# On Simultaneous Long-Short Stock Trading Controllers with Cross-Coupling

Atul Deshpande  John A. Gubner  B. Ross Barmish  Department of Electrical and
Computer Engineering,
University of Wisconsin, Madison, WI 53706.
email<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Electrical
and Computer Engineering,
Boston University, Boston, MA 02215.
email<EMAIL_ADDRESS>
###### Abstract

The _Simultaneous Long-Short_ (SLS) controller for trading a single stock is
known to guarantee positive expected value of the resulting gain-loss function
with respect to a large class of stock price dynamics. In the literature, this
is known as the _Robust Positive Expectation_ (RPE) property. An obvious way
to extend this theory to the trading of two stocks is to trade each one of
them using its own independent SLS controller. Motivated by the fact that such
a scheme does not exploit any correlation between the two stocks, we study the
case when the relative sign between the drifts of the two stocks is known. The
main contributions of this paper are three-fold: First, we put forward a novel
architecture in which we cross-couple two SLS controllers for the two-stock
case. Second, we derive a closed-form expression for the expected value of the
gain-loss function. Third, we use this closed-form expression to prove that
the RPE property is guaranteed with respect to a large class of stock-price
dynamics. When more information over and above the relative sign is assumed,
additional benefits of the new architecture are seen. For example, when bounds
or precise values for the means and covariances of the stock returns are
included in the model, numerical simulations suggest that our new controller
can achieve lower trading risk than a pair of decoupled SLS controllers for
the same level of expected trading gain.

###### keywords:

Finance, Robustness, Stochastic Control, Uncertain Dynamic Systems.

## 1\. Introduction

The starting point for this paper is the fact that the so-called Simultaneous
Long-Short (SLS) stock trading controller, see Barmish (2011); Barmish and
Primbs (2016, 2011); Baumann and Grüne (2017); Malekpour and Barmish (2016);
Deshpande and Barmish (2018), guarantees satisfaction of the Robust Positive
Expectation (RPE) property, which means the following: The trading gain-loss
function is guaranteed to have positive expected value for a broad class of
stock-price processes. In this paper, we go beyond the single-stock results
cited above and pursue this theme in a more general two-stock trading
scenario. To this end, we introduce Cross-Coupled SLS controllers to exploit a
“correlation” between the stocks. For this new controller, we prove an RPE
theorem which holds when the sign of the cross-coupling coefficient is chosen
appropriately. When additional information about the price processes is
assumed, our numerical examples suggest that cross-coupled SLS controllers can
achieve lower trading risk than a pair of decoupled SLS controllers for the
same target value of the expected trading gain.
For an SLS controller trading a single stock, the key idea in existing
literature involves generating the investment level in the stock from
calculations based on hypothetically holding both long and short positions at
the same time. This is accomplished using two complementary controllers. The
associated RPE results, obtained using linear feedback, differ from earlier
work which is based on sample paths such as Dokuchaev and Savkin (2002);
Dokuchaev (2002); Dokuchaev and Savkin (2004), and model-specific trading
strategies such as Markowitz (1952); Zhang (2001); Song and Zhang (2013);
Deshpande and Barmish (2016). In addition to the work in Barmish (2011);
Barmish and Primbs (2016, 2011); Baumann and Grüne (2017); Malekpour and
Barmish (2016); Deshpande and Barmish (2018), other contributions to the SLS
theory involve robustness results with respect to stock prices having time-
varying drift and volatility Primbs and Barmish (2017), prices generated from
Merton’s diffusion model Baumann (2017), generalization to the case of PI
controllers Malekpour et al. (2018), and discrete-time systems with delays
Malekpour and Barmish (2016). More recently, in O’Brien et al. (2018), the
authors generalize the RPE Theorem to the case of an SLS controller which can
have different parameters for the long and short sides of the trade and
suggest procedures for controller parameter selection to minimize trading risk
based on historical data. In Maroni et al. (2019), the authors start with a
problem formulation which treats prices as if they are disturbances, as in
Barmish (2008), and obtain the SLS controller parameters as the solution of an
appropriately constructed ${\mathcal{H}_{\infty}}$ optimization problem.
A noticeable attribute of the SLS literature is that it addresses single-stock
trading scenarios. For the multi-stock case, the obvious approach for using
existing results would be to independently trade each stock using its own SLS
controller, without exploiting any information about their price correlation.
Influenced by these considerations, the innovation in Deshpande and Barmish
(2018) is to trade one stock with a long-only linear feedback and the other
with a short-only linear feedback instead of using two separate SLS
controllers. Additionally, a strong assumption on the price relationships
between the two stocks is made. In contrast to the aforementioned, in this
paper we impose a much weaker assumption. Specifically, we assume that only
the relative sign between the underlying drifts of the two stock prices is
known. We then put forward a new architecture that cross-couples two SLS
controllers to take advantage of this relative-sign information.
In comparison to existing SLS literature, our new control architecture
includes an extra degree of freedom: a cross-coupling feedback parameter
${\gamma}$, which forces interactions between the two SLS controllers. In
addition to the introduction of this novel trading architecture, the main
theoretical contributions in this paper are results related to the expected
value of the trading gain-loss function. First, we provide a closed-form
expression for the expected value of the trading gain-loss function.
Subsequently, we prove that for a range of ${\gamma}$, satisfaction of the RPE
property is guaranteed with respect to a large class of stochastic processes
for the stock prices. We also establish a recursive formula to calculate the
variance of the gain-loss function.
Under strengthened assumptions that additional information about the stock
prices is known over and above the relative sign between the mean returns, we
demonstrate via a numerical simulation example that there can be performance
benefits due to the use of cross-coupling. In our example, using an assumed
price model with known means and variances of the stock-price returns, an
optimized cross-coupled architecture achieves lower risk than two similarly
optimized independent SLS controllers. The methodology used in the numerical
example can be easily adapted to evaluate performance benefits when the price
model assumed is not precise. For example, when the drifts and volatilities of
the price processes are characterized with bounds instead of precise values,
minimax optimization of the controller design is still possible.
Additionally, in our numerical example, we consider the account leverage
resulting from the use of the cross-coupled controller. For the specific case
of Geometric Brownian Motion prices, along most sample paths, trading with the
optimal cross-coupled controller results in lower account leverage than that
obtained with optimal independent SLS controllers. For scenarios with a limit
on the trading account leverage, we see that a “saturated” implementation of
our cross-coupled controller still results in trading gains with a positive
sample mean.

## 2\. Two-Stock Trading Scenario

In this section, we describe our two-stock trading setup, including the
assumptions which are in force.

### Stock Price Dynamics

We consider two stocks with stochastically varying prices ${S_{1}(k)}$ and
${S_{2}(k)}$ having associated returns

$\rho_{1}(k)\doteq\frac{S_{1}(k+1)-S_{1}(k)}{S_{1}(k)};\;\;\rho_{2}(k)\doteq\frac{S_{2}(k+1)-S_{2}(k)}{S_{2}(k)}$

at stages ${k=0,1,\dots,N-1}$, with the assumption that the return vectors
$[\rho_{1}(k)\;\;\rho_{2}(k)]^{T}$ are independent and identically
distributed. The mean values of the returns
$\mu_{1}\doteq\mathbb{E}[\rho_{1}(k)]$ and
$\mu_{2}\doteq\mathbb{E}[\rho_{2}(k)]$ are unknown to the trader. Only the
relative sign between the two means, namely ${\text{sign}(\mu_{1}\mu_{2})}$,
is assumed to be known. For instance, if the two stocks are from the same
sector, it is often the case that they tend to move in the same direction;
i.e., ${{\rm sign}(\mu_{1}\mu_{2})=1}$, over the medium to long term; e.g.,
see King (1966). Similarly, when assets in a portfolio are negatively
correlated; e.g., see Irwin and Landa (1987), we assume ${{\rm
sign}(\mu_{1}\mu_{2})=-1}$.

### Idealized Market Assumptions

In the theory to follow, consistent with existing SLS literature, an idealized
market is assumed. That is, transaction costs such as brokerage, commissions,
taxes, or fees levied by the stock exchange, are not incurred. In addition, we
assume perfect liquidity so that there is no gap between the bid and ask
prices, and the trader can buy or sell any number of shares, including
fractions, at the market price. These assumptions are similar to those made in
the context of “frictionless markets” in finance literature; e.g., see Merton
(1990).

### Leverage and Interest

In practice, the broker usually imposes limits on the trading account
leverage. For our theoretical analysis, however, we assume that leverage
limits are not in play. That is, the trader has sufficient account resources
to hold any desired position in the stocks. In Section 8, when we provide a
numerical example, we study the practical implications of a leverage
constraint and suggest further research on this issue in Section 9. We also
assume that the margin interest and the risk-free rates of return are zero; we
defer consideration of nonzero rates to future research.

## 3\. Two Independent SLS Controllers

To provide context for the analysis and main results to follow, we first
elaborate on the obvious way that existing single-stock SLS theory might be
applied to the two-stock case. As discussed in Section 1, one can simply
design two decoupled SLS controllers: one for the first stock and another for
the second. Proceeding in this manner, the net investment levels ${I_{1}(k)}$
and ${I_{2}(k)}$ in the stocks at stage $k$ are obtained as sums

$I_{1}(k)=I_{{}_{1L}}(k)+I_{{}_{1S}}(k);\;\;\;I_{2}(k)=I_{{}_{2L}}(k)+I_{{}_{2S}}(k),$

where ${I_{{}_{iL}}(k)}$ and ${I_{{}_{iS}}(k)}$ for ${i=1,2,}$ are the
nominally long and short positions in the $i$-th stock, each obtained using a
linear feedback controller. That is, with initial investment levels
${I_{01}>0}$ and ${I_{02}>0}$ and feedback parameters ${K_{1}>0}$ and
${K_{2}>0}$, the long and short investment functions are given respectively as

$\displaystyle
I_{iL}(k)=I_{0i}+K_{i}g_{{}_{iL}}(k);\;\;\;I_{iS}(k)=-I_{0i}-K_{i}g_{{}_{iS}}(k),$

for ${i=1,2}$, where the cumulative gain-loss functions resulting from
individual long and short positions in each stock are obtained using the gain-
loss update equations

$\displaystyle g_{{}_{iL}}(k+1)$
$\displaystyle=g_{{}_{iL}}(k)+I_{{}_{iL}}(k)\rho_{i}(k);$ $\displaystyle
g_{{}_{iS}}(k+1)$ $\displaystyle=g_{{}_{iS}}(k)+I_{{}_{iS}}(k)\rho_{i}(k)$

for ${i=1,2}$, with
${g_{{}_{1L}}(0)=g_{{}_{1S}}(0)=g_{{}_{2L}}(0)=g_{{}_{2S}}(0)=0}$. In the
sequel, we refer to the above as the 2-SLS controller.
Now, the overall trading gain-loss function ${g(k)}$ for this setup is given
by

$g(k)\doteq g_{{}_{1L}}(k)+g_{{}_{1S}}(k)+g_{{}_{2L}}(k)+g_{{}_{2S}}(k).$

Applying existing results as in Deshpande and Barmish (2018) to each of the
two stocks individually, we arrive at

$\mathbb{E}\left[g(N)\right]=\sum_{i=1}^{2}{\frac{I_{0i}}{K_{i}}\left[(1+K_{i}\mu_{i})^{N}+(1-K_{i}\mu_{i})^{N}-2\right]},$

which is positive for ${N>1}$, and ${\mu_{1}}$, ${\mu_{2}}$ not both zero.

## 4\. Cross-Coupling and State Equations

In this section, we describe the main technical novelty of this paper: a new
architecture for trading which involves cross-coupling between two single-
stock SLS controllers. This is achieved by augmenting each of the four linear
investment functions of Section 3 with a coupling term having feedback gain
${-1<\gamma<1}$ to obtain

$\displaystyle I_{iL}(k)$ $\displaystyle=I_{0i}+K_{i}g_{{}_{iL}}(k)-\gamma
K_{j}g_{{}_{jS}}(k);$ $\displaystyle I_{iS}(k)$
$\displaystyle=-I_{0i}-K_{i}g_{{}_{iS}}(k)+\gamma K_{j}g_{{}_{jL}}(k).$

for ${i,j\in\\{1,2\\}}$; ${i\neq j}$. We refer to this as the Cross-Coupled
SLS (CC-SLS) controller. When $\gamma=0$, we recover the decoupled 2-SLS
controller and the corresponding results stated in Section 3.
To provide some insight into the operation of the CC-SLS controller, we
consider the case when the drifts ${\mu_{1}}$ and ${\mu_{2}}$ as well as
${\gamma}$ are positive. For this case, we expect ${g_{{}_{1L}}(k)}$ and
${g_{{}_{2L}}(k)}$ to both be positive, with ${g_{{}_{1S}}(k)}$ and
${g_{{}_{2S}}(k)}$ both negative. This leads to ${I_{1L}(k)}$ and
${I_{2L}(k)}$ being greater than would be the case if ${\gamma=0}$; i.e.,
greater than the long-investment levels in the 2-SLS counterpart. Another case
which can be similarly analyzed is encountered when the first stock follows a
sample path with a positive trend, but the second stock does not, so that
${g_{2S}(k)}$ is positive; resulting in a smaller ${I_{1L}(k)}$ than would be
the case without coupling. More generally, the CC-SLS controller invests more
aggressively or less aggressively than its 2-SLS counterpart, depending on the
extent to which the stock behaviors are consistent with their drifts.

### Gain-Loss Update Equations

Substituting the formulae for the relevant CC-SLS investment functions into
each of the four gain-loss update equations in Section 3, we obtain the
closed-loop equations

$\displaystyle{g_{{}_{{iL}}}(k+1)}=(1+K_{i}\rho_{i}(k)){g_{{}_{iL}}(k)}-\gamma
K_{j}\rho_{i}(k){g_{{}_{jS}}(k)}+I_{{}_{0i}}\rho_{i}(k);$
$\displaystyle{g_{{}_{iS}}(k+1)}=(1-K_{i}\rho_{i}(k)){g_{{}_{iS}}(k)}+\gamma
K_{j}\rho_{i}(k){g_{{}_{jL}}(k)}-I_{{}_{0i}}\rho_{i}(k)$

for ${i,j\in\\{1,2\\}}$; ${i\neq j}$, with overall trading gain-loss function
$g(k)=g_{{}_{1L}}(k)+g_{{}_{1S}}(k)+g_{{}_{2L}}(k)+g_{{}_{2S}}(k).$

### State-Space Representation

We work with a state-space representation of the CC-SLS controller to derive
our main results on robust positivity of ${\mathbb{E}\left[g(N)\right]}$. With
state

${x}(k)\doteq\begin{bmatrix}{g_{{}_{1L}}(k)}&{g_{{}_{2S}}(k)}&{g_{{}_{1S}}(k)}&{g_{{}_{2L}}(k)}\end{bmatrix}^{T}$

and ${c\doteq[1\;\;1\;\;1\;\;1]^{T}}$, the output of interest is

$g(k)=c^{T}x(k).$

Then, the gain-loss update equations become

$\displaystyle{x}(k+1)$ $\displaystyle=A(k){x}(k)+{b}(k){u}(k);$
$\displaystyle{y}(k)$ $\displaystyle\doteq{c}^{T}{x}(k)$

where

$\displaystyle A(k)=\begin{bmatrix}1+K_{1}\rho_{1}(k)&-\gamma
K_{2}\rho_{1}(k)&0&0\\\ \gamma K_{1}\rho_{2}(k)&1-K_{2}\rho_{2}(k)&0&0\\\
0&0&1-K_{1}\rho_{1}(k)&\gamma K_{2}\rho_{1}(k)&\\\ 0&0&-\gamma
K_{1}\rho_{2}(k)&1+K_{2}\rho_{2}(k)\\\ \end{bmatrix},$
$\displaystyle{b}(k)=\begin{bmatrix}I_{{}_{01}}\rho_{1}(k)&-I_{{}_{02}}\rho_{2}(k)&-I_{{}_{01}}\rho_{1}(k)&I_{{}_{02}}\rho_{2}(k)\end{bmatrix}^{T},$

and constant input ${{u}(k)\equiv 1.}$ Since ${x(0)=\mathbf{0}}$, the standard
solution for ${{g(N)}}$ is

$\displaystyle{g(N)}={y}(N)={c}^{T}\sum_{k=0}^{N-1}\Phi(N,k+1){b}(k){u}(k),$

where the state transition matrix $\Phi(k,k_{0})$ from stage ${k_{0}}$ to
${k\geq k_{0}}$ is given by

$\Phi(k,k_{0})\doteq\begin{cases}A(k-1)\cdots A(k_{0}+1)\cdot A(k_{0})&\text{
for }k>k_{0};\\\ I_{4\times 4}&\text{ for }k=k_{0}.\end{cases}$

Taking expectations, we obtain

$\displaystyle\mathbb{E}\left[g(N)\right]={c}^{T}\sum_{k=0}^{N-1}\bar{A}^{N-1-k}{\bar{b}},$

where

$\displaystyle\bar{A}\doteq\mathbb{E}\left[A(k)\right]=\begin{bmatrix}1+K_{1}\mu_{1}&-\gamma
K_{2}\mu_{1}&0&0\\\ \gamma K_{1}\mu_{2}&1-K_{2}\mu_{2}&0&0\\\
0&0&1-K_{1}\mu_{1}&\gamma K_{2}\mu_{1}&\\\ 0&0&-\gamma
K_{1}\mu_{2}&1+K_{2}\mu_{2}\\\ \end{bmatrix}$

and

$\bar{b}\doteq\mathbb{E}\left[b(k)\right]=\begin{bmatrix}I_{{}_{01}}\mu_{1}&-I_{{}_{02}}\mu_{2}&-I_{{}_{01}}\mu_{1}&I_{{}_{02}}\mu_{2}\end{bmatrix}^{T}.\\\
$

## 5\. Main Results

In this section, we provide two theorems related to the expected value
${\mathbb{E}\left[g(N)\right]}$ of the overall gain-loss function at stage
${N}$. The first theorem gives us the formula for
${\mathbb{E}\left[g(N)\right]}$. Following this, the second theorem gives us
conditions under which ${\mathbb{E}\left[g(N)\right]>0}$. For simplicity of
the proofs, we assume that both ${\mu_{1}}$ and ${\mu_{2}}$ are nonzero.
However, by separately considering the case when one of these two drifts
vanish, it is easy to see that ${\mathbb{E}\left[g(N)\right]>0}$ in this
situation, except for the break-even case when both ${\mu_{1}}$ and
${\mu_{2}}$ are zero.
To obtain a formula for ${\mathbb{E}\left[g(N)\right]}$, we use the following
notation. For ${K_{1}>0}$, ${K_{2}>0}$, and ${0<|\gamma|<1}$, we define

$\displaystyle\theta$
$\displaystyle\doteq\sqrt{(K_{1}\mu_{1}+K_{2}\mu_{2})^{2}-4\gamma^{2}K_{1}K_{2}\mu_{1}\mu_{2}},$
$\displaystyle\alpha_{1}\ $ $\displaystyle\doteq(\theta-
K_{1}\mu_{1}+K_{2}\mu_{2})/2,$ $\displaystyle\alpha_{2}\ $
$\displaystyle\doteq(\theta+K_{1}\mu_{1}-K_{2}\mu_{2})/2,$
$\displaystyle\beta_{1}$
$\displaystyle\doteq(K_{1}\mu_{1}+K_{2}\mu_{2}+\theta),$
$\displaystyle\beta_{2}$
$\displaystyle\doteq(K_{1}\mu_{1}+K_{2}\mu_{2}-\theta),$

and the function

$\phi_{{}_{N}}(x)\doteq(1+x)^{N}+(1-x)^{N}-2,$

which is positive for all ${x\neq 0}$ and ${N>1}$.

### Expected Value Theorem

_Suppose two stocks with stochastically varying prices ${S_{1}(k)}$ and
${S_{2}(k)}$ with mean returns ${\mu_{1}}$ and ${\mu_{2}}$ are traded using
the Cross-Coupled SLS controller with ${K_{1}>0}$, ${K_{2}>0}$, and coupling
coefficient satisfying ${0<|\gamma|<1}$. Then, for ${\mu_{1}}$ and ${\mu_{2}}$
nonzero, the expected value of the gain-loss function is given by_

$\displaystyle\mathbb{E}\left[g(N)\right]=\frac{1}{2\theta}\Bigg{[}\frac{2\gamma\mu_{1}\mu_{2}(I_{{}_{01}}K_{1}+I_{{}_{02}}K_{2})+I_{{}_{02}}\mu_{2}\beta_{1}+I_{{}_{01}}\mu_{1}\beta_{2}}{\alpha_{1}}\cdot{\phi_{{}_{N}}(\alpha_{1})}$
$\displaystyle\qquad+\frac{2\gamma\mu_{1}\mu_{2}(I_{{}_{01}}K_{1}+I_{{}_{02}}K_{2})+I_{{}_{02}}\mu_{2}\beta_{2}+I_{{}_{01}}\mu_{1}\beta_{1}}{\alpha_{2}}\cdot{\phi_{{}_{N}}(\alpha_{2})}\Bigg{]}.$

### Proof

For ${\mu_{1},\mu_{2}\neq 0}$ and coupling coefficient ${0<|\gamma|<1}$, we
diagonalize the block-diagonal matrix ${\bar{A}}$ defined in Section 4 to
obtain

$\bar{A}=P\Lambda P^{-1},$

where

$P\doteq\begin{bmatrix}\frac{\beta_{2}}{2\gamma
K_{1}\mu_{2}}&\frac{\beta_{1}}{2\gamma K_{1}\mu_{2}}&0&0\\\ 1&1&0&0\\\
0&0&\frac{\beta_{1}}{2\gamma K_{1}\mu_{2}}&\frac{\beta_{2}}{2\gamma
K_{1}\mu_{2}}\\\ 0&0&1&1\end{bmatrix}$

is composed of the eigenvectors of ${\bar{A}}$ as its columns, and

$\Lambda\doteq\mathop{\rm
diag}\nolimits\left(1-\alpha_{1},\;\;1+\alpha_{2},\;\;1-\alpha_{2},\;\;1+\alpha_{1}\right)$

is the diagonal matrix with the corresponding eigenvalues of ${\bar{A}}$. Note
that the standing assumptions assure that ${\alpha_{1}}$ and ${\alpha_{2}}$
are nonzero.
Rewriting the expression for ${\mathbb{E}\left[g(N)\right]}$ in terms of ${P}$
and ${\Lambda}$, we obtain

$\displaystyle\mathbb{E}\left[g(N)\right]$
$\displaystyle=c^{T}\sum_{k=0}^{N-1}\bar{A}^{N-1-k}\bar{b}$
$\displaystyle=c^{T}\sum_{i=0}^{N-1}\left(P\Lambda^{N-1-i}P^{-1}\right)\bar{b}$
$\displaystyle=c^{T}P\Lambda_{{}_{S}}P^{-1}\bar{b},$

where the diagonal matrix ${\Lambda_{{}_{S}}}$ is given by

$\displaystyle\Lambda_{{}_{S}}\doteq\mathop{\rm
diag}\nolimits\left(\frac{(1-\alpha_{1})^{N}-1}{-\alpha_{1}},\;\;\frac{(1+\alpha_{2})^{N}-1}{\alpha_{2}},\;\;\frac{(1-\alpha_{2})^{N}-1}{-\alpha_{2}},\;\;\frac{(1+\alpha_{1})^{N}-1}{\alpha_{1}}\right).$

Since ${\mathbb{E}\left[g(N)\right]}$ is a scalar, we write

$\mathbb{E}\left[g(N)\right]=c^{T}P\Lambda_{{}_{S}}P^{-1}\bar{b}=\text{tr}\left(c^{T}P\Lambda_{{}_{S}}P^{-1}\bar{b}\right),$

where ${\text{tr}(\cdot)}$ is the trace operator, and its cyclic property
Strang (2009) gives us

$\displaystyle\mathbb{E}\left[g(N)\right]$
$\displaystyle=\text{tr}(c^{T}P\Lambda_{{}_{S}}P^{-1}\bar{b})$
$\displaystyle=\text{tr}(P\Lambda_{{}_{S}}P^{-1}\bar{b}c^{T})$
$\displaystyle=\text{tr}(\Lambda_{{}_{S}}P^{-1}\bar{b}c^{T}P)$
$\displaystyle=\sum_{i}(\Lambda_{{}_{S}})_{ii}(P^{-1}\bar{b}c^{T}P)_{ii},$

where ${(\Lambda_{{}_{S}})_{ii}}$ and ${(P^{-1}\bar{b}c^{T}P)_{ii}}$ denote
the diagonal entries of the respective matrices. For ${\Lambda_{{}_{S}}}$ as
given above, we need only find the values of ${(P^{-1}\bar{b}c^{T}P)_{ii}}$,
which we collect in the vector

$\displaystyle D$
$\displaystyle\doteq\begin{bmatrix}(P^{-1}\bar{b}c^{T}P)_{11}\\\
(P^{-1}\bar{b}c^{T}P)_{22}\\\ (P^{-1}\bar{b}c^{T}P)_{33}\\\
(P^{-1}\bar{b}c^{T}P)_{44}\end{bmatrix}.$

Further simplification using
${\beta_{1}\beta_{2}=4\gamma^{2}K_{1}K_{2}\mu_{1}\mu_{2}}$ yields

$D=\frac{1}{2\theta}\begin{bmatrix}-I_{{}_{02}}\mu_{2}(\beta_{1}+2\gamma
K_{2}\mu_{1})-I_{{}_{01}}\mu_{1}(\beta_{2}+2\gamma K_{1}\mu_{2})\\\
I_{{}_{02}}\mu_{2}(\beta_{2}+2\gamma
K_{2}\mu_{1})+I_{{}_{01}}\mu_{1}(\beta_{1}+2\gamma K_{1}\mu_{2})\\\
-I_{{}_{02}}\mu_{2}(\beta_{2}+2\gamma
K_{2}\mu_{1})-I_{{}_{01}}\mu_{1}(\beta_{1}+2\gamma K_{1}\mu_{2})\\\
I_{{}_{02}}\mu_{2}(\beta_{1}+2\gamma
K_{2}\mu_{1})+I_{{}_{01}}\mu_{1}(\beta_{2}+2\gamma
K_{1}\mu_{2})\end{bmatrix}.$

Then the summation above for ${\mathbb{E}\left[g(N)\right]}$ simplifies to the
claimed closed-form expression. ∎

### Remarks

We observe that the expected value of the gain-loss function is of the form

$\mathbb{E}\left[g(N)\right]=C_{1}\cdot\phi_{{}_{N}}(\alpha_{1})+C_{2}\cdot\phi_{{}_{N}}(\alpha_{2}),$

where ${C_{1}}$ and ${C_{2}}$ are independent of ${N}$. This is similar in
form to the result given for two independent SLS controllers in Section 3,
that is,

$\mathbb{E}\left[g(N)\right]\Bigl{|}_{\mbox{\footnotesize{2-SLS}}}=\frac{I_{{}_{01}}}{K_{1}}\phi_{N}(K_{1}\mu_{1})+\frac{I_{{}_{02}}}{K_{2}}\phi_{N}(K_{2}\mu_{2}).$

In the theorem to follow, we use the closed-form expression for
${\mathbb{E}\left[g(N)\right]}$ to prove that if ${\rm sign}(\gamma)={\rm
sign}(\mu_{1}\mu_{2}),$ the RPE property of the CC-SLS controller is
guaranteed.

### Robust Positive Expectation Theorem

_Suppose two stocks with stochastically varying prices ${S_{1}(k)}$ and
${S_{2}(k)}$ with nonzero mean returns ${\mu_{1}}$ and ${\mu_{2}}$ are traded
using the Cross-Coupled SLS controller with coupling coefficient satisfying
${0<|\gamma|<1}$ and ${{\rm sign}(\gamma)={\rm sign}(\mu_{1}\mu_{2})}$. Then,
for any ${N>1}$, robust satisfaction of the condition_

$\mathbb{E}\left[g(N)\right]>0$

_is guaranteed._

Scenario | Bounds on $\theta$ | ${\alpha_{1}}$ | ${\alpha_{2}}$ | ${\beta_{1}}$ | ${\beta_{2}}$
---|---|---|---|---|---
${\mu_{1}>0;\mu_{2}>0}$ | ${|K_{1}\mu_{1}-K_{2}\mu_{2}|<\theta<K_{1}\mu_{1}+K_{2}\mu_{2}}$ | ${\alpha_{1}>0}$ | ${\alpha_{2}>0}$ | ${\beta_{1}>0}$ | ${\beta_{2}>0}$
${\mu_{1}<0;\mu_{2}<0}$ | ${|K_{1}\mu_{1}-K_{2}\mu_{2}|<\theta<|K_{1}\mu_{1}+K_{2}\mu_{2}|}$ | ${\alpha_{1}>0}$ | ${\alpha_{2}>0}$ | ${\beta_{1}<0}$ | ${\beta_{2}<0}$
${\mu_{1}>0;\mu_{2}<0}$ | ${|K_{1}\mu_{1}+K_{2}\mu_{2}|<\theta<K_{1}\mu_{1}-K_{2}\mu_{2}}$ | ${\alpha_{1}<0}$ | ${\alpha_{2}>0}$ | ${\beta_{1}>0}$ | ${\beta_{2}<0}$
${\mu_{1}<0;\mu_{2}>0}$ | ${|K_{1}\mu_{1}+K_{2}\mu_{2}|<\theta<K_{2}\mu_{2}-K_{1}\mu_{1}}$ | ${\alpha_{1}>0}$ | ${\alpha_{2}<0}$ | ${\beta_{1}>0}$ | ${\beta_{2}<0}$
Table 1: Satisfaction of the First Two Inequalities for All Combinations of
Signs of ${\mu_{1}}$ and ${\mu_{2}}$

### Proof

Beginning with the formula obtained for ${\mathbb{E}\left[g(N)\right]}$ and
rearranging, it suffices to show that the following three inequalities hold:

$\displaystyle\frac{\phi_{{}_{N}}(\alpha_{1})\cdot(I_{{}_{02}}\mu_{2}\beta_{1}+I_{{}_{01}}\mu_{1}\beta_{2})}{\alpha_{1}}$
$\displaystyle>0;$
$\displaystyle\frac{\phi_{{}_{N}}(\alpha_{2})\cdot(I_{{}_{02}}\mu_{2}\beta_{2}+I_{{}_{01}}\mu_{1}\beta_{1})}{\alpha_{2}}$
$\displaystyle>0;$ $\displaystyle
2\gamma\mu_{1}\mu_{2}(I_{{}_{01}}K_{1}+I_{{}_{02}}K_{2})\Bigg{(}\frac{\phi_{{}_{N}}(\alpha_{1})}{\alpha_{1}}+\frac{\phi_{{}_{N}}(\alpha_{2})}{\alpha_{2}}\Bigg{)}$
$\displaystyle\geq 0.$

For arbitrary admissible pair ${\mu_{1}\neq 0}$ and ${\mu_{2}\neq 0}$, we
verify the satisfaction of the first two inequalities above for the cases
enumerated in Table 1. In each row of the table, we consider a possible
combination of the signs of ${\mu_{1}}$ and ${\mu_{2}}$. Each combination
determines the range of possible values ${\theta}$ can take, which in turn
dictates the signs of ${\alpha_{1},\alpha_{2},\beta_{1}}$ and ${\beta_{2}}$.
Using these in conjunction with the positivity of ${\phi_{N}(\alpha_{i})}$ for
${N>1}$ establishes the first two inequalities.
To prove the third inequality holds, given that

$\text{sign}(\mu_{1}\mu_{2})=\text{sign}(\gamma),$

we readily see that
$2\gamma\mu_{1}\mu_{2}(I_{{}_{01}}K_{1}+I_{{}_{02}}K_{2})>0.$ Thus, to
complete the proof, it suffices to show that

$\frac{\phi_{{}_{N}}(\alpha_{1})}{\alpha_{1}}+\frac{\phi_{{}_{N}}(\alpha_{2})}{\alpha_{2}}\geq
0.$

To this end, we consider the following two cases:
Case 1: If ${\text{sign}(\mu_{1})=\text{sign}(\mu_{2})}$, we see from Table 1
that ${\alpha_{1}>0}$ and ${\alpha_{2}>0}$. Combined with the fact that the
function ${\phi_{N}(x)>0}$ for all ${x\neq 0}$, it follows that
${\phi_{N}(\alpha_{1})>0}$ and ${\phi_{N}(\alpha_{2})>0}$. Hence,

$\frac{\phi_{{}_{N}}(\alpha_{1})}{\alpha_{1}}+\frac{\phi_{{}_{N}}(\alpha_{2})}{\alpha_{2}}>0.\\\
$

Case 2: If ${\text{sign}(\mu_{1})=-\text{sign}(\mu_{2})}$, we see from Table 1
that ${\text{sign}(\alpha_{1})=-\text{sign}(\alpha_{2})}$. Without loss of
generality, assuming ${\mu_{1}>0>\mu_{2}}$, we obtain
${\alpha_{2}>0>\alpha_{1}}$ with ${|\alpha_{2}|\geq|\alpha_{1}|}$ and we use
this condition to arrive at

$\frac{\phi_{{}_{N}}(\alpha_{1})}{\alpha_{1}}+\frac{\phi_{{}_{N}}(\alpha_{2})}{\alpha_{2}}=\frac{|\alpha_{1}|\phi_{{}_{N}}(\alpha_{2})-|\alpha_{2}|\phi_{{}_{N}}(\alpha_{1})}{|\alpha_{1}\alpha_{2}|}.$

Since ${N>1}$, it is easily shown that

$|\alpha_{1}|\phi_{{}_{N}}(\alpha_{2})>|\alpha_{2}|\phi_{{}_{N}}(\alpha_{1}).$

This completes the proof of the theorem. ∎

## 6\. Variance of the Gain-Loss Function

Assuming that the mean vector

$\mu\doteq[\mu_{1}\;\;\mu_{2}]^{T}\doteq\mathbb{E}\left[\rho_{1}(k)\;\;\rho_{2}(k)\right]^{T},$

and covariance matrix

${\sigma}\doteq\begin{bmatrix}\sigma_{1}^{2}&\sigma_{12}\\\
\sigma_{12}&\sigma_{2}^{2}\\\ \end{bmatrix}\doteq{\rm
cov}([\rho_{1}(k)\;\;\rho_{2}(k)]^{T})$

of the returns of the two stocks are known, we now derive a recursion to
calculate ${\mathop{\sf var}\nolimits(g(N))}$. Recalling the state-update
equation from Section 4, we first rewrite the state matrix as

$A(k)=A_{0}\rho_{0}(k)+A_{1}\rho_{1}(k)+A_{2}\rho_{2}(k)$

where ${\rho_{0}(k)\equiv 1}$, $A_{0}$ is the identity matrix ${I}$,

$\displaystyle A_{1}\doteq\begin{bmatrix}K_{1}&-\gamma K_{2}&0&0\\\ 0&0&0&0\\\
0&0&-K_{1}&\gamma K_{2}&\\\ 0&0&0&0\\\
\end{bmatrix},\;\;A_{2}\doteq\begin{bmatrix}0&0&0&0\\\ \gamma
K_{1}&-K_{2}&0&0\\\ 0&0&0&0&\\\ 0&0&-\gamma K_{1}&K_{2}\\\ \end{bmatrix},$

and we rewrite ${b(k)}$ as

$b(k)=b_{0}\rho_{0}(k)+b_{1}\rho_{1}(k)+b_{2}\rho_{2}(k),$

where

$\displaystyle
b_{0}\doteq[0\;\;0\;\;0\;\;0]^{T};b_{1}\doteq[I_{{}_{01}}\;\;0\;\;-I_{{}_{01}}\;\;0]^{T};b_{2}\doteq[0\;\;-I_{{}_{02}}\;\;0\;\;I_{{}_{02}}]^{T}.$

With this notation, the state update equation becomes

$\displaystyle
x(k+1)=(A_{0}\rho_{0}(k)+A_{1}\rho_{1}(k)+A_{2}\rho_{2}(k))x(k)$
$\displaystyle\qquad\qquad\qquad+(b_{0}\rho_{0}(k)+b_{1}\rho_{1}(k)+b_{2}\rho_{2}(k)),$

with initial value ${x(0)={0}}$. It follows that

$\mathbb{E}[x(k+1)]=\bar{A}\mathbb{E}[x(k)]+\bar{b}.$

To calculate the variance of the gain-loss function, we first define
${\rho(k)\doteq[\rho_{0}(k)\;\;\rho_{1}(k)\;\;\rho_{2}(k)]^{T}}$, and
subsequently

$\displaystyle R(\mu,{\sigma})$
$\displaystyle\doteq\mathbb{E}\left[\rho(k)\rho(k)^{T}\right]$
$\displaystyle=\begin{bmatrix}1&\mu_{1}&\mu_{2}\\\
\mu_{1}&\sigma_{1}^{2}+\mu_{1}^{2}&\sigma_{12}+\mu_{1}\mu_{2}\\\
\mu_{2}&\sigma_{12}+\mu_{1}\mu_{2}&\sigma_{2}^{2}+\mu_{2}^{2}\\\
\end{bmatrix}.$

We now obtain the recursion

$\displaystyle\mathbb{E}[x(k+1)x^{T}(k+1)]=\mathbb{E}[(A(k)x(k)+b(k))(A(k)x(k)+b(k))^{T}],$

Substituting the various quantities into the recursion above for
${\mathbb{E}[x(k)x^{T}(k)]}$ and denoting the ($i,j$)-th element of the matrix
${R(\mu,{\sigma})}$ as ${R_{i,j}(\mu,{\sigma})}$, it follows that

$\displaystyle\mathbb{E}[x(k+1)x^{T}(k+1)]=\sum_{i=0}^{2}\sum_{j=0}^{2}R_{i+1,j+1}(\mu,{\sigma})\Bigl{[}A_{i}\mathbb{E}[x(k)x^{T}(k)]A_{j}^{T}$
$\displaystyle\qquad\qquad\qquad+A_{i}\mathbb{E}[x(k)]b_{j}^{T}+b_{i}\mathbb{E}[x^{T}(k)]A_{j}^{T}+b_{i}b_{j}^{T}\Bigl{]}.$

Starting with initial value ${\mathbb{E}\left[x(0)x^{T}(0)\right]=0}$, we use
the above recursion for ${0\leq k\leq N-1}$ to obtain
${\mathbb{E}[x(N)x^{T}(N)]}$. Recalling that the gain-loss function
${g(N)=c^{T}x(N)}$ for ${c=[1\;\;1\;\;1\;\;1]^{T}}$, we now calculate

$\displaystyle\mathbb{E}\left[g(N)\right]=c^{T}\mathbb{E}\left[x(N)\right];\;\;\mathbb{E}\left[g^{2}(N)\right]=c^{T}\mathbb{E}\left[x(N)x(N)^{T}\right]c,$

and subsequently,

$\mathop{\sf
var}\nolimits(g(N))=\mathbb{E}\left[g^{2}(N)\right]-\mathbb{E}^{2}[g(N)].$

From this, we calculate the standard deviation of ${g(N)}$, which is used in
the numerical example in Section 8.

## 7\. Risk Mitigation via Cross-Coupling

To augment the analysis of the CC-SLS controller in Section 4, we now analyze
a trading scenario where a cross-coupling can result in lower trading risk
when compared to two independent SLS controllers. This analysis is performed
under the strengthened assumption that the mean returns and covariances of the
two stock prices are known. Then in the spirit of modern portfolio theory, for
the classical case when the controller parameters
${I_{{}_{01}},I_{{}_{02}},K_{1},K_{2}}$ and ${\gamma}$ are optimized with
respect to these assumed price models, we compare the mean and standard
deviation of ${g(N)}$ obtained by the CC-SLS controller against the ones
obtained by the 2-SLS controller. We demonstrate how, for a given target
return, the CC-SLS controller architecture can lead to lower trading risk than
that of the 2-SLS controller. Namely, the standard deviation of ${g(N)}$
resulting from the use of the CC-SLS is lower than that obtained using 2-SLS.
Associated with the two controllers described in Sections 3 and 4, whenever
convenient, we emphasize the dependence of various quantities on the
controller parameter vector

$d\doteq{(I_{{}_{01}},I_{{}_{02}},K_{1},K_{2},\gamma)}$

by including it as an argument in mathematical functions of interest; e.g., we
write ${g(d,k)}$ instead of ${g(k)}$ for the gain-loss function at stage
${k}$. Thus, given initial trading account value ${V_{0}}$, the account value
${V(k)}$ at stage ${k}$ is given by

$V(d,k)=V_{0}+g(d,k)$

and depends on ${d}$. Without loss of generality, we assume ${V_{0}=1}$ so
that the cumulative return

$\frac{V(d,N)-V_{0}}{V_{0}}$

is equal to ${g(d,N)}$, and the risk-return pair is

$\bigl{(}\mathop{\sf
std}\nolimits(g(d,N)),{\mathbb{E}\left[g(d,N)\right]}\bigr{)}.\\\ $

## 8\. Numerical Example

To illustrate the ideas in Section 7, we work with an assumed stochastic model
of the stock-price processes with independent returns having respective mean
values ${\mu_{1}=0.023374}$ and ${\mu_{2}=0.031014},$ and known variances
${\sigma_{1}^{2}=8.3333\times 10^{-3}}$ and ${\sigma_{2}^{2}=16.333\times
10^{-3}}.$ Then, for both the CC-SLS and the 2-SLS controllers, we consider
5000 candidate parameter vectors $d$ selected by using the uniform
distribution to generate points

$\displaystyle
I_{{}_{01}},I_{{}_{02}}\in(0,3];\;\;K_{1},K_{2}\in(0,3];\;\;\gamma\in[0,0.99],$

noting that ${\gamma=1}$ is inadmissible in the RPE Theorem. For each vector
${d}$ selected above for the CC-SLS controller, we force ${\gamma=0}$ to get a
corresponding 2-SLS parameter vector. Then for each ${d}$, and ${N=30}$, we
calculate the risk-return pair $\left({\mathop{\sf
std}\nolimits(g(d,N))},\mathbb{E}\left[g(d,N)\right]\right)$ for each of the
two controllers. In Figure 1, for the CC-SLS controller, each such pair is
denoted with a blue dot and for the 2-SLS controller, a green dot is used.

Figure 1: Expected Value vs. Standard Deviation of ${g(d,N)}$

For each of the two architectures, for a given target return ${G}$, we seek a
parameter vector ${d}$ that solves

$\displaystyle\min_{d}\mathop{\sf std}\nolimits(g(d,N))\text{ subject to
}\mathbb{E}\left[g(d,N)\right]\geq G.$

In terms of Figure 1, this means that the CC-SLS optimum parameter vector
corresponds to the leftmost blue dot along the line with
$\mathbb{E}\left[g(d,N)\right]\approx G.$ Similarly for 2-SLS, the optimum
parameter vector corresponds to the leftmost green dot along the same line.
From the figure, we see that for target return ${G}$, the optimal CC-SLS
controller appears to guarantee a lower level of the risk ${\mathop{\sf
std}\nolimits(g(d,N))}$ versus that obtained for the optimal 2-SLS controller.
To make the above more concrete, for ${G=2}$, using the search sets for the
controller parameters above, an optimal 2-SLS controller parameter vector is
found to be

$d^{*}_{2sls}=(3,3,0.713,0.381)$

and results in

$\left(\mathop{\sf
std}\nolimits(g(d^{*}_{2sls},N)),\mathbb{E}\left[g(d^{*}_{2sls},N)\right]\right)\approx(2.399,2.00).$

Similarly, an optimal CC-SLS parameter vector is found to be

$d^{*}_{ccsls}=(3,2.58,0.339,0.234,0.990),$

and results in

$\left(\mathop{\sf
std}\nolimits(g(d^{*}_{ccsls},N)),\mathbb{E}\left[g(d^{*}_{ccsls},N)\right]\right)\approx(2.225,2.00).$

Consistent with the discussion above, this CC-SLS controller leads to lower
risk than its 2-SLS counterpart.

### Account Leverage Considerations

As stated in Section 2, the broker typically imposes limits on the account
leverage to ensure that investment levels are commensurate with the account
value. Thus, to supplement the foregoing risk-return analysis, we study the
leverage used by the two optimal controllers above. Given that one or both
stocks may be sold short with the corresponding net investments
${I_{i}(k)<0}$, consistent with practice, we work with leverage ratio

$L(k)\doteq\frac{|I_{{}_{1}}(k)|+|I_{{}_{2}}(k)|}{V(k)},$

and study $L_{\max}\doteq\displaystyle\max_{0\leq k\leq N-1}L(k)$ using one
million sample paths. For simulating these paths, however, we need to know the
joint distribution of the returns, not just their means and covariances. To
this end, we assume that the prices are obtained from two independent
Geometric Brownian Motion models which are consistent with the
${(\mu_{i},\sigma_{i}^{2})}$ used for the two controller optimization tasks
above. Since the total returns ${S_{i}(k+1)/S_{i}(k)}$ are log-normally
distributed, we generate these prices using the update equations

$\displaystyle S_{1}(k+1)$
$\displaystyle=S_{1}(k)\cdot\exp\left(0.019142+0.08903w_{1}(k)\right);$
$\displaystyle S_{2}(k+1)$
$\displaystyle=S_{2}(k)\cdot\exp\left(0.022918+0.12349w_{2}(k)\right),$

where ${w_{1}(k)\sim\mathcal{N}(0,1)}$ and ${w_{2}(k)\sim\mathcal{N}(0,1)}$.
Without loss of generality, we assume that the initial prices in the above
update equations are ${S_{1}(0)=S_{2}(0)=1}$. Subsequently, for ${N=30}$, we
estimate that for the CC-SLS controller, $L_{\max}\leq 3.44$ about 95% of the
time, and that for the 2-SLS controller, using that same 95% figure of merit,
we estimate ${L}_{\max}\leq 6.94.$ Furthermore, out of the one million sample
paths, the optimal CC-SLS controller results in a bankruptcy, characterized by
account value ${V(k)\leq 0}$, in only $715$ sample paths as compared to
$15,006$ bankruptcies for the 2-SLS controller. For such sample paths, we
record the maximum leverage to be ${L_{\max}=\infty.}$ To summarize, the
optimal CC-SLS controller not only leads to lower risk than its optimal 2-SLS
controller, but it also results in a much lower account leverage almost all
the time and leads to a lower probability of account bankruptcy.

### Saturated 2-SLS and CC-SLS Controllers

Noting that the account leverage for both controllers can far exceed the
limits imposed by stock brokers, to illustrate how one might conform with
common practice, we revisit our simulation with the added constraint
${L(k)\leq 2}$, and “saturate” the investments of the 2-SLS and the CC-SLS
controllers whenever this inequality is violated. More precisely, we take

$I_{{}_{iL}}(k)=\begin{cases}I_{{}_{iL}}(k)&\text{ when }L(k)\leq 2;\\\
I_{{}_{iL}}(k)\cdot 2/L(k)&\text{ otherwise},\end{cases}$

and

$I_{{}_{iS}}(k)=\begin{cases}I_{{}_{iS}}(k)&\text{ when }L(k)\leq 2;\\\
I_{{}_{iS}}(k)\cdot 2/L(k)&\text{ otherwise}.\end{cases}$

Although such a scheme ensures that ${L(k)\leq 2}$ for all ${k}$, the
theoretical guarantee of robust positive expectation is no longer available.
Nonetheless, from the one million sample paths of the GBM prices described
above, we statistically estimate ${\mathbb{E}\left[g(N)\right]\approx 1.86}$
and ${{\mathop{\sf std}\nolimits}({g}(N))\approx 2.15}$, and encounter no
bankruptcies using the saturated CC-SLS controller. In comparison, for the
saturated 2-SLS controller, we estimate ${\mathbb{E}\left[g(N)\right]\approx
1.74}$, with ${{\mathop{\sf std}\nolimits}({g}(N))=2.27}$ and face account
bankruptcy in 12 sample paths. Thus, for this more practical scenario, the
saturated CC-SLS and 2-SLS controllers yield positive average trading gains
while conforming to the leverage constraints imposed on them.

## 9\. Conclusion

In this paper, we introduced the notion of cross-coupled SLS controllers for
trading two stocks. We derived a closed-form expression for the expected
trading gain-loss function resulting from this new architecture and, based on
this formula, our new Robust Positive Expectation Theorem provides conditions
under which positivity of the expected gain-loss function is guaranteed. We
also provided simulations which suggest, under strengthened hypotheses, that
our new CC-SLS architecture enables controller designs that achieve lower
trading risk than two independent SLS controllers for the same target expected
gain. Finally, in our numerical simulations, we found that the cross-coupled
SLS controller results in lower trading account leverage than two decoupled
SLS controllers. Similarly, we showed that using “saturated” implementations
of both controller designs which guarantee compliance with a leverage limit
imposed by the broker, the mean-variance performance of the cross-coupled SLS
controller is better than that obtained with two decoupled SLS controllers.
Three directions for future research immediately present themselves: The first
involves the introduction of cross-coupling into different variants of the SLS
controller found in the literature; e.g., in Malekpour and Barmish (2016), an
SLS controller with delays is considered. A second direction involves
extending the theory presented in this paper to trading scenarios involving
more than two stocks. Finally, the third possible direction for future
research is motivated by the fact that the practitioner invariably faces
leverage restrictions. Hence, it would be of interest to see if the CC-SLS
controller still leads to a guarantee of robust positive expectation when a
saturation scheme, along the lines studied in the previous section, is used.
Results given in Barmish and Primbs (2016) for a standalone SLS controller
provide motivation for further work.

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