reduce to the same pure sequence. This defines an equivalence relation $\simeq$, e.g., $${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}}.$$

(iv) In a similar way we can define a **subsequence** relation $\subseteq$ where we treat blocks of signs as if they were single signs. E.g. we have $${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}} \subseteq {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{-}}}\bm{]}} \subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$

(v) A subsequence which also preserves the initial sign is called a **head** and a subsequence which preserves the terminal sign is called a **tail**. We write $${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}}{\overset{\scriptscriptstyle{H}}{\subseteq}}{\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}}, \qquad {\bm{[}{\textup{\texttt{+}}}\bm{]}}{\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$$ for the respective relations.

(vi) Sign sequences should only keep track of ‘strong’ sign changes.[^1] Therefore we add the convention that sign sequences **containing zeroes** can be reduced to a pure sequence by simply omitting all zeroes and then applying the reduction rules described above. E.g. we have $${\bm{[}{\textup{\texttt{+}}}0{\textup{\texttt{+}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}0{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}, \qquad {\bm{[}0{\textup{\texttt{-}}}00{\textup{\texttt{-}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{-}}}\bm{]}}.$$ Note that all strong sign changes (and their direction) are preserved under the described reduction.

(vii) If a variable, say $a$, appears inside a sign sequence, it should be interpreted as ‘sign of $a$’. E.g. the sign sequence ${\bm{[}ab\bm{]}}$ evaluates to ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ if $a = 6$ and $b = -1$ and to ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$ if $a = -1$, $b = 0$.

(viii)  Let $f$ be a continuous function, defined on a subset $X$ of ${\mathbb{R}}$ and not constantly zero. The **sign sequence of $f$** is the sequence of signs that $f$ takes on between its zeroes. Only functions with finite sign sequences will be considered and we denote the sign sequence of such a function $f$ by ${\mathrm{sseq}}(f)$. For example $$f(x) = x^2 -1\,\text{, defined on $X = [0,\infty)$} \qquad \Longrightarrow \qquad {\mathrm{sseq}}(f) = {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$

In some cases, the first and the last sign in the sign sequence of $f$ will be of particular interest. For them, we use the notation $$S_\text{init}(f)\qquad \text{and} \qquad S_\text{term}(f).$$

Total positivity and Descartes systems {#sec:tot_pos}
--------------------------------------

We introduce some definitions and key results from the theory of total positivity. For background and further details we refer to [@karlin1966tchebycheff; @karlin1968total] and [@borwein1995polynomials].

Let $X, Y \subseteq {\mathbb{R}}$ and let $K$ be a function (‘kernel’) from $X \times Y$ to ${\mathbb{R}}$. If $$\label{eq:K}
\det \begin{pmatrix} K(x_1,y_1) & K(x_1, y_2) & \dotsc &  K(x_1, y_m) \\ \vdots & \vdots && \vdots\\  K(x_m, y_1) &  K(x_m,y_2) & \dotsc &  K(x_m,y_m)\end{pmatrix} \ge 0$$ for any $m \in {\mathbb{N}}$, $x_1 < x_2 < \dotsm < x_m$ in $X$ and $i_1 < i_2  < \dotsm < i_m$ in $Y$, then $K(x,y)$ is called **totally positive**. If strict equality holds in , the kernel is called **strictly totally positive**.

(i) The kernels $K(x,y) = e^{xy}$ and $K(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$ are examples of totally positive kernels on ${\mathbb{R}}^2$ (or on any $X \times Y$ with $X, Y \subseteq {\mathbb{R}}$); see [@karlin1968total Ch.1 §2] and [@karlin1968total Ch.3, Eq. (1.13)ff]. The first kernel is even strictly totally positive.

(ii) A totally positive kernel on $X=Y= {\left\{1, \dotsc, n\right\}}$ can be written as a matrix; accordingly such matrices are also called totally positive, cf. [@ando1987totally] or [@hogben2013handbook Ch.29].

A crucial property of totally positive kernels is the following:

\[thm:vardim\_K\] Let $K$ be a totally positive kernel on $X \times Y$, such that $\int_Y K(x,y) dy < \infty$ for all $x \in X$. Let $f: Y \to {\mathbb{R}}$ be a bounded continuous function with finite sign sequence and set $$g(x) := \int_Y K(x,y) f(y) dy.$$ Then $${\mathrm{sseq}}(g)  \subseteq {\mathrm{sseq}}(f).$$

This result is a particular case of [@karlin1968total Ch. 5, Thm. 3.1], formulated in the language of sign sequences. It can be extended from integration with respect to Lebesgue measure $dy$ to a large class of $\sigma$-finite measures $d\mu(y)$ on $Y$. These extensions, however, will not be needed here.\
Next, we discuss a closely related definition, which applies to families of functions.

Let $X$ be a subinterval of ${\mathbb{R}}$ and let ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ be a family of continuous functions from $X$ to ${\mathbb{R}}$. If $$\label{eq:Phi}
\det \begin{pmatrix} \phi_{i_1}(x_1) & \phi_{i_2}(x_1) & \dotsc &  \phi_{i_m}(x_1)\\ \vdots & \vdots && \vdots\\  \phi_{i_1}(x_m) &  \phi_{i_2}(x_m) & \dotsc &  \phi_{i_m}(x_m)\end{pmatrix} > 0$$ for any $m \le n$, $x_1 < x_2 < \dotsm < x_m$ in $X$ and $i_1 < i_2  < \dotsm < i_m$ in ${\left\{1, \dotsc, n\right\}}$, then ${\mathcal{D}}$ is called a **Descartes system** on $X$.

\[rem:Descartes\]

(i) The order of the functions $\phi_1, \dotsc, \phi_n$ matters and a permutation of a Descartes system need not be a Descartes system.

(ii) A Descartes system can seen as a strictly totally positive kernel on $X \times {\left\{1, \dotsc, n\right\}}$

(iii) The family of monomials $(1,x,x^2,x^3, \dotsc, x^n)$ is a Descartes system.

(iv) The family of exponential functions $(e^{x\gamma_1}, \dotsc, e^{x \gamma_n})$ is a Descartes system if and only if $\gamma_1 < \gamma_2 < \dotsm < \gamma_n$ \[item:exp\]

Also Descartes systems enjoy variation-diminishing properties:

\[thm:vardim\] Let $(\phi_1, \dots, \phi_n)$ be a Descartes system and let $(a_1, \dotsc, a_n) \subseteq {\mathbb{R}}^n$. Then $$\label{eq:vardim}
{\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right) \subseteq {\bm{[}a_1 a_2, \dotsm a_n\bm{]}}.$$

(i) This theorem is [@karlin1966tchebycheff Thm. 3.1, 4.4] (see also [@borwein1995polynomials Thm. 3.2.4]), translated into the language of sign sequences.

(ii) The well-known ‘Descartes’ rule of signs’ for polynomials follows by applying this theorem to the Descartes system $(1, x, x^2, \dotsc, x^n)$; see [@borwein1995polynomials 3.2.E7].

Given a Descartes system ${\mathcal{D}}= (\phi_1, \dots, \phi_n)$, a function of the form $$\phi(x) := \sum_{i=1}^n a_i \phi_i(x)$$ is called a **D-polynomial** in ${\mathcal{D}}$. We call $\phi$ **extremal**, if equality is attained in . The next result concerns the interpolation properties of D-polynomials:

\[thm:extremal\] Let $(\phi_1, \dots, \phi_n)$ be a Descartes system on $X$ and let $r_1 < r_2 < \dotsm < r_{n-1}$ be $n-1$ distinct points in $X$. Then there exists a $D$-polynomial $\phi(x) = \sum_{i =1}^n a_i \phi_i(x)$ with all $a_i$ non-zero, which satisfies:

-   $\phi(r_i) = 0$ for all $i \in 1, \dotsc, n-1$;

-   $\phi$ has a strong sign change at each $r_i$ in the interior of $X$.

If all $r_i$ are interior points of $X$, then $\phi$ is extremal, i.e.,

-   ${\mathrm{sseq}}(\phi) \simeq {\bm{[}a_1\,a_2\dotsm a_n\bm{]}}$.

This result follows from [@karlin1966tchebycheff Ch. I, Thm. 5.1] or [@borwein1995polynomials 3.1.E11], but we provide a self-contained proof and some related results in Sec. \[app:interpolation\], \[app:interpolation2\].

The Vasicek model
=================

The Vasicek model, originally introduced by [@vasicek1977equilibrium] as a single-factor model, has been extended to multiple factors by [@dai2000specification] within the framework of affine term structure models. A detailed study of the two-dimensional case can also be found in [@brigo2007interest].

The Vasicek model with multiple factors
---------------------------------------

The $d$-dimensional Vasicek model is based on a factor process $Z = (Z^1, \dotsc, Z^d)$, with components given under a risk-neutral measure ${\mathbb{Q}}$ by $$\begin{aligned}
\label{eq:vasicek}
dZ^i_t = -\lambda_i ( Z_t^i - \theta_i) \, dt + \sigma_i dB_t^i, \qquad i \in {\left\{1, \dotsc, d\right\}}.\end{aligned}$$ The long-term rates $\theta = (\theta_1, \dotsc, \theta_d)$ are real and the Brownian motions $B^1, \dotsc, B^d$ may be correlated with the covariation matrix of $(\sigma_1 B^1, \dotsc, \sigma_d B^d)$ denoted by $\Sigma$. Moreover, we assume that the mean-reversion speeds are strictly positive and ordered as $$\lambda_1 < \lambda_2 < \dotsm < \lambda_d.$$ Thus, the speed of mean reversion increases with the index $i$ and $Z^1$ is the factor with the slowest mean-reversion. Given a vector $\kappa = (\kappa_1, \dotsc, \kappa_d)$ of strictly positive numbers and $\kappa_0 \in {\mathbb{R}}$, the short rate $r$ of the $d$-dimensional Vasicek model is defined as $$r_t = \kappa_0 + Z_t^\top \kappa = \kappa_0 + \kappa_1 Z_t^1 + \dotsm \kappa_d Z_t^d.$$ From [@dai2000specification], the bond price in this multivariate Vasicek model can be written as $$\label{eq:bond}
P(t,t+x) = {\mathbb{E}^{{\mathbb{Q}}}\left[\left.\exp\left(- \int_t^{t+x} r_s ds\right)\right|{\mathcal{F}}_t\right]} = \exp \left(A(x) + Z_t^\top B(x)\right)$$ where $A$ and $B$ are given as solutions of the ODE system

\[eq:Riccati\] $$\begin{aligned}
A'(x) &= F(B(x)), \qquad A(0) = 0\label{eq:Riccati_A}\\
B'(x) &= R(B(s)), \qquad B(0) = 0\label{eq:Riccati_B}\end{aligned}$$

with

\[eq:FR\_def\] $$\begin{aligned}
F(b) &= \mu^\top b + \frac{1}{2} b^\top \Sigma b- \kappa_0, \qquad \quad \mu^\top = \begin{pmatrix} \lambda_1 \theta_1, \dotsc, \lambda_d \theta_d \end{pmatrix},\label{eq:F_def}\\
R(b) &= -{\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d) b - \kappa.\label{eq:R_def}\end{aligned}$$

The differential equations decouple into scalar linear equations, which can be solved explicitly with solutions given by $$B_i(x) = \frac{\kappa_i}{\lambda_i} \left(e^{-\lambda_i x} - 1\right), \quad i \in {\left\{1,\dotsc, d\right\}}.$$ The explicit form of $A$ can be determined from , but will never be needed here and is therefore omitted.

Yield and forward curves
------------------------

The yield and forward curves in the Vasicek model are easily computed from and as $$\begin{aligned}
f(x;Z_t) &= -\partial_x \log P(t,t+x) = -F(B(x)) - Z_t^\top R(B(x))\label{eq:forward},\\
Y(x;Z_t) &= -\frac{1}{x} \log P(t,t+x) = - \frac{A(x)}{x} - Z_t^\top \frac{B(x)}{x}. \label{eq:yield}\end{aligned}$$ When we want to emphasize the dependency of these curves on the state vector $z$ and on some parameter $p$, we write $f(x;z,p)$ and $Y(x;z,p)$. We use the same notation for all quantities derived from $f$ and $Y$.\
To study the shapes of the yield and forward curve, in particular their local extrema, we need to consider their derivatives.

\[lem:curve\_diff\]The derivatives of the forward and the yield curve in the Vasicek model are given by $$\begin{aligned}
l(x) := \partial_x f(x;z) &= - \mu^\top B'(x) - B(x)^\top \Sigma B'(x) + z ^\top {\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d) B'(x) \label{eq:forward_d}\\
\intertext{and}
m(x) := \partial_x Y(x;z) &= \frac{1}{x^2} \left\{ (A(x) - xF(B(x))) + z^\top (B(x) - xR(B(x))) \right\} = \notag\\
&= \frac{1}{x^2}\int_0^x yl(y) dy.\label{eq:yield_integral}
 \end{aligned}$$

First, we calculate the gradient of $F$ and the Jacobian of $R$ as $$\begin{aligned}
\nabla F(b) &= \mu^\top + b^\top \Sigma\\
J_R(b) &= -{\mathrm{diag}}(\lambda_1, \dotsc, \lambda_d).\end{aligned}$$ For the forward curve, differentiation of gives $$l(x) = \partial_x f(x;z) = -\left(\nabla F(B(x)) +z^\top J_R(B(x))\right) B'(x),$$ which is . For the yield curve, differentiation of directly gives the first part of . Taking another derivative of $$x^2 m(x) = \left(A(x) - xF(B(x))\right) + z^\top \left(B(x) - xR(B(x))\right)$$ we obtain after some cancellations that $$\partial_x(x^2 m(x)) = - x \left(\nabla F(B(x)) + z^\top J_R(B(x))\right) B'(x)  = x l(x),$$ which yields the integral representation in .

A first application of total positivity
---------------------------------------

The first application of total positivity concerns the relation between yield and forward curves: We show that $m$ is a totally positive transformation of $l$.

\[lem:K\_tot\_pos\] The functions $l$ and $m$ of Lemma \[lem:curve\_diff\] are related by $$m(x) = \int_0^\infty K(x,y) l(y) dy,$$ where the kernel $K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{y \le x\right\}}}$ is totally positive on $(0,\infty) \times (0,\infty)$.

The kernel $K(x,y)$ is of the form $K(x,y) = \phi(x)\psi(y)L(x,y)$, where $\phi(x) = \frac{1}{x^2}$, $\psi(y) = y$ are strictly positive on $(0,\infty)$ and where $L(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$. The total positivity of $L(x,y) = {\mathbf{1}_{\left\{y \le x\right\}}}$ is shown in [@karlin1968total Ch. 3, Eq.(1.10)ff]. The total positivity of the composed kernel $K(x,y)$ now follows from [@karlin1968total Ch. 1, Thm. 2.1].

From the variation-diminishing property of $K$ (cf. Theorem \[thm:vardim\_K\]) we can immediately conclude that ${\mathrm{sseq}}(m) \subseteq {\mathrm{sseq}}(l)$ and hence, that the number of local extrema of the yield curve is bounded by the number of local extrema of the forward curve. This result can be slightly strengthened by also considering the initial signs of $l$ and $m$, which are easily obtained from Lemma \[lem:curve\_diff\]. Taking into account that $B(0) = 0$ and $B'(0) = -\kappa$, we obtain from that $$l(0) = \mu^\top \kappa - z^\top {\mathrm{diag}}(\lambda) \kappa.$$ Applying l’Hospital twice to yields $$m(0) = \frac{l(0)}{2},$$ and we conclude that the initial sign of $l$ and $m$ is always the same. Hence, we have shown the following lemma:

The sign sequences of $l$ and $m$ satisfy $${\mathrm{sseq}}(m){\overset{\scriptscriptstyle{H}}{\subseteq}}{\mathrm{sseq}}(l).$$

Reformulating the lemma in terms of term structure shapes, we conclude the following:

\[thm:fw\_yield\] In the multivariate Vasicek model

(a) the initial slope of yield and forward curve has the same sign;

(b) the number of local extrema of the yield curve is less or equal to the number of local extrema of the forward curve;

(c) if the number of local extrema is the same, then also the sequence of types (hump/dip) coincides.

We give an example which demonstrates how concrete restrictions of the yield curve can be derived from this result: Suppose for instance that the forward curve is [`humped`]{}. Then (b) and (c) leave as possible yield curve shapes [`inverse`]{}, [`normal`]{}, and [`humped`]{}. Restriction (a) further eliminates [`inverse`]{}, and the possible forward curve shapes are [`normal`]{}, [`humped`]{}. This is consistent with the analysis of the Vasicek model in the one-dimensional case (which is discussed in more detail below), but – as we have just shown – it also applies in the multivariate case.

The one-dimensional case revisited
----------------------------------

The classification of term structure shapes in the one-dimensional Vasicek model has already been discussed in [@vasicek1977equilibrium] (See also [@keller-ressel2008yield; @keller-ressel2018correction] for the case of general one-dimensional affine short rate models). We revisit this classification problem from the perspective of total positivity. First we calculate $l$, the derivative of the forward curve from Lemma \[lem:curve\_diff\] as $$l(x) =  \frac{\sigma^2 \kappa^2}{\lambda} e^{-2\lambda x} + \left\{\kappa \lambda (\theta - z) - \frac{\sigma^2 \kappa^2}{\lambda}\right\} e^{-\lambda x} .$$ From Rem. \[rem:Descartes\] we know that ${\mathcal{D}}= (e^{-2\lambda x}, e^{-\lambda x})$ is a Descartes system. Thus, $l$ is a D-polynomial with coefficients $$\begin{aligned}
u = \frac{\sigma^2 \kappa^2}{\lambda} \qquad \text{and} \qquad w = \kappa \lambda (\theta - z) - \frac{\sigma^2 \kappa^2}{\lambda}.\end{aligned}$$ From the variation-diminishing property of Descartes systems (Thm. \[thm:vardim\]), we conclude that $l$ has at most a single sign change. Thus, only the shapes [`normal`]{}, [`inverse`]{}, [`humped`]{} and [`dipped`]{} are possible. The shape [`dipped`]{} corresponds to a sign sequence ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ of $l$, which is not compatible with the positive sign of $u$. We conclude that the forward curve can only attain the shapes [`normal`]{}, [`inverse`]{}, and [`humped`]{}; the same must be true for the yield curve by Theorem \[thm:fw\_yield\]. This result corresponds to [@keller-ressel2008yield Thm. 3.9] (see also [@keller-ressel2018correction Thm. 2.1]).\
To find the regions of the state space associated to the different shapes it suffices to analyze the initial and terminal sign of $l$ and $m$. First we find $$S_\text{init}(l) = S_\text{init}(m) = {\mathrm{sign}}(\theta - z).$$ For the terminal sign, we find $$\begin{aligned}
S_\text{term}(l) &= {\mathrm{sign}}\left(\lim_{x \to \infty}e^{\lambda x} l(x) \right) = {\mathrm{sign}}\left(\theta - \frac{\sigma^2 \kappa}{\lambda^2} - z\right) \\
S_\text{term}(m) &= {\mathrm{sign}}\left(\lim_{x \to \infty}x^2 m(x) \right) = {\mathrm{sign}}\left(\int_0^x l(x) dx \right) = {\mathrm{sign}}\left(\theta - \frac{3 \sigma^2 \kappa}{4 \lambda^2} - z\right).\end{aligned}$$ Thus we conclude that the forward curve $f(x;Z_t)$ is

-   [`normal`]{}, if $Z_t \le \theta -  \frac{\sigma^2 \kappa}{\lambda^2}$;

-   [`humped`]{}, if $\theta -  \frac{\sigma^2 \kappa}{\lambda^2} < Z_t < \theta$; and

-   [`inverse`]{}, if $Z_t \ge \theta$.

For the yield curve $Y(x;Z_t)$, we conclude that it is

-   [`normal`]{}, if $Z_t \le \theta -  \frac{3\sigma^2 \kappa}{4\lambda^2}$;

-   [`humped`]{}, if $\theta -  \frac{3 \sigma^2 \kappa}{4\lambda^2} < Z_t < \theta$; and

-   [`inverse`]{}, if $Z_t \ge \theta$.

These results are consistent with [@vasicek1977equilibrium p. 186f], (see also [@desmettre2018moderne Satz 2.53], [@keller-ressel2018correction Thms. 2.1, 2.3]).

Classification of term structure shapes in the two-dim. case
============================================================

The main result
---------------

Let $\bm{P}$ denote the full parameter space of the two-dimensional Vasicek model, i.e. $$\bm{P} = {\left\{\begin{pmatrix}\theta_1\\\theta_2\end{pmatrix} \in {\mathbb{R}}^2, \kappa_0 \in {\mathbb{R}}, \begin{pmatrix}\kappa_1\\\kappa_2\end{pmatrix} \in (0,\infty)^2, \begin{pmatrix}\sigma_1\\\sigma_2\end{pmatrix} \in [0,\infty)^2, \rho \in [-1,1], 0 < \lambda_1 < \lambda_2\right\}},$$ We start with several definitions related to the attainability of term structure shapes.

\[def:attainable\]

(a) A shape $\mathsf{S}$ of the forward curve is called **attainable**, if we can find a parameter vector $p \in \bm{P}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, p)$ has shape $\mathsf{S}$.

(b) The shape $\mathsf{S}$ of the forward curve is called **strictly attainable**, if we can find a parameter vector $p \in \bm{P}$, such that $x \mapsto f(x; Z_t, p)$ attains shape $\mathsf{S}$ with strictly positive probability for all $t > 0$.

(c) A shape $\mathsf{S}$ of the forward curve with $k$ local extrema is called **strongly attainable**, if for any $0 < r_1 < \dotsm < r_k$, we can find a parameter vector $p \in \bm{P}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, p)$ has shape $\mathsf{S}$, with its local extrema located at $r_1, \dotsc, r_k$.

The same terminology is applied to the yield curve $x \mapsto Y(x;z,p)$.

We remark that in (b) it makes no difference whether probabilities under the risk-neutral measure ${\mathbb{Q}}$ or probabilities under the statistical measure ${\mathbb{P}}$ are considered, as ${\mathbb{Q}}$ and ${\mathbb{P}}$ are equivalent. It also makes no difference whether ‘all $t > 0$’ or ‘some $t > 0$’ are considered, as in the Vasicek model also the laws of $Z_t$ and $Z_{t'}$ are equivalent for any $t,t' > 0$.

It turns out that stronger attainability results can be obtained, in the sense that not all parameters in $\bm{P}$, but only a subset, need to be varied in order to attain a given shape. To formulate these results, we write $\bm{P}'$ for $\bm{P}$ with the volatility parameters $(\sigma_1, \sigma_2, \rho)$ removed, and introduce the parameter space of covariance matrices $$\bm{\Sigma} := {\left\{\Sigma  = \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix}, \sigma_1, \sigma_2 \in [0,\infty), \rho \in [-1,1]\right\}}.$$ Additional restrictions on $\bm{\Sigma}$ are denoted by $\bm{\Sigma}_{\rho <0}$, $\bm{\Sigma}_{\rho = 0}$, etc. We can now introduce the following stronger notion of attainability:

A shape $\mathsf{S}$ of the forward curve is called **$\Sigma$-attainable**, if for any parameter vector $p' \in \bm{P}'$, we can find a covariance matrix $\Sigma \in \bm{\Sigma}$ and a state vector $z \in {\mathbb{R}}^2$, such that $x \mapsto f(x; z, (p',\Sigma))$ has shape $\mathsf{S}$.\
The same terminology is applied to the yield curve $x \mapsto Y(x;z, (p',\Sigma))$.

Combining with Definition \[def:attainable\] we naturally obtain the notions of **strict and strong $\Sigma$-attainability**.\
Our third and final definition concerns the separation of scales in the two-dimensional Vasicek model. Recall that the mean-reversion speeds are ordered as $\lambda_1 < \lambda_2$. Thus, intuitively, $Z_t^1$ controls the long end of the term structure, while $Z_t^2$ controls the short end. The degree of separation between these two effects is captured by the following definition:

The two-dimensional Vasicek model is called

-   **scale-separated**, if $2 \lambda_1 < \lambda_2$,

-   **scale-proximal**, if $2 \lambda_1 > \lambda_2$, and

-   **scale-critical**, if $2 \lambda_1 = \lambda_2$.

We can now formulate our main result on the classification of term structure shapes:

\[thm:main\] Consider the two-dimensional Vasicek model.

(a) In the scale-separated or scale-critical case, the following yield and forward curve shapes are attainable:

    > [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{}, [`DH`]{}, [`HDH`]{};

    no other shapes are attainable.

(b) In the scale-proximal case, the following yield and forward curve shapes are attainable with $\rho \ge 0$:

    > [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{};

    no other shapes are attainable with $\rho \ge 0$.

(c) In the scale-proximal case, the following yield and forward curve shapes are attainable with $\rho < 0$:

    > [`normal`]{}, [`inverse`]{}, [`humped`]{}, [`dipped`]{}, [`HD`]{}, [`DH`]{}, [`HDH`]{}, [`DHD`]{}, [`HDHD`]{};

    no other shapes are attainable with $\rho < 0$.

To attain the listed shapes, it suffices to vary only the correlation and volatility parameters and the state vector:

\[cor:strong\] In all cases of Theorem \[thm:main\], the given shapes are $\Sigma$-attainable. In cases (a) and (b) the shapes are also strongly $\Sigma$- and $\Sigma_{\rho=0}$-attainable. In case (c), all shapes except possibly [`DH`]{}, [`HDH`]{}, [`DHD`]{}, and [`HDHD`]{} are also strongly $\Sigma$- and $\Sigma_{\rho<0}$-attainable.

The second corollary concerns the *strict* attainability in the sense of Def. \[def:attainable\].

\[cor:strict\]

(a) The shapes of Theorem \[thm:main\](a) are strictly $\Sigma_{\rho > 0}$-, $\Sigma_{\rho = 0}$-, and $\Sigma_{\rho < 0}$-attainable.

(b) The shapes of Theorem \[thm:main\](b) are strictly $\Sigma_{\rho > 0}$- and $\Sigma_{\rho = 0}$-attainable

(c) The shapes of Theorem \[thm:main\](c), with possible exception of ${\texttt{DH}}$ and ${\texttt{DHD}}{}$, are strictly $\Sigma_{\rho < 0}$-attainable.

\[rem:sigma\_att\]

(i) In all except the strong attainability results, the sets $\bm{\Sigma}_{\rho = 0}$, etc. can be further restricted to the subsets of *regular* covariance matrices.\[item:regular\]

(ii) In case (c) it remains an open question whether ${\texttt{DH}}{}$ and ${\texttt{DHD}}{}$ are strictly attainable and whether [`DH`]{}, [`HDH`]{}, [`DHD`]{}, and [`HDHD`]{} are strongly attainable.

(iii) We emphasize that these are *theoretical* attainability results. It is for instance not clear whether the more complex shapes can be attained within realistic ranges of parameter values or whether the local extrema that are generated are pronounced enough to be of practical relevance. This is especially true for the cases where a strong attainability result (which allows us to control the locations of extrema) is lacking.

### Role of covariance and correlation

We can immediately make some interesting observations on the role of the correlation parameter:

-   In the scale-separated case, the correlation parameter $\rho$ has no effect on the scope of attainable term structure shapes. In fact with $\rho = 0$ the same shapes can be attained as with $\rho > 0$ and $\rho < 0$.

-   In comparison, the scope of attainable term structure shapes in the scale-proximal case shrinks for $\rho \ge 0$, but grows for $\rho < 0$. Intuitively, the effects of the long-range factor $Z^1$ and the short-range factor $Z^2$ interact in the scale-proximal case, with positive correlation leading to congruence and negative correlation leading to interference.

Finally, we give a heuristic argument, which supports the conclusion of Cor. \[cor:strong\] and \[cor:strict\], that the variation of the (co-)variance parameters and the state vector is sufficient to attain the listed shapes: For strong attainability of [`HDH`]{}, the most complex shape in case (a), four degrees of freedom are needed: Three for the local extrema and an additional degree of freedom to select between [`HDH`]{} and [`DHD`]{}. The parameter space $\bm{\Sigma}_{\rho = 0}$ has two degrees of freedom and the state space ${\mathbb{R}}^2$ also has two, matching the required four degrees. In case (c) the most complex shape, [`HDHD`]{}, needs five degrees of freedom. The parameter space $\bm{\Sigma}_{\rho < 0}$ provides three of them and the state space ${\mathbb{R}}^2$ provides two. Finally, one could argue that in case (b) the congruent interaction of the two state processes prohibits the full utilization of all five degrees of freedom.\
The proof of Theorem \[thm:main\] and its corollaries rests on the introduction of Descartes systems related to yield and forward curves in the two-dimensional Vasicek model. These Descartes systems are given in Section \[sec:Descartes\] below and allow to apply the results from the theory of total positivity, which were discussed in Section \[sec:tot\_pos\]. The actual proof of Theorem \[thm:main\] is then given in two parts: First, in Section \[sec:admissible\], we show necessity, i.e., that no term structure shapes outside of the lists given in Theorem \[thm:main\] can be attained. Then we show sufficiency, i.e., that all listed shapes are actually attainable. This more difficult part is done in Section \[sec:attainable\].

Descartes systems for the Vasicek model {#sec:Descartes}
---------------------------------------

We introduce several Descartes systems associated to the Vasicek model. As we will show, the derivatives of forward and yield curve, i.e. the functions $l(x)$ and $m(x)$ introduced in Lemma \[lem:curve\_diff\], can be written as D-polynomials in these systems. The next Lemma follows directly from Remark \[rem:Descartes\](\[item:exp\]) and from the ordering of exponents that is implied by the scale-separation properties:

\[lem:Descartes\_f\] The following families of functions are Descartes systems on $[0,\infty)$: $$\begin{aligned}
{\mathcal{D}}_\text{sep} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-\lambda_2 x}, e^{-2 \lambda_1 x},  e^{-\lambda_1 x}) \quad &&\text{if} \quad  2 \lambda_1 < \lambda_2\\
{\mathcal{D}}_\text{prox} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-2 \lambda_1 x}, e^{-\lambda_2 x},  e^{-\lambda_1 x}) \quad &&\text{if} \quad  2 \lambda_1 > \lambda_2\\
{\mathcal{D}}_\text{crit} &= (e^{-2 \lambda_2 x}, e^{-(\lambda_1 + \lambda_2)x}, e^{-\lambda_2 x}, e^{-\lambda_1 x}) \quad &&\text{if} \quad  2 \lambda_1 = \lambda_2\end{aligned}$$

Note that the only difference between ${\mathcal{D}}_\text{prox}$ and ${\mathcal{D}}_\text{sep}$ are the order of the third and the fourth element. Collapsing these cases yields the boundary case ${\mathcal{D}}_\text{crit}$.\
For the analysis of yield curve shapes a slightly different Descartes system is needed:

\[lem:Descartes\_g\] Set $$g_\alpha(x) = \frac{1}{x^2}\int_0^x y e^{-\alpha y} dy = \tfrac{1}{ \alpha^2 x^2} \left(e^{-\alpha x} - 1 + \alpha x e^{-\alpha x}\right).$$ The following families of functions are Descartes systems on $[0,\infty)$: $$\begin{aligned}
{\mathcal{E}}_\text{sep} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2},  g_{\lambda_2}, g_{2 \lambda _1}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 < \lambda_2\\
{\mathcal{E}}_\text{prox} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2}, g_{2 \lambda _1}, g_{\lambda_2}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 > \lambda_2\\
{\mathcal{E}}_\text{crit} &= \left(g_{2 \lambda_2}, g_{\lambda_1 + \lambda_2}, g_{\lambda_2}, g_{\lambda_1}\right) \quad &&\text{if} \quad 2 \lambda_1 = \lambda_2\end{aligned}$$

Note that $g_\alpha(x)$ can be written as $$\label{eq:g_rep}
g_\alpha(x) = \int_0^\infty K(x,y) e^{-\alpha y} dy, \quad \text{where} \quad K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{y \le x\right\}}}.$$ Essentially, Lemma \[lem:Descartes\_g\] follows from the total positivity of $K(x,y)$ (see Lemma \[lem:K\_tot\_pos\]); the details are given in Sec. \[app:Descartes\_E\].

Necessary conditions for attainability {#sec:admissible}
--------------------------------------

To derive necessary conditions for attainability of term structure shapes, we write $l$ and $m$ as D-polynomials in the Descartes systems introduced in Lemmas \[lem:Descartes\_f\] and \[lem:Descartes\_g\] and determine their coefficients. Specializing to the case $d=2$, we obtain $$\begin{aligned}
l(x) &=  \begin{pmatrix} \kappa_1e^{-\lambda_1 x} \\\kappa_2e^{-\lambda_2 x} \end{pmatrix}^\top 
\left\{ - \begin{pmatrix}\lambda_1 \theta_1\\\lambda_2 \theta_2 \end{pmatrix} -  \begin{pmatrix}\sigma_1^2 & \rho \sigma_1 \sigma_2\\\rho \sigma_1 \sigma_2 &  \sigma_2^2\end{pmatrix}  \begin{pmatrix}\frac{\kappa_1}{\lambda_1} (e^{-\lambda_1 x} - 1)\\\frac{\kappa_2}{\lambda_2} (e^{-\lambda_2 x} - 1)\end{pmatrix} + \begin{pmatrix}z_1\\z_2 \end{pmatrix} \right\}.\end{aligned}$$ This can be expanded into $$\label{eq:l_poly}
l(x) = u_2 f_{2\lambda_2}(x) + c f_{\lambda_1 + \lambda_2}(x) + w_2 f_{\lambda_2}(x) +  u_1 f_{2\lambda_1}(x) + w_1 f_{\lambda_1}(x),$$ with $f_\alpha(x) = e^{-\alpha x}$ and coefficients given, for $j \in {\left\{1,2\right\}}$, by $$\begin{aligned}
u_j &= \frac{\sigma_j^2 \kappa_j^2}{\lambda_j} \ge 0\\
w_j &= w_j(z_j) = \kappa_j \lambda_j \left(\theta_j - z_j\right) - \frac{\sigma_j^2 \kappa_j^2}{\lambda_j} - \rho \lambda_j \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}
\intertext{and}
c &= \rho (\lambda_1 + \lambda_2) \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}.\end{aligned}$$ Using the representation $m(x) = \frac{1}{x^2} \int_0^x y l(y) dy$ from Lemma \[lem:curve\_diff\], it is obvious that also holds for $m(x)$, with $f_\alpha$ replaced by $g_\alpha$. Thus, we have shown the following.

\[lem:coef\]The functions $l(x)$ and $m(x)$ are D-polynomials in the Descartes systems ${\mathcal{D}}$ and ${\mathcal{E}}$ respectively, with coefficients given by

-   $(u_2, c, u_1, w_2, w_1)$ in the scale-proximal case,

-   $(u_2, c, w_2, u_1, w_1)$ in the scale-separated case,

-   $(u_2, c, w_2 + u_1, w_1)$ in the scale-critical case.

We can now use the variation-diminishing property of Descartes systems to derive restrictions on attainable forward and yield curve shapes.

\[thm:sise\_m\]If $\rho \ge 0$, then the sign sequence of $q \in {\left\{l,m\right\}}$, the derivatives of forward and yield curve, satisfies $$\begin{aligned}
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}w_2 w_1\bm{]}} \quad &&\text{(under scale-proximity)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}w_2 {\textup{\texttt{+}}}w_1\bm{]}} \quad &&\text{(under scale-separation)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}(u_1 + w_2) w_1\bm{]}} &&\quad \text{(under scale-criticality)}.\end{aligned}$$ If $\rho < 0$ then the sign sequence of $q \in {\left\{l,m\right\}}$ satsifies $$\begin{aligned}
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}w_2 w_1\bm{]}} \quad &&\text{(under scale-proximity)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}w_2 {\textup{\texttt{+}}}w_1\bm{]}} \quad &&\text{(under scale-separation)}\\
&{\mathrm{sseq}}(q)\subseteq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}(u_1 + w_2) w_1\bm{]}} &&\quad \text{(under scale-criticality)}.\end{aligned}$$

For forward curves this result can be strengthened by using additional information from the terminal sign of $m$.

\[cor:sise\_l\] In Theorem \[thm:sise\_m\] ‘$\subseteq$’ can be replaced by ‘${\overset{\scriptscriptstyle{T}}{\subseteq}}$’ whenever the sign sequence of $l$ is considered.

Theorem \[thm:sise\_m\] follows by applying Theorem \[thm:vardim\] to the coefficients given in Lemma \[lem:coef\]. In doing so, we take into account that $u_j$ has positive sign regardless of the choice of parameters, and apply the reductions of sign sequences described in Sec. 1.1(ii) to arrive at the expressions on the right hand sides.\
For the corollary, the obtained relations can be strengthened from $\subseteq$ to ${\overset{\scriptscriptstyle{T}}{\subseteq}}$ by analyzing the terminal sign of $l$. From we first obtain that $\lim_{x \to \infty} l(x) = 0$, which, however, yields no information on the terminal sign. Rather, the terminal sign of $l$ must be determined by the component with the slowest decay, which is $w_1 f_{\lambda_1}(x) = w_1 e^{-\lambda_1 x}$. Thus, the terminal sign of $l$ is equal to the sign of $w_1$, which is the last sign in all sequences of Lemma \[lem:coef\]. We conclude that ${\mathrm{sseq}}(l)$ is not just a subset, but rather a tail of all the sign sequences that were obtained on the right hand sides.[^2]

Using Theorem \[thm:sise\_m\] we obtain the first part of Therorem \[thm:main\].

Consider the case of the forward curve. The shape of the forward curve is determined by the sign sequence of $l$, and this sign sequence is controlled by the results of Corollary \[cor:sise\_l\]. Hence, restrictions on attainable term structure shapes can be obtained by iterating through all cases of Corollary \[cor:sise\_l\] and through the four possible sign combinations of $w_1$ and $w_2$. Note that we only need to consider the strict signs ${\textup{\texttt{+}}}$ and ${\textup{\texttt{-}}}$, because zeroes can be omitted from sign sequences and do not lead to additional shapes. Instead of listing all possible combinations, we discuss two exemplary cases:

-   Suppose that $\rho \ge 0$, $w_1 > 0$ and $w_2 < 0$. In the scale-proximal case we obtain from Corollary \[cor:sise\_l\], that $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}.$$ The possible tail sequences of ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ are ${\bm{[}{\textup{\texttt{+}}}\bm{]}}, {\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ itself. These cases correspond to the shapes [`normal`]{}, [`humped`]{} and [`HD`]{}, and we conclude that no other forward curve shapes can be attainable under the given parameter restrictions. Switching to scale-separation, Corollary \[cor:sise\_l\] yields $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{+}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}},$$ and the same admissible shapes are obtained as in the scale-separated case.

-   Now suppose that $\rho \ge 0$, $w_1 < 0$ and $w_2 < 0$. In the scale-proximal case Corollary \[cor:sise\_l\] yields $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{-}}}\bm{]}} \simeq {\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}},$$ which leaves the shapes [`inverse`]{}, [`humped`]{} as potentially attainable shapes. In the scale-separated case we obtain $${\mathrm{sseq}}(l){\overset{\scriptscriptstyle{T}}{\subseteq}}{\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}},$$ which, in addition, leaves [`DH`]{} and [`HDH`]{} as potentially attainable.

Applying the same procedure to all other cases produces the lists given in the theorem, in the case of forward curves. The scale-critical case can be treated like the scale-proximal case. For yield curves, we apply Theorem \[thm:sise\_m\] to $m$ in the same manner. Despite the weaker constraint $\subseteq$ instead of ${\overset{\scriptscriptstyle{T}}{\subseteq}}$, it turns out (after iterating through all cases) that the same lists of shapes are obtained.

Sufficient conditions for attainability {#sec:attainable}
---------------------------------------

To complete the proof of Theorem \[thm:main\], we need to show sufficiency, i.e., that all listed shapes are actually attainable. Before going into details, we describe the general strategy of the proof: Let a shape $\mathsf{S}$ of the forward curve with $k$ local extrema be given. Choosing a suitable Descartes-subsystem ${\mathcal{D}}'$ of ${\mathcal{D}}$ with $k+1$ elements, we can apply Theorem \[thm:extremal\] and find a D-polynomial $f$ in ${\mathcal{D}}'$, such that $f$ has a sign sequence with $k$ sign changes, which corresponds to the shape $\mathsf{S}$. Padding the list of coefficients with zeroes, we can write $f$ as a D-polynomial in the full system ${\mathcal{D}}$, i.e. as $$f(x) = a_{2\lambda_2} f_{2\lambda_2}(x) + a_{\lambda_1 + \lambda_2} f_{\lambda_1 + \lambda_2}(x) + a_{\lambda_2} f_{\lambda_2}(x) +  a_{2 \lambda_1} f_{2\lambda_1}(x) + a_1 f_{\lambda_1}(x),$$ where we have labeled the coefficients $a$ consistently with the basis functions of ${\mathcal{D}}$. Comparing coefficients with , we can conclude that the shape $\mathsf{S}$ is attainable[^3] in the Vasicek-model, if we can show that the system of equations

\[eq:key\] $$\begin{aligned}
\frac{\sigma_1^2 \kappa_1^2}{\lambda_1} &= a_{2\lambda_1} \label{eq:key_sigma1}\\
\frac{\sigma_2^2 \kappa_2^2}{\lambda_2} &= a_{2\lambda_2} \label{eq:key_sigma2}\\
\rho (\lambda_1 + \lambda_2) \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2} &= a_{\lambda_1 + \lambda_2} \label{eq:rho}\\
\kappa_1 \lambda_1 \left(\theta_1 -z_1\right) - \frac{\sigma_1^2 \kappa_1^2}{\lambda_1} - \rho \lambda_1 \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}  &= a_{\lambda_1}\label{eq:key_z1}\\
\kappa_2 \lambda_2 \left(\theta_2 -z_2\right) - \frac{\sigma_2^2 \kappa_2^2}{\lambda_2} - \rho \lambda_2 \frac{\sigma_1\sigma_2\kappa_1\kappa_2}{\lambda_1\lambda_2}  &= a_{\lambda_2}\label{eq:key_z2}\end{aligned}$$

has a solution $(\sigma_1, \sigma_2, \rho, z_1, z_2) \in [0,\infty)^2 \times [-1,1] \times {\mathbb{R}}^2$. The argument for yield curves is analogous, using the appropriate Descartes system ${\mathcal{E}}$ from Lemma \[lem:Descartes\_g\].\
Having reduced the attainability problem to the equation system , we need to discuss its solvability: Clearly, whenever – can be solved for $(\sigma_1, \sigma_2, \rho)$, then also and can be solved for $(z_1, z_2)$. Moreover, the solvability of and only depends on the signs of $a_{2\lambda_1}$ and $a_{2\lambda_2}$. It is therefore only for which solvability is nontrivial, due to the restriction $\rho \in [-1,1]$. These elementary observations are summarized in the following Lemma:

\[lem:key\] Consider the system of equations given in

(a) If $a_{2\lambda_1} < 0$ or $a_{2\lambda_2} < 0$, then has no solution.

(b) If $a_{2\lambda_1} = 0$ and $a_{2\lambda_2} \ge 0$, or if $a_{2\lambda_1} \ge 0$ and $a_{2\lambda_2} = 0$ then has a solution. In this solution $\sigma_1 = \rho = 0$ or $\sigma_2 = \rho = 0$ or both.

(c) If $a_{2\lambda_1} > 0$ and $a_{2\lambda_2} > 0$, then has a solution if and only if $$\rho := \frac{\sqrt{\lambda_1 \lambda_2}}{\lambda_1  + \lambda_2 } \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2\lambda_1}a_{2\lambda_1}}} \quad \text{is in $[-1,1]$.}$$

To complete the proof of Theorem \[thm:main\] we apply the strategy outlined above on a case-by-case basis to the different shapes:

We partition the proof according to the number $k$ of local extrema of the term structure curve; later we also need to distinguish between the cases (a), (b) and (c) given in Theorem \[thm:main\].

(i) For $k=0$ we use the system ${\mathcal{D}}_1 = (f_{\lambda_1})$. We set $a^\pm_{\lambda_1} = \pm 1$ and all other coefficients to zero. This yields the D-polynomials $f_\pm(x) = \pm f_{\lambda_1}(x) = \pm e^{-\lambda_1 x}$ with sign sequences ${\bm{[}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}\bm{]}}$. Setting $z_2 = \sigma_1 = \sigma_2 = \rho = 0$ the system can be solved for $z_1$ in both cases. We conclude that the shapes [`normal`]{} and [`inverse`]{} are attainable.

(ii) For $k=1$ we use the system ${\mathcal{D}}_2 = (f_{\lambda_2}, f_{\lambda_1})$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{\lambda_2}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$. Setting $\sigma_1 = \sigma_2 = \rho = 0$ the system can be solved for $(z_2, z_1)$ in both cases. We conclude that the shapes [`dipped`]{} and [`humped`]{} are attainable.

(iii) For $k=2$ we use the system ${\mathcal{D}}_3 = (f_{2 \lambda_2}, f_{\lambda_2}, f_{\lambda_1})$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{2 \lambda_2}, a^\pm_{\lambda_2}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$. Setting $\sigma_1 = \rho = 0$ the system can be solved for $(\sigma_2, z_2, z_1)$ in the case of $f_+$. In the case of $f_-$ the system cannot be solved, because $a^-_{2 \lambda_2} < 0$. We conclude that the shape [`HD`]{} is attainable.

At this point we have already covered all attainable shapes in the scale-proximal case with $\rho \ge 0$, i.e., part (b) of the theorem. Next we complete part (a), i.e., the scale-separated case:

(i) For $k=2$ we can alternatively use the system ${\mathcal{D}}_{3,sep} = (f_{\lambda_2}, f_{2 \lambda_1}, f_{\lambda_1})$, which is a subsystem of ${\mathcal{D}}_\text{sep}$.[^4] By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{\lambda_2}, a^\pm_{2 \lambda_1}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$. Setting $\sigma_2 = \rho = 0$ the system can be solved for $(z_2, \sigma_1, z_1)$ in the case of $f_-$. In the case of $f_+$ the system cannot be solved, because $a^-_{2 \lambda_1} < 0$. We conclude that the shape [`DH`]{} is attainable.

(ii) For $k=3$, we use the system ${\mathcal{D}}_\text{4,sep} =  (f_{2 \lambda_2} f_{\lambda_2}, f_{2 \lambda_1}, f_{\lambda_1})$, which is a subsystem of ${\mathcal{D}}_\text{sep}$. By Theorem \[thm:extremal\] we can find two extremal D-polynomials $f_+, f_-$ with coefficients $(a^\pm_{2 \lambda_2}, a^\pm_{\lambda_2}, a^\pm_{2 \lambda_1}, a^\pm_{\lambda_1})$ and sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$. Setting $\rho = 0$ the system can be solved for $(\sigma_2, z_2, \sigma_1, z_1)$ in the case of $f_+$. In the case of $f_-$ the system cannot be solved, because $a^-_{2 \lambda_1} < 0$ and $a^-_{2 \lambda_2} < 0$. We conclude that [`HDH`]{} is attainable.

At this point we have also covered all attainable shapes in the scale-separated case (with arbitrary $\rho$) and thus part (a) is complete. The most difficult case is part (c), i.e., the scale-proximal case with $\rho < 0$. Here, Theorem \[thm:extremal\] is not sufficient to find suitable D-polynomials $f_\pm$ and we have to use the more specialized result Lemma \[lem:special\] instead.

(i) For $k=3$ we use the system ${\mathcal{D}}_{4,prox} = (f_{2 \lambda_2}, f_{\lambda_1 + \lambda_2}, f_{2 \lambda_1}, f_{\lambda_2})$, which is a subsystem of ${\mathcal{D}}_\text{prox}$. By Lemma \[lem:special\] we can find two sets of real numbers $0 < r_1^+ < r_2^+ < r_3^+$ and $0 = r_1^0 < r_2^0 < r_3^0$ as well as D-polynomials $f_+$ and $f_0$ with the following properties:

    -   The zeroes of $f_+$ and $f_0$ are located exactly at the points $r_1^+, r_2^+, r_3^+$ and $r_1^0 = 0, r_2^0, r_3^0$;

    -   the sign sequence of $f_+ $ is ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$ and the sign sequence of $f_0$ is ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$;

    -   the coefficients of both $f_+$ and $f_0$ have sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}\bm{]}}$.

    Moreover, the coefficients (of both $f_+$ and $f_0$) satisfy $$\left| \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2 \lambda_1}a_{2 \lambda_2}}} \right| < 2;$$ see . Thus, applying the geometric-arithmetic-mean inequality, we obtain $$\label{eq:rho_solvable}
    |\rho| \le \frac{\sqrt{\lambda_1 \lambda_2}}{\lambda_1 + \lambda_2} \left| \frac{a_{\lambda_1 + \lambda_2}}{\sqrt{a_{2 \lambda_1}a_{2 \lambda_2}}} \right| < 1.$$ By Lemma \[lem:key\], this implies that the system of equations is solvable. We conclude that the shapes [`HDH`]{} and [`DH`]{} are attainable.

(ii) For $k=4$ we use the full system ${\mathcal{D}}_\text{prox}$. As in the previous case, we can apply Lemma \[lem:special\] to find two D-polynomials $f_+$ and $f_0$ with prescribed zeroes and with sign sequences ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and ${\bm{[}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ respectively. The first zero of $f_0$ is located at the boundary point $r_1^0 = 0$. Moreover, the coefficients of both $f_+$ and $f_0$ have sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and inequality holds. Thus, Lemma \[lem:key\] implies that the system of equations is solvable and we conclude that the shapes [`HDHD`]{} and [`DHD`]{} are attainable.

Having completed part (c), the last case, the proof of Theorem \[thm:main\] is finished.

Strict and strong attainability
-------------------------------

We now discuss how the stronger conclusions of Corollary \[cor:strong\] and \[cor:strict\] can be obtained from the proof of Theorem \[thm:main\] that was given above. First, observe that in all steps (i) - (vii) of the proof, we have shown that the system of equations could be solved by choosing suitable covariance parameters $(\sigma_1, \sigma_2, \rho)$ and state vectors $(z_1, z_2)$ and that it was not necessary to modify any of the remaining parameters in $\bm{P}'$. This shows that attainability can be strengthened to *$\Sigma$-attainability* in all cases.\
Next, observe that that in steps (i) - (v) of the proof we have used Theorem \[thm:extremal\] to find a D-polynomial $f_+$ or $f_-$, which, after solving , equates to $l$, the derivative of the forward curve. Theorem \[thm:extremal\] allows us to predetermine all zeroes $r_1 < \dotsm < r_k$ of $f_\pm$, and hence the locations of the extrema of the forward curve. The same is true for $m$, the derivative of the yield curve. This shows that in cases (i) -(v) we obtain *strong* $\Sigma$-attainability. In addition, note that it was sufficient to choose $\rho = 0$ in all cases (i) - (v). Thus, we even get *strong $\Sigma_{\rho = 0}$-attainability*. This completes the arguments needed for Cor. \[cor:strong\].\
The contents of Cor \[cor:strict\] follow from a perturbation argument. Consider for instance case (iii) in the proof of Thm. \[thm:main\]: There, we have shown that we can find parameters $\sigma_1 = \rho = 0$, $\sigma_2 > 0$ and a state vector $(z_1,z_2) \in {\mathbb{R}}$, which produces the sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ corresponding to shape ${\texttt{HD}}{}$. Suppose that a perturbation $$\sigma_1^\epsilon = \epsilon, \quad \rho^\epsilon = \pm \epsilon \quad \text{and} \quad z_1^\epsilon = z_1 \pm \epsilon, \quad z_2^\epsilon = z_2 \pm \epsilon$$ with $\epsilon$ in some small set $[0,\delta)$ still produces the same sign sequence ${\bm{[}{\textup{\texttt{+}}}{\textup{\texttt{-}}}{\textup{\texttt{+}}}\bm{]}}$ and shape ${\texttt{HD}}{}$. Then, we may conclude

-   that the shape ${\texttt{HD}}{}$ is strictly $\Sigma$-attainable, as $(Z_t^1, Z_t^2)$ visits any small neighborhood of $(z_1,z_2)$ with strictly positive probability;

-   that ${\texttt{HD}}{}$ is also $\Sigma_{\rho > 0}$- and $\Sigma_{\rho < 0}$-attainable, as we have relaxed the condition $\rho = 0$ to $\rho^\epsilon = \pm \epsilon$; and

-   that it is sufficient to consider regular matrices $\Sigma$, as we have relaxed the condition $\sigma_1 = 0$ to $\sigma_1^\epsilon = \epsilon$.

The necessary perturbation Lemma is given below. Applying the same argument to each of the cases (i) - (v) in the proof yields part (a) and (b) of Cor. \[cor:strict\]. For cases (vi) and (vii) note that the Lemma can only be applied to the D-polynomials $f_+$, but not to $f_0$, which has a zero at the boundary of $[0,\infty)$ and is not an extremal D-polynomial. This yields part (c) of Cor. \[cor:strict\] and Rem. \[rem:sigma\_att\](i).

Let $\phi = \sum_{i=1}^n a_i \phi_i$ be a non-vanishing D-polynomial in a Descartes system ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ on a subinterval $X \subset {\mathbb{R}}$ which satisfies $${\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right) \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}$$ and has no zeroes on the boundary of $X$. Then there exist $(b_i)_{i=1\dotsc n} \in {\left\{-1,+1\right\}}$ and $\delta > 0$, such that $${\mathrm{sseq}}\left(\sum_{i=1}^n a_i^\epsilon \phi_i\right) \simeq {\mathrm{sseq}}\left(\sum_{i=1}^n a_i \phi_i\right)$$ for all $\epsilon \in [0,\delta)$ and with $a_i^\epsilon = a_i + \epsilon b_i$.

First, we show that the sequence $(b_i)$ can be chosen such that $${\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}.$$ To this end define $b_1, \dotsc, b_n$ as follows: $$\begin{aligned}
a_i  > 0 \quad &\Longrightarrow &\quad &b_i := +1\\
a_i < 0  \quad &\Longrightarrow &\quad &b_i := -1\\
a_i = 0 \quad &\Longrightarrow &\quad &b_i := \begin{cases}+1 \quad &\text{if the block of zeroes containing $a_i$ borders}\\&\;\text{on at least one $a_j > 0$,} \\-1 \quad &\text{else.}\end{cases}\end{aligned}$$ It is easy to see that the number and direction of strong sign changes in $(a_1^\epsilon, \dotsc, a_n^\epsilon)$ is the same as in $(a_1, \dotsc, a_n)$ for all $\epsilon \ge 0$, i.e., we have $${\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}}, \quad \forall\,\epsilon \ge 0.$$ Set $\phi^\epsilon = \sum_{i=1}^n a_i^\epsilon \phi_i$. Then by Theorem \[thm:vardim\] $$\label{eq:sise_sub}
{\mathrm{sseq}}(\phi^\epsilon)  \subseteq {\bm{[}a_1^\epsilon \dotsc a_n^\epsilon\bm{]}} \simeq {\bm{[}a_1 \dotsc a_n\bm{]}} \simeq {\mathrm{sseq}}(\phi),$$ for all $\epsilon \ge 0$, and we have shown that $\phi^\epsilon$ cannot have *more* sign changes than $\phi$. It remains to show that equivalence holds for small enough $\epsilon$. Let $k$ be number of strong sign changes of $\phi$. Clearly, we can find $r_0, \dotsc, r_k$ such that the sequence $\phi(r_i)_{i=0, \dotsc, k}$ is of alternating signs. Each interval $(r_i, r_{i+1})$ must contain exactly one zero of $\phi$. Set $$\delta := \frac{\min_{i=0, \dotsc, k} |\phi(r_i)|}{\sum_{j=1}^n \max_{i=0, \dotsc, k} |\phi_j(r_i)|}.$$ Then, $\delta > 0$ and for all $\epsilon \in [0,\delta)$ $$\begin{aligned}
\left|1 - \frac{\phi^\epsilon(r_i)}{\phi(r_i)} \right| &= \left|\frac{\phi(r_i) - \phi^\epsilon(r_i)}{\phi(r_i)} \right| \le \frac{1}{|\phi(r_i)|} \left| \sum_{j=1}^n \epsilon b_j \phi_j(r_i) \right| \le \\ &\le \epsilon \frac{\sum_{j=1}^n |\phi_j(r_i)|}{|\phi(r_i)|} < 1.\end{aligned}$$ This shows that the sequence $\phi^\epsilon(r_i)_{i=0, \dotsc, k}$ has the same alternating signs as $\phi(r_i)_{i=0, \dotsc, k}$ and hence that $\phi^\epsilon$ has at least the same number of zeroes as $\phi$, for all $\epsilon \in [0,\delta)$. Together with , this completes the proof.

Discussion and Outlook
======================

We have shown that the theory of total positivity, in particular the notion of Descartes systems, can be applied to the problem of classifying term structure shapes in the one- and two-dimensional Vasicek model. In principle, this analysis can be extended to Vasicek models with three and more factors, presumably at the expense of even more cases of ‘scale-separation’ and correlation links between the factors that need to be distinguished. In the two-dimensional case, a natural next step that builds on the results given above, is a ‘state-space analysis’ of term structure shapes, i.e., to determine and classify the regions of the state space in which a particular shape of the term structure is produced. Finally, it would be interesting to see, whether the theory of total positivity can also be applied to non-Gaussian affine (or even non-affine) interest rate models, such as those of [@dai2000specification].

Additional results on Descartes systems
=======================================

Let a family $(\phi_1, \dotsc, \phi_k)$ of functions on $X \subseteq {\mathbb{R}}$ be given. We set $\bm{x} = (x_1, \dotsc, x_k) \in X^k$ and $$\Delta_k(X) := {\left\{\bm{x} \in X^k: x_1 < \dotsc < x_k\right\}}.$$ From [@borwein1995polynomials] we adopt the compact notation $$\label{eq:det}
D\begin{pmatrix}\phi_1, \dotsc ,\phi_k\\x_1, \dotsc, x_k\end{pmatrix} := \det \begin{pmatrix} \phi_{1}(x_1) & \phi_{2}(x_1) & \dotsc &  \phi_{k}(x_1)\\ \vdots & \vdots && \vdots\\  \phi_{1}(x_k) &  \phi_{2}(x_k) & \dotsc &  \phi_{k}(x_k)\end{pmatrix}.$$ An important special case is the Vandermonde determinant, which for any real $(\gamma_i)_{i=1, \dotsc, k}$ evaluates as $$\label{eq:Vandermonde}
D\begin{pmatrix}1, x ,x^2, \dotsc, x^{k-1}\\ \gamma_1, \dotsc, \gamma_k\end{pmatrix} = \prod_{j=1}^{k-1} (\gamma_j - \gamma_{j-1}),$$ see e.g. [@hogben2013handbook Ch. 22.4]. For sufficiently differentiable functions $\phi_1, \dotsc, \phi_k$, we also introduce the Wronskian determinant (or simply Wronskian) $$\label{eq:wronskian}
W(\phi_1, \dotsc, \phi_k)(x) = \det \begin{pmatrix} 1 & \phi_1(x) & \phi_1'(x) & \dotsm & \phi_1^{(k)}(x) \\ 1 & \phi_2(x) & \phi_2'(x) & \dotsm & \phi_2^{(k)}(x) \\ \vdots & \vdots & \vdots & & \vdots\\ 1 & \phi_k(x) & \phi_k'(x) & \dotsm & \phi_k^{(k)}(x) \\\end{pmatrix}.$$ In [@karlin1968total Ch. 2, §2] relations between the two determinants in and as well as intermediate notions of ‘derived determinants’ are discussed.

D-polynomials with prescribed zeroes {#app:interpolation}
------------------------------------

Let a Descartes system ${\mathcal{D}}= (\phi_1, \dotsc, \phi_n)$ on $X$ and a set of prescribed zeroes $\bm{r} =(r_1, \dotsc, r_{n-1}) \in \Delta_{n-1}(X)$ be given. We show that the D-polynomial $$\label{eq:interpol}
\phi(x;\bm{r}) = D\begin{pmatrix} \phi_1, &\phi_2, &\dots, &\phi_n\\ x, &r_1, &\dotsc ,&r_{n-1} \end{pmatrix}$$ is the desired interpolation polynomial of Theorem \[thm:extremal\]. First, observe that the determinant vanishes whenever $x = r_i$ for any $i = 1, \dotsc, n-1$, and hence $\phi(x,\bm{r})$ possesses a zero at each $r_i$, which shows (a). Second, as ${\mathcal{D}}$ is a Descartes system, the determinant must be non-zero at all other points in $X$. The point $x$ crossing an interior zero $r_i$ changes the order of two columns in the determinant and hence flips the sign of $\phi(x;\bm{r})$, which shows (b). Claim (c) now follows from Theorem \[thm:vardim\] – because $\phi(x;\bm{r})$ has $n-1$ sign changes, equivalence must hold in .

To prepare for additional results, we remark that the coefficients $a_1, \dotsc, a_n$ of the interpolation D-polynomial $\phi(x;\bm{r})$ can be determined directly from . Expanding the determinant in the first column yields $$\phi(x,\bm{r}) = \sum_{i=1}^n a_i(\bm{r}) \phi_i(x),$$ where $$\label{eq:coefficients}
a_i(\bm{r}) = (-1)^{1+i} D\begin{pmatrix}\phi_1, \dotsc, \phi_{i-1}, \; \phi_{i+1}, \dotsc, \phi_n \\ r_1, \dotsc, r_{n-1}\end{pmatrix}.$$ Because $(\phi_1, \dotsc, \phi_n)$ is a Descartes system, the determinant on the right hand side is strictly positive. This shows that the coefficients of $\phi(x,\bm{r})$ must have alternating signs, starting with ${\textup{\texttt{+}}}$.

The Descartes property of ${\mathcal{E}}$ {#app:Descartes_E}
-----------------------------------------

To show that ${\mathcal{E}}_\text{sep}, {\mathcal{E}}_\text{prox}$ and ${\mathcal{E}}_\text{crit}$ are Descartes systems on $[0,\infty)$, it is sufficient to show that $$D\begin{pmatrix}g_{\alpha_k}, \dotsc, g_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} > 0$$ for any $\alpha_k > \dotsc > \alpha_1 \ge 0$ and $\bm{x} = (x_1, \dotsc, x_k) \in \Delta_{k-1}[0,\infty)$. Our starting point is the representation of $g_\alpha$ as an integral of $f_\alpha(x) = e^{-\alpha x}$ with respect to the totally positive kernel $$K(x,y) = \frac{y}{x^2}{\mathbf{1}_{\left\{x \le y\right\}}}.$$ From [@karlin1968total Ch. 3, Eq. (1.11)ff] and with $K_i := K(x,y_i)$ we obtain that $$D\begin{pmatrix}K_1, \dotsc, K_k\\ x_1, \dotsc, x_k\end{pmatrix}  = \frac{y_1 \dotsm y_{k}}{x_1^2 \dotsm x_k^2} {\mathbf{1}_{\left\{0 \le y_1 \le x_1 \le y_2 \le x_2 \dotsm \le x_k\right\}}}$$ for any $\bm{x}, \bm{y} \in \Delta_k(0,\infty)$. Combining this with the composition formula [@karlin1968total Ch. 3, Eq. (1.2)] we obtain $$\begin{gathered}
D\begin{pmatrix}g_{\alpha_k}, \dotsc, g_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} = \\ = \int_0^{x_1} \int_{x_1}^{x_2} \dotsm \int_{x_{k-1}}^{x_k} \frac{y_1  \dotsm y_{k}}{x_1^2 \dotsm x_k^2} D\begin{pmatrix}f_{\alpha_k}, \dotsc, f_{\alpha_1}\\ x_1, \dotsc, x_k\end{pmatrix} \,dy_1 dy_2 \dotsm dy_k.\end{gathered}$$ Because $(f_{\alpha_k}, \dotsc, f_{\alpha_1})$ is a Descartes system, the integrand is strictly positive. Moreover, the domain of integration has strictly positive measure. We conclude that the left hand side is strictly positive for any $\bm{x} = (x_1, \dotsc, x_k) \in \Delta_k(0,\infty)$, and hence that ${\mathcal{E}}$ is a Descartes system on $(0,\infty)$. It remains to extend this property to the left-closed interval $[0,\infty)$. By [@karlin1968total Ch. 2, Thm. 2.3] it is sufficient to show the Wronskian $W(g_{\alpha_k}, \dotsc, g_{\alpha_1})(0)$ is strictly positive for any $k$. We first calculate the Taylor expansion $$g_\alpha(x) = \frac{1}{x^2} \int_0^x y e^{-\alpha y} dy = \sum_{k=0}^\infty \frac{(-\alpha)^k}{(k +2)} \frac{x^k}{k!},$$ which follows from the Taylor expansion of the exponential function. We conclude that the $k$-th derivative of $g_\alpha$ at zero is given by $$\label{eq:g_der}
g_\alpha^{(k)}(0) = \frac{(-\alpha)^k}{k+2}.$$ Thus we obtain that the Wronskian at zero is given by $$\label{eq:g_Wronskian}
W(g_{\alpha_k}, \dotsc, g_{\alpha_1})(0) = (k+1)!^{-k} D\begin{pmatrix}1, x, x^2 \dotsc, x^{k-1}\\ -\alpha_k, \dotsc, -\alpha_{1}\end{pmatrix}.$$ The latter is a Vandermonde determinant, which evaluates to $\prod_{j=1}^{k-1}(\alpha_{j+1} - \alpha_j)$ and is therefore strictly positive.

Further results on interpolation polynomials {#app:interpolation2}
--------------------------------------------

\[lem:coef\_ratio\] Let $\alpha_n > \dotsc > \alpha_1 \ge 0$ be given and consider the Descartes system $${\mathcal{D}}= (f_{\alpha_n}, \dotsc, f_{\alpha_1}), \quad \text{where} \quad f_\alpha(x) = e^{-\alpha x}.$$ Let $f(x,\bm{r}) = \sum_{i=1}^n a_i(\bm{r}) f_{\alpha_i}(x)$ be the interpolation D-polynomial  of $\bm{r} \in \Delta_{n-1}$. Then its coefficients satisfy, for any $i, j \in {\left\{1, \dotsc, n\right\}}$, $$\label{eq:coef_asymp}
\lim_{\bm{r} \to \bm{0}} \frac{a_i(\bm{r})}{a_j(\bm{r})}  = (-1)^{(i-j)}\frac{\alpha_{i+1} - \alpha_{i-1}}{(\alpha_{i+1} - \alpha_{i})(\alpha_{i} - \alpha_{i-1})} \frac{(\alpha_{j+1} - \alpha_{j})(\alpha_{j} - \alpha_{j-1})} {\alpha_{j+1} - \alpha_{j-1}}$$ with the convention that terms containing $\alpha_0$ or $\alpha_{n+1}$ shall be omitted. The same result holds for ${\mathcal{D}}$ replaced with $${\mathcal{E}}= (g_{\alpha_n}, \dotsc, g_{\alpha_1}), \quad \text{where} \quad g_\alpha(x) = \frac{1}{x^2}\int_0^x y e^{-\alpha y} dy.$$

Combining with [@karlin1968total Ch. 6, Eqs.(1.3), (1.4)], we obtain $$\begin{aligned}
\lim_{\bm{r} \to \bm{0}} \frac{a_i(\bm{r})}{a_j(\bm{r})} &= (-1)^{(i-j)} \lim_{\bm{r} \to \bm{0}}  \frac{
D\begin{pmatrix}f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \\ r_1, \dotsc, r_{n-1}\end{pmatrix}
}{
D\begin{pmatrix}f_n, \dotsc, f_{j+1}, \; f_{j-1}, \dotsc, f_1 \\ r_1, \dotsc, r_{n-1}\end{pmatrix}
} = \\
&= (-1)^{(i-j)} \frac{
W\left(f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \right)(0)
}{
W\left(f_n, \dotsc, f_{j+1}, \; f_{j-1}, \dotsc, f_1 \right)(0).
} \end{aligned}$$ As $f_\alpha(x) = e^{-\alpha x}$, the Wronskian determinants become Vandermonde determinants, i.e. $$\begin{aligned}
&W\left(f_n, \dotsc, f_{i+1}, \; f_{i-1}, \dotsc, f_1 \right)(0) = D\begin{pmatrix}1, x ,x^2, \dotsc, x^{n-1}\\ -\alpha_n, \dotsc, -\alpha_{i+1}, \; -\alpha_{i-1}, \dotsc, -\alpha_1 \end{pmatrix} = \\ &\qquad \frac{\alpha_{i+1} - \alpha_{i-1}}{(\alpha_{i+1} - \alpha_{i})(\alpha_{i} - \alpha_{i-1})} \prod_{k=1}^{n-1} (\alpha_k - \alpha_{k-1}),\end{aligned}$$ and similarly for $j$. Evaluating their ratio, is obtained. For $g$ the proof is analogous, using to evaluate the Wronskians.

\[lem:special\] Consider the Descartes system ${\mathcal{D}}_{4,\text{prox}}  = (f_{2\lambda_2}, f_{\lambda_2 + \lambda_1}, f_{2\lambda_1}, f_{\lambda_2})$ on $[0,\infty)$. There exists a neighborhood $N$ of $\bm{0}$ in $[0,\infty)^3$, such that the coefficients of the interpolation D-polynomial $$f(x;\bm{r}) = a_{2\lambda_2}(\bm{r})f_{2\lambda_2}(x) +  a_{\lambda_1 + \lambda_2}(\bm{r})f_{\lambda_1 + \lambda_2}(x) + a_{2\lambda_1}(\bm{r})f_{2\lambda_1}(x) + a_{\lambda_2}(\bm{r})f_{\lambda_2}(x)$$ satisfy $$\label{eq:coef_ineq}
\left| \frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{\sqrt{a_{2 \lambda_1}(\bm{r}) a_{2 \lambda_2}(\bm{r})}}\right| < 2 \qquad \forall \,\bm{r} \in N \cap \Delta_3[0,\infty).$$ The same holds for ${\mathcal{D}}, {\mathcal{E}}_{4,\text{prox}}$ and ${\mathcal{E}}$.

Applying Lemma \[lem:coef\_ratio\] to ${\mathcal{D}}_{4,\text{prox}}$, we calculate the limits $$\begin{aligned}
\lim_{\bm{r} \to \bm{0}} \left|\frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{a_{\lambda_1}(\bm{r})}\right| &= 2 \left(2 - \frac{\lambda_2}{\lambda_1}\right)\\
\lim_{\bm{r} \to \bm{0}} \left|\frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{a_{\lambda_2}(\bm{r})}\right| &= 2\end{aligned}$$ Taking square roots and multiplying, we obtain $$\lim_{\bm{r} \to \bm{0}}\left| \frac{a_{\lambda_1 + \lambda_2}(\bm{r})}{\sqrt{a_{2 \lambda_1}(\bm{r}) a_{2 \lambda_2}(\bm{r})}}\right| = 2 \sqrt{2 - \frac{\lambda_2}{\lambda_1}}.$$ As $\lambda_1 < \lambda_2 < 2 \lambda_1$, the right hand side is contained strictly between $0$ and $2$. Due to , the coefficients of the interpolation D-polynomial $f(x;\bm{r})$ depend continuously on $\bm{r} \in \Delta_3[0,\infty)$, and follows. The proof for ${\mathcal{D}}, {\mathcal{E}}_{4,\text{prox}}$ and ${\mathcal{E}}$ is analogous.

[^1]: A sign change from ${\textup{\texttt{+}}}$ to $0$ and back to ${\textup{\texttt{+}}}$, for example, is not considered a strong sign change, whereas a sign change from ${\textup{\texttt{+}}}$ to $0$ and then to ${\textup{\texttt{-}}}$ is.

[^2]: Note that the same approach does not work for $m$ due to the different asymptotic behaviour as $x$ tends to infinity.

[^3]: In fact even $\Sigma$-attainable.

[^4]: But not a Descartes subsystem of ${\mathcal{D}}_\text{prox}$!
