of the cross-validation period. Once we found the optimal hyper-parameters
during cross-validation, we learn the parameters of our portfolios again on
the $T_{ins}$ days prior to the testing period (using the obtained hyper-
parameters). We then evaluate the performance of the learned portfolios during
the testing-period.

##### Optimization under Transaction Costs

In practice, we re-balance our portfolio-weights once a day since our market
data is also of daily frequency. In this setting, we would like to incorporate
transaction costs during optimization.

As explained in detail in Section 6, including transaction costs is not
trivial in our setting. In particular, since we have shown that (6.1) does not
necessarily have a (unique) solution, we cannot aim for any closed-form.
Therefore, incorporating the exact amount of transaction-costs to be paid
during the optimization would render the optimization task infeasible.
However, we propose the following penalization which preserves the form of a
convex quadratic optimization problem. Moreover, our empirical results in
Subsection 7.2.3 verify that this penalization is effective in all cases and
indeed useful for learning portfolios which perform well even under 5% of
transaction costs.

######  Corollary 7.5.

Given a convex quadratic optimization problem of the form

$\min_{\mathbf{l}}\,\frac{1}{2}\mathbf{l}^{\mathsf{T}}\mathbf{Q}(T)\mathbf{l}+\mathbf{c}(T)^{\mathsf{T}}\mathbf{l}$

adding the penalization for linear path-functional portfolios $\pi$

$+\frac{\beta}{T}\sum_{t=0}^{T-1}\sum_{i\in\mathcal{U}}\left(\frac{\pi^{i}_{t+1}}{\mu^{\mathcal{U},i}_{t+1}}-\frac{\pi^{i}_{t}}{\mu^{\mathcal{U},i}_{t}}\right)^{2}$

preserves the form of a convex quadratic optimization problem. Here, $\beta$
is a hyperparameter that has to be chosen appropriately.

###### Proof.

Clearly the penalization is quadratic in $\pi$ and therefore quadratic in the
parameters $\\{l_{\nu}^{i}\\}_{i\in\mathcal{U},\nu\in\mathcal{V}}$. Moreover,
the penalization is positive for all
$\\{l_{\nu}^{i}\\}_{i\in\mathcal{U},\nu\in\mathcal{V}}$ and hence the
convexity is preserved as well. ∎

The motivation for this penalization is, first of all, that the universe
portfolio $\mu^{\mathcal{U}}$ is not punished, which should be the case
because this portfolio has no transaction costs at all. Moreover, the
penalization punishes changes in the weights which exceed changes in the
market weights. Note that those are exactly the changes that lead to
transaction costs.

Algorithm 1 Optimization Under Transaction Costs

1:function Optimize_Under_TC(beta):

2: $l\leftarrow$ Solve_Convex_Quadratic(beta)

3: weights $\leftarrow$ Get_Insample_Weights($l$)

4: wealth $\leftarrow$ Wealth_Under_TC(weights)

5: return $-1\cdot$ value

6:end function

7:optimal_beta

8:while ($-1\cdot$Optimize_Under_TC(beta_0)) $<10^{-4}$ do

9: beta_0+=0.5

10:end while

11:optimal_beta $\leftarrow$Minimize(Optimize_Under_TC, beta_0)

12:optimal_$l$ $\leftarrow$ Solve_Convex_Quadratic(optimal_beta)

The downside of this penalization is, that for a given level of transaction
costs, we do not know how to choose $\beta$. To find an appropriate $\beta$,
we propose the procedure described in Algorithm 1. That is, we solve the
quadratic optimization problem for a given $\beta$ and calculate the in-sample
performance _under the true transaction costs_ for the found optimal
portfolio. We then minimize over $\beta$ in order to find the one which best
penalizes for transaction costs over the in-sample period. Note that choosing
the initial value $\beta_{0}$ can be delicate. Namely, if in a neighbourhood
around $\beta_{0}$ all optimal strategies lead to ruin under transaction
costs, one may not move away from the initial value. Therefore, we test if
$\beta_{0}$ leads to ruin under transaction cost and, if so, increase it, in
Algorithm 1. Note that the condition in Line 8 must be false for some
$\beta_{0}$ because the universe portfolio itself is included in the set of
portfolios we optimize over.

##### Calculating the JL-Signature

In order to calculate the JL-signature, the true truncated signature needs to
be calculated before applying the JL-projection. However, calculating the true
signature can be computationally too heavy in large markets. We therefore
propose the following memory-efficient procedure (Algorithm 2), which is based
on the observation that for each component of the signature at level $l$, at
most $l$ components of the underlying path appear in the integrals. Hence, one
can instead compute the signature of combinations of $l$ components of the
path and apply the projection ”batch-wise”. It is important to note that by
doing so, some words are computed multiple times, for example

$\int_{0}^{t}\int_{0}^{s}\circ dX_{u}^{1}\circ dX_{s}^{1}$

can arise from the combination $(X^{1},X^{2},X^{3})$ and from
$(X^{1},X^{4},X^{5})$ and from many more. Therefore, one needs to be careful
which words to keep and our proposed algorithm takes care of that. Moreover,
it is important to store the random projection matrix, once a realization is
computed.

Algorithm 2 Calculate JL-Signature

1:function JL_Signature(path, level):

2: Initialize A_list=list()

3: for $l\in\\{1,\dots,\textrm{level}\\}$ do

4: combos $\leftarrow$ Combinations(dim(path), $l$)

5: for $c\in\textrm{combos}$ do

6: A_slice $\sim$ N(0, 1/k)

7: sig, words $\leftarrow$ Signature(path[c], level)

8: words_to_keep= list()

9: for word $\in$ words do

10: if length(set(word))== $l$ then

11: words_to_keep.append(word)

12: end if

13: end for

14: sig= sig[words_to_keep]

15: A_slice= A_slice[ : , words_to_keep]

16: randomized_sig+=matrix_product(A_slice, sig)

17: end for

18: end for

19:end function

##### Parametrization of Time-Components

We parameterize the time-component of the time augmentation in the following
way: Let $\hat{X}_{t}=(\varphi_{T}(t),X_{t})$. For a given trading horizon
$T_{hor}$, we consider the parametrization
$\varphi_{T_{hor}}(t)=\frac{t}{T_{hor}}$. Concretely, for the in-sample period
of 2000 days, we set $T_{hor}=2000$ and for the out-of-sample period we set
$T_{hor}=750$. This time-augmentation therefore contains information about the
amount of time that is left (or has passed) in the current trading period.
This is to compensate for the different training and testing periods. Another
way to deal with this would be to use signature portfolios with rolling
windows as defined in Remark 3.8.

##### Performance Metrics

Recall that we re-balance our portfolios once a day and follow a buy-and-hold
strategy in between. We evaluate the out-of-sample performance of portfolios
by their log-relative wealth

$\log\left(V^{b\&h}_{T}\right)=\log\left(\prod_{t=t_{0}}^{T-1}\sum_{i\in\mathcal{U}}\pi^{i}_{t}\frac{\mu^{\mathcal{U},i}_{t+1}}{\mu^{\mathcal{U},i}_{t}}\right)$

and their log-wealth

$\log\left(W^{b\&h}_{T}\right)=\log\left(\prod_{t=t_{0}}^{T-1}\sum_{i\in\mathcal{U}}\pi^{i}_{t}\frac{S^{\mathcal{U},i}_{t+1}}{S^{\mathcal{U},i}_{t}}\right).$

####  7.2.2 Rank-based approach: NASDAQ

We consider the _ranked_ NASDAQ market and choose as a universe the stocks
with ranks $1,\dots,100$. We obtained the data from the CRSP database.777The
raw/processed data required to reproduce our findings cannot be shared at this
time due to legal reasons. We train JL-signature portfolios of dimension
$(50,3)$ of type $I$ and randomized signature portfolios of dimension 50 of
type $I$ investing in the universe of the 100 largest stocks and use the
universe portfolio as the auxiliary portfolio. The portfolio is daily re-
balanced. We train such portfolios in two ways, once in the log-wealth
optimization and once in the mean-variance optimization, where in both
optimization problems, we measure the performance of the portfolios by the
log-relative wealth and the log-wealth achieved in the out-of-sample period.

We choose $t_{0}=100$ and take as an in-sample period $2000$ trading days and
as an out-of-sample period the following $750$ trading days. For the mean-
variance optimization task we set the $L^{2}$-regularization parameter
$\gamma_{MV}=10^{-6}$ and for the log-wealth optimization task we found
$\gamma_{LV}=4.849\cdot 10^{-3}$ for JL-signature portfolios and
$\gamma_{LV}=1\cdot 10^{-2}$ for randomize-signature portfolios during the
cross-validation. The grid-search for $\gamma_{LV}$ was performed over 100
equally-distant points in $[10^{-6},10^{-2}]$.

We report the out-of-sample performance of the trained portfolios in Table 2
in terms of log-(relative)-wealth with respect to the universe portfolio. All
but one of the trained portfolios outperform the universe portfolio.

Optimization | JL-Signature | Randomized Signature
---|---|---
Task | $\log V_{T}^{b\&h}$ | $\log W_{T}^{b\&h}$ | $\log V_{T}^{b\&h}$ | $\log W_{T}^{b\&h}$
Log-Opt. | 0.1045 | 0.6622 | 0.0002 | 0.5579
MV $\lambda=4$ | 0.3395 | 0.8972 | -0.0045 | 0.5531
MV $\lambda=3$ | 0.3173 | 0.8750 | 0.0081 | 0.5658
MV $\lambda=2$ | 0.2531 | 0.8108 | 0.0131 | 0.5707
MV $\lambda=1$ | 0.1472 | 0.7049 | 0.0104 | 0.5680
MV $\lambda=0.75$ | 0.1143 | 0.6720 | 0.0085 | 0.5662
MV $\lambda=0.5$ | 0.0788 | 0.6364 | 0.0061 | 0.5638
MV $\lambda=0.05$ | 0.0083 | 0.5660 | 0.0007 | 0.5584
Table 2:  Performance of the trained portfolios over the out-of-sample period
in terms of log-(relative)-wealth. We ran the mean-variance optimization task
for seven different risk-factors $\lambda$. All but one of the optimized
portfolios out-performed the universe portfolio over the out-of-sample period.

In Figure 2(a) and Figure 2(b) we display the wealth processes of the trained
portfolio, the one of the universe portfolio and as a benchmark also of the
equally-weighted portfolio. The wealth processes of the mean-variance
portfolio are the more volatile, the higher the risk-factor is, as one would
expect. Moreover, the wealth-processes of the randomized signature portfolios
are more tamed than their JL-counterparts.

We present the average values (in time) of the portfolio weights for each rank
in Figure 3. The trained portfolios mainly take over- and under-weighted
positions in the largest stocks. While the positions are extremer, the higher
the risk-factor. Although we do not enforce any long-only constraints or
regularization, we do not observe any extreme short-selling positions.

(a) Wealth processes of the learned JL-signature portfolios.

(b) Wealth processes of the randomized signature portfolios.

Figure 2:  Wealth processes of NASDAQ the learned signature portfolios, the
universe portfolio and the equally-weighted portfolio, over the out-of-sample
period.

(a) Average weights of the JL-signature portfolios.

(b) Average weights of the randomized signature portfolios.

Figure 3:  Average weights in the NASDAQ of the signature portfolios and the
universe portfolio, averaged over the out-of-sample period.

####  7.2.3 Name-based approach: SMI and S&P500

In the name-based setting we tackle the mean-variance and log-relative wealth
optimization problems under transaction costs. We do this in two markets; the
Swiss Market Index (SMI) and S&P500 Index. For the SMI, we consider the
universe of stocks which survived between 2000-2022 and for the S&P500 those
that survived between 2001-2022. For both markets, we obtained the data from
Reuters Datastream.888The raw/processed data required to reproduce our
findings cannot be shared at this time due to legal reasons. We compare the
performance of the trained portfolios with the universe portfolio. Note that
the universe portfolio benefits from the same survivor ship bias as the
signature portfolio and hence we do not have an unfair advantage. Again, we
choose $t_{0}=100$, an in-sample period of 2000 days and an out-of-sample
period of 750 days. When we include the regularization cost, we choose
$\beta_{0}=0.5$ and enforce a lower bound $\beta\geq 10^{-8}$. We want to
emphasize, that even with transaction costs our portfolios are re-balanced
daily.

##### SMI Market

We train signature portfolios $\pi(\mu^{\mathcal{U}},\hat{\mu}^{\mathcal{U}})$
of type $I$ degree two. We set $\gamma_{MV}=10^{-8}$ and found
$\gamma_{LV}=2.02\cdot 10^{-3}$ during cross-validation, where the grid-search
was performed over 100 equally-distant points in $[10^{-8},10^{-2}]$. We
present the out-of-sample performance in terms of log-relative wealth in Table
3(a). The first three columns show the performance of portfolios with and
without transaction costs, where we added no regularization for transaction
costs. The next four columns show the performance with and without
transactions costs, but with a regularization for transaction costs at the
respective level. The corresponding regularization parameters $\beta$ which we
found during the in-sample training are shown in Table 4. We note that the
lower the risk-factor $\lambda$ the less regularization for transaction costs
is needed. However, adding such a regularization, the signature portfolios
trained under the mean-variance optimization always outperform the universe
portfolio under transaction costs during the out-of-sample period. This is
remarkable, since the universe portfolio does not pay any transaction costs.
However, the portfolio trained under log-relative wealth optimization does not
out-perform the universe portfolio under transaction costs. Nevertheless, the
regularization for transaction cost proved effective, since all portfolios
performed better with the regularization than without, under the respective
level of transaction costs.

We show the wealth processes of the signature portfolios for the SMI universe
in Figure 4. It is obvious that the higher $\lambda$ is, the more volatile the
portfolios are, which lead to some signature portfolios under-performing the
universe portfolio under transaction costs during some parts of the out-of-
sample period. The signature portfolio with $\lambda=0.05$, however, did
rather well over the entire period even under transaction costs, which we
highlight in Figures 4(d) & 4(e).

##### S&P500 Market

We trained JL-signature portfolios
$\pi(\mu^{\mathcal{U}},\hat{\mu}^{\mathcal{U}})$ of dimension (30,2) and type
$I$ as well as randomized-signature portfolios
$\pi(\mu^{\mathcal{U}},\hat{\mu}^{\mathcal{U}})$ of dimension 30 and type $I$
in the S&P500 market. For the mean-variance optimization, we set
$\gamma_{MV}=10^{-6}$ and for obtained $\gamma_{LV}=1.02\cdot 10^{-4}$ for
both the JL-signature portfolio and the randomized signature portfolio during
cross-validation over an equally-spaced grid of 100 points in
$[10^{-6},10^{-2}]$. We report the out-of-sample results of the optimized
portfolios in Table 3(b) & 3(c) for the JL- and randomized-signature
portfolios respectively. However, it is remarkable that many portfolios also
out-performed the universe portfolio without any regularization for
transaction costs. Indeed, in many cases the optimization yielded
$\beta=10^{-8}$, as we report in Table 4. In the cases where such a
regularization was needed, the performance with transaction costs was always
improved with the regularization. Apart from the log-wealth optimized
portfolio under 5% of transaction costs, the learned portfolio all out-
performed the universe portfolio under transaction costs. We show the
corresponding wealth processes in Figure 6.

(a) Wealth process without transaction costs

(b) Wealth process with 1% proportional transaction costs

(c) Wealth process with 5% proportional transaction costs

(d) Wealth process with 1% proportional transaction costs

(e) Wealth Process with 5% Proportional Transaction Costs

Figure 4:  SMI wealth processes of the signature portfolios, the universe
portfolio and the equally-weighted portfolio with and without transaction
costs. Note that signature portfolios in the settings with transaction cost
are those, where we included the regularization for transaction costs.

Opt. Task |  $\log{V_{T}}$ without Reg. |  $\log{V_{T}}$ with Reg. $\beta_{1\%}$ |  $\log{V_{T}}$ with Reg. $\beta_{5\%}$
---|---|---|---
Prop. TC | 0% | 1% | 5% | 0% | 1% | 0% | 5%
Log-Opt. | 0.0458 | -0.0494 | -0.4333 | 0.0461 | -0.0293 | 0.0540 | -0.0158
MV $\lambda=1$ | 1.7896 | -1.9354 |  -$\infty$ | 1.0514 | 0.1760 | 0.0257 | 0.2804
MV $\lambda=0.75$ | 1.4767 | -0.6214 | -14.6792 | 1.1547 | 0.0601 | 0.2724 | 0.0247
MV $\lambda=0.5$ | 1.0748 | 0.1155 | -4.3536 | 1.0539 | 0.1409 | 0.2620 | 0.0220
MV $\lambda=0.05$ | 0.1240 | 0.0973 | -0.0096 | 0.1229 | 0.0979 | 0.1056 | 0.0381
(a) Log-relative wealth of the signature portfolios with and without
transaction costs trading in the SMI universe.

Opt. Task |  $\log{V_{T}}$ without Reg. |  $\log{V_{T}}$ with Reg. $\beta_{1\%}$ |  $\log{V_{T}}$ with Reg. $\beta_{5\%}$
---|---|---|---
Prop. TC | 0% | 1% | 5% | 0% | 1% | 0% | 5%
Log-Opt. | 1.7442 | -2.5700 |  -$\infty$ | 0.9282 | 0.0402 | 0.2287 | -0.1564
MV $\lambda=1$ | 0.3089 | 0.2657 | 0.0944 | 0.3089 | 0.2657 | 0.2292 | 0.0987
MV $\lambda=0.75$ | 0.2342 | 0.2092 | 0.1101 | 0.2342 | 0.2092 | 0.2179 | 0.1078
MV $\lambda=0.5$ | 0.1578 | 0.1460 | 0.0990 | 0.1578 | 0.1460 | 0.1578 | 0.0990
MV $\lambda=0.05$ | 0.0161 | 0.0156 | 0.0137 | 0.0161 | 0.0156 | 0.0161 | 0.0137
(b) Log-relative wealth of the JL-signature portfolios with and without
transaction costs trading in the S&P500 universe.

Opt. Task |  $\log{V_{T}}$ without Reg. |  $\log{V_{T}}$ with Reg. $\beta_{1\%}$ |  $\log{V_{T}}$ with Reg. $\beta_{5\%}$
---|---|---|---
Prop. TC | 0% | 1% | 5% | 0% | 1% | 0% | 5%
Log-Opt. | 1.0296 | 0.1415 | -3.9870 | 0.7823 | 0.2411 | 0.1789 | -0.0820
MV $\lambda=1$ | 0.0988 | 0.0903 | 0.0566 | 0.0988 | 0.0903 | 0.0988 | 0.0566
MV $\lambda=0.75$ | 0.0743 | 0.0686 | 0.0455 | 0.0743 | 0.0686 | 0.0743 | 0.0455
MV $\lambda=0.5$ | 0.0497 | 0.0462 | 0.0319 | 0.0497 | 0.0462 | 0.0497 | 0.0319
MV $\lambda=0.05$ | 0.0050 | 0.0047 | 0.0033 | 0.0050 | 0.0047 | 0.0050 | 0.0033
(c) Log-relative wealth of the randomized-signature portfolios with and
without transaction costs trading in the S&P500 universe.

Table 3:  The first three columns show the performance with and without transaction costs of signature portfolios with no regularization for transaction costs. The next four column show the performance with and without transaction costs of portfolios trained with a regularization for transaction costs at the respective level.  Universe | SMI | S&P500 (JL) | S&P500 (rand.)
---|---|---|---
Beta | $\beta_{1\%}$ | $\beta_{5\%}$ | $\beta_{1\%}$ | $\beta_{5\%}$ | $\beta_{1\%}$ | $\beta_{5\%}$
Log-Opt. | 27.9579 | 847.6125 | 9.6437 | 97.1598 | 2.9908 | 65.2854
MV $\lambda=1$ | 0.0879 | 1.1661 | $10^{-8}$ | 0.0059 | $10^{-8}$ | $10^{-8}$
MV $\lambda=0.75$ | 0.0240 | 0.8063 | $10^{-8}$ | 0.0012 | $10^{-8}$ | $10^{-8}$
MV $\lambda=0.5$ | 0.0008 | 0.4537 | $10^{-8}$ | $10^{-8}$ | $10^{-8}$ | $10^{-8}$
MV $\lambda=0.05$ | 0.0003 | 0.0095 | $10^{-8}$ | $10^{-8}$ | $10^{-8}$ | $10^{-8}$
Table 4:  Optimal regularization parameters $\beta$ found during in-sample
training.

(a) Wealth process without transaction costs for JL-signature portfolios
(left) and randomized-signature portfolios (right)

(b) Wealth process with 1% proportional transaction costs for JL-signature
portfolios (left) and randomized-signature portfolios (right)

(c) Wealth process with 5% proportional transaction costs for JL-signature
portfolios (left) and randomized-signature portfolios (right)

Figure 6:  S&P500 wealth processes of the JL-signature portfolios (left),
randomized-signature portfolios (right), universe portfolio and equally-
weighted portfolio with and without transaction costs. Note that signature
portfolios in the settings with transaction cost are those, where we included
the regularization for transaction costs.

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##  Appendix A Proofs of Section 2

Since the space of lifted stopped paths as introduced in Definition 2.7
contains paths defined on different time-intervals, we need the following
definition.

######  Definition A.1 (Projection on the final value).

We denote by $(\cdot)_{\odot}$ the projection to the value at the final time,
i.e. for each $t\in[0,T]$

$(x_{[0,t]})_{\odot}:=x_{t}.$

For this projection on the final value we have the following continuity
result.

######  Lemma A.2.

The function

$\displaystyle\lambda:\,[0,T]\times\mathcal{G}_{T}^{N}(\varphi,x)$
$\displaystyle\rightarrow\Lambda_{T}^{N}(\varphi,x)$
$\displaystyle(t,\hat{\mathbb{X}}^{N}_{[0,T]}(\omega))$
$\displaystyle\mapsto\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)$

is continuous.

###### Proof.

Take an arbitrary but fixed $(s,\hat{\mathbb{Y}}^{N}_{[0,T]}(\omega_{y}))$. We
show that $\lambda$ is continuous at
$(s,\hat{\mathbb{Y}}^{N}_{[0,T]}(\omega_{y}))$ but since this point is
arbitrary, $\lambda$ is continuous everywhere. For simplicity, we write
$\hat{\mathbf{x}}_{[0,t]}=\hat{\mathbb{X}}_{[0,t]}^{N}(\omega_{x})$,
$\hat{\mathbf{y}}_{[0,t]}=\hat{\mathbb{Y}}_{[0,t]}^{N}(\omega_{y})$ and
$\widehat{\mathbf{y}^{s}}_{[0,t]}=\widehat{\,\mathbb{Y}^{s}}_{[0,t]}^{N}(\omega_{y})$.
For any $(t,\hat{\mathbf{x}}_{[0,T]})\in[0,T]\times\mathcal{G}_{T}^{N}$ it
holds that

$\displaystyle
d_{\Lambda}\left(\lambda\left(t,\hat{\mathbf{x}}_{[0,T]}\right),\lambda\left(s,\hat{\mathbf{y}}_{[0,T]}\right)\right)$
$\displaystyle\quad=d_{\Lambda}\left(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,s]}\right)$
$\displaystyle\quad\leq
d_{\Lambda}\left(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,t]}\right)+d_{\Lambda}\left(\hat{\mathbf{y}}_{[0,t]},\hat{\mathbf{y}}_{[0,s]}\right)$
$\displaystyle\quad=d_{2-var;[0,t]}\left(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,t]}\right)+|t-s|+d_{2-var;[0,t\vee
s]}\left(\widehat{\,\mathbf{y}^{t}}_{[0,t\vee
s]},\widehat{\mathbf{y}^{s}}_{[0,t\vee s]}\right)$ $\displaystyle\quad\leq
d_{2-var;[0,t]}\left(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,t]}\right)+|t-s|+\|\hat{\mathbf{y}}\|_{2-var;[t\wedge
s,t\vee s]},$

where $\|\cdot\|_{2-var;[0,T]}$ is the norm induced by $d_{2-var;[0,T]}$. Let
now $\epsilon>0$ and choose $\delta>0$ such that

$\delta+\|\hat{\mathbf{y}}\|_{2-var;[s-\delta,s+\delta]}<\epsilon.$

Note that this is possible because
$\|\hat{\mathbf{y}}\|_{2-var;[s-\delta,s+\delta]}\to 0$ as $\delta\to 0$.

Therefore, for all
$(t,\hat{\mathbf{x}}_{[0,T]})\in[0,T]\times\mathcal{G}_{T}^{N}$ (equipped with
the product topology) satisfying

$\displaystyle|t-s|+d_{2-var;[0,T]}\left(\hat{\mathbf{x}}_{[0,T]},\hat{\mathbf{y}}_{[0,T]}\right)<\delta,$

we have

$\displaystyle|t-s|+d_{2-var;[0,t]}\left(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,t]}\right)+\|\hat{\mathbf{y}}\|_{2-var;[t\wedge
s,t\vee s]}$ $\displaystyle<\delta+\|\hat{\mathbf{y}}\|_{2-var;[t\wedge
s,t\vee s]}$
$\displaystyle\leq\delta+\|\hat{\mathbf{y}}\|_{2-var;[s-\delta,s+\delta]}$
$\displaystyle<\epsilon,$

which proves continuity of $\lambda$.

∎

The following lemma can be proved by means of Corollary 2.10.

######  Lemma A.3.

For each multiindex $I$ the map

$\Lambda_{T}^{2}(\varphi,x)\ni\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mapsto\langle
e_{I},S^{|I|}(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))_{\odot}\rangle=\langle
e_{I},\hat{\mathbb{X}}^{N}_{t}(\omega)\rangle\in\mathbb{R}$

is continuous on bounded sets.

###### Proof.

Let us show the statement for an arbitrary but fixed multiindex $I$. For $N=2$
we apply the same notation as in the proof of Lemma A.2 above, i.e.
$\hat{\mathbf{x}}_{[0,t]}=\hat{\mathbb{X}}_{[0,t]}^{2}(\omega_{x})$, etc. We
show continuity at some fixed $\hat{\mathbf{y}}_{[0,s]}$ in some bounded set
of $\Lambda_{T}^{2}(\varphi,x)$. Recall that by Corollary 2.10

$\hat{\mathbf{x}}_{[0,t]}\mapsto\langle
e_{I},S^{|I|}(\hat{\mathbf{x}}_{[0,t]})\rangle$

is continuous on bounded sets of $\mathcal{G}_{t}^{2}(\varphi,x)$ for each
$t\in[0,T]$. Thus, for every $\epsilon>0$ there exists some $\delta_{1}>0$
such that for all $\hat{\mathbf{x}}_{[0,t]}$ satisfying

$d_{2-var,[0,t]}(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,t]})<\delta_{1}$

we have

$\|\langle e_{I},S^{|I|}(\hat{\mathbf{x}}_{[0,t]})\rangle-\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,t]})\rangle\|_{2-var,[0,t]}<\frac{\epsilon}{2}.$

Here, $\|\cdot\|_{2-var,[0,t]}$ denotes the $2$-variation distance on
$\mathbb{R}$. Moreover, the map

$[0,T]\ni t\mapsto\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,t]})_{\odot}\rangle\in\mathbb{R}$

is continuous for each fixed $\hat{\mathbf{y}}$. Hence, there exists some
$\delta_{2}>0$ such that for all $t$ with $|t-s|<\delta_{2}$

$|\langle e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,t]})_{\odot}\rangle-\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,s]})_{\odot}\rangle|<\frac{\epsilon}{2}.$

Choose now $\delta=\min(\delta_{1},\delta_{2})$ and let
$\hat{\mathbf{x}}_{[0,t]}$ (in the bounded set) be such that

$\displaystyle\delta>d_{\Lambda}(\hat{\mathbf{x}}_{[0,t]},\hat{\mathbf{y}}_{[0,s]})$
$\displaystyle=|t-s|+d_{2-var,[0,t\vee s]}(\widehat{\mathbf{x}^{t}}_{[0,t\vee
s]},\widehat{\mathbf{y}^{s}}_{[0,t\vee s]})$
$\displaystyle\geq|t-s|+d_{2-var,[0,t]}(\hat{\mathbf{x}}_{[0,t]},\widehat{\mathbf{y}^{t\wedge
s}}_{[0,t]}).$

Then, by the above

$\displaystyle|\langle
e_{I},S^{|I|}(\hat{\mathbf{x}}_{[0,t]})_{\odot}\rangle-\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,s]})_{\odot}\rangle|\leq\,$
$\displaystyle|\langle
e_{I},S^{|I|}(\hat{\mathbf{x}}_{[0,t]})_{\odot}\rangle-\langle
e_{I},S^{|I|}(\widehat{\mathbf{y}^{t\wedge s}}_{[0,t]})_{\odot}\rangle|$
$\displaystyle+|\langle e_{I},S^{|I|}(\widehat{\mathbf{y}^{t\wedge
s}}_{[0,t]})_{\odot}\rangle-\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,s]})_{\odot}\rangle|$
$\displaystyle\leq\|\langle
e_{I},S^{|I|}(\hat{\mathbf{x}}_{[0,t]})\rangle-\langle
e_{I},S^{|I|}(\widehat{\mathbf{y}^{t\wedge
s}}_{[0,t]})\rangle\|_{2-var,[0,t]}$ $\displaystyle\quad+|\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,t]})_{\odot}\rangle-\langle
e_{I},S^{|I|}(\hat{\mathbf{y}}_{[0,s]})_{\odot}\rangle|$
$\displaystyle<\epsilon,$

which proves the assertion.

∎

We are now prepared to prove Theorem 2.11.

###### Proof of Theorem 2.11.

Consider the function $\lambda$ as introduced in Lemma A.2 and note that for
any compact subset $K\subset\mathcal{G}_{T}^{2}(\varphi,x)$ the set

$\mathcal{K}_{\Lambda}=\lambda([0,T]\times
K)=\left\\{\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mid\,t\in[0,T]\textrm{ and
}\hat{\mathbb{X}}^{2}_{[0,T]}(\omega)\in K\right\\}$

is a compact subset of $\Lambda_{T}^{2}(\varphi,x)$ due to the continuity of
$\lambda$ as proved in Lemma A.2. For each
$\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\Lambda_{T}^{2}(\varphi,x)$ and $N\geq
2$ it holds that
$S^{N}(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))_{\odot}=\hat{\mathbb{X}}^{N}_{t}(\omega)$
a.s. Hence, we can associate with every linear function on the signature $L$ a
non-anticipative path-functional
$f_{L}:\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mapsto
L(\hat{\mathbb{X}}_{t}(\omega))$. Moreover, for every linear function on the
signature $L$ it holds that $f_{L}\in
C(\Lambda_{T}^{2}(\varphi,x);\mathbb{R})$, which follows from Lemma A.3. Let
us denote by

$\displaystyle\mathfrak{F}_{\mathfrak{L}}:=\left\\{f_{L}\mid
f_{L}:\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mapsto
L(\hat{\mathbb{X}}_{t}(\omega))\textrm{ for
}\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\Lambda_{T}^{2}(\varphi,x),\,L\in\mathfrak{L}\right\\}$

the set of path functionals induced by linear functions on the signature and
by

$\displaystyle\mathfrak{F}_{\mathfrak{L}}\rvert_{\mathcal{K}_{\Lambda}}:=\left\\{f_{L}\mid
f_{L}:\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\mapsto
L(\hat{\mathbb{X}}_{t}(\omega))\textrm{ for
}\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\mathcal{K}_{\Lambda},\,L\in\mathfrak{L}\right\\}$

its restriction to lifted paths in $\mathcal{K}_{\Lambda}$. We will now apply
the Stone-Weierstrass Theorem to show that
$\mathfrak{F}_{\mathfrak{L}}\rvert_{\mathcal{K}_{\Lambda}}$ is dense in
$C(\mathcal{K}_{\Lambda};\mathbb{R})$. To this end we need to show that
$\mathfrak{F}_{\mathfrak{L}}\rvert_{\mathcal{K}_{\Lambda}}$ is a sub-algebra
of $C(\mathcal{K}_{\Lambda};\mathbb{R})$ which separates points and vanishes
nowhere. The algebra property follows from the fact that for each
$\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\Lambda_{T}^{2}(\varphi,x)$ and for
any two $f_{L_{1}},f_{L_{2}}\in\mathfrak{F_{L}}$ it holds that

$\displaystyle\left(f_{L_{1}}\cdot
f_{L_{2}}\right)\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right)=f_{L_{1}}\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right)f_{L_{2}}\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right)$
$\displaystyle\quad=L_{1}\left(\hat{\mathbb{X}}_{t}(\omega)\right)L_{2}\left(\hat{\mathbb{X}}_{t}(\omega)\right)=L\left(\hat{\mathbb{X}}_{t}(\omega)\right)=f_{L}\left(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\right),$

where the second last equality follows from the shuffle product property. We
now show that $\mathfrak{F}_{\mathfrak{L}}$ separates points. To this end take
$\hat{\mathbb{X}}^{2}_{[0,t]}(\omega_{x}),\hat{\mathbb{Y}}^{2}_{[0,s]}(\omega_{y})\in\Lambda_{T}^{2}(\varphi,x)$
with
$\hat{\mathbb{X}}^{2}_{[0,t]}(\omega_{x})\neq\hat{\mathbb{Y}}^{2}_{[0,s]}(\omega_{y})$.
If $t\neq s$, the two points are separated in the time-component, i.e.
$\langle e_{1},\hat{\mathbb{X}}_{t}(\omega_{x})\rangle\neq\langle
e_{1},\hat{\mathbb{Y}}_{s}(\omega_{y})\rangle$. If $t=s$, the point-separation
follows from Lemma 2.4. The algebra vanishes nowhere because
$f_{\emptyset}(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))=\langle
e_{\emptyset},\hat{\mathbb{X}}_{t}(\omega)\rangle\equiv 1$ for all
$\hat{\mathbb{X}}^{2}_{[0,t]}(\omega)\in\Lambda^{2}_{T}$, where $\emptyset$
denotes the empty word. The desired density statement follows by the Stone-
Weierstrass Theorem, i.e. for any continuous non-anticipative path-functional
$f:\Lambda_{T}^{2}(\varphi,x)\rightarrow\mathbb{R}$ and for every $\epsilon>0$
there exists a path-functional $f_{L}\in\mathfrak{F_{L}}$ such that for almost
all $\omega\in\Omega$

$\sup_{(t,\hat{\mathbb{X}}^{2}_{[0,T]}(\omega))\in[0,T]\times
K}|f(\hat{\mathbb{X}}^{2}_{[0,t]}(\omega))-f_{L}(\hat{\mathbb{X}}_{t})(\omega)|<\epsilon,$

which is equivalent to the claim of Theorem 2.11. ∎

The proof of the global UAT stated in Theorem 2.18 is based on the fact that
$\Lambda_{T}^{2,\alpha^{\prime}}(\varphi,x)$ with
$0\leq\alpha^{\prime}<\alpha<\frac{1}{2}$ is a weighted space. This is proved
below.

###### Proof of Lemma 2.17.

We have to prove that $K_{R}=\psi^{-1}((0,R])$ is compact for any $R>0$. To
this end, note that the set

$K_{R}^{\prime}=\left\\{\widehat{\,\mathbb{X}^{t}}^{N}_{[0,T]}(\omega)\Bigg{\lvert}\,\psi\left(\hat{\mathbb{X}}^{N}_{[0,t]}(\omega)\right)\in[0,R)\right\\}$

equipped with the $d_{\alpha^{\prime},[0,T]}$-topology is for any $R>0$ a
$\|\cdot\|_{\alpha,[0,T]}$-bounded subset of $\mathcal{G}^{N}_{T}(\varphi,x)$
and hence compact for $0\leq\alpha^{\prime}<\alpha$ due to the compact
embedding of $\alpha$-Hölder spaces into $\alpha^{\prime}$-Hölder spaces, see
[22, Theorem A.3]. Moreover, $K_{R}=\lambda([0,T]\times K_{R}^{\prime})$ and
therefore compact for any $R>0$ due to Tychonoff’s theorem and the continuity
of $\lambda$. ∎

###### Proof of Theorem 2.18.

Using Lemma 2.17 the proof is analogous to the proof of [22, Theorem 5.4]. The
only difference here is that we can choose $\xi>1$ and do not need to require
$\xi>2$ in the weight function $\psi$ as it would be the case in the setting
of [22, Theorem 5.4]. The reason for this is that we consider here
semimartingales which almost surely satisfy the ‘RIE’ property introduced in
[58] (see also [3, Proposition 4.1]). Hence, we can construct the Stratonovic
lifts pathwise [58, Lemma 4.18] which then almost surely coincide with the
stochastic lifts. Due to this pathwise construction, the first level of the
paths uniquely determines the second order lifts and therefore the set

$\tilde{\mathcal{A}}=\textrm{span}\left\\{\hat{X}_{[0,t]}(\omega)\mapsto\langle(e_{i}\shuffle
e_{1}^{\otimes k})\otimes
e_{1},\hat{\mathbb{X}}_{t}(\omega)\rangle\,\Big{\lvert}\,i\in\\{1,\dots,n\\},\,k\in\mathbb{N}\right\\}$

suffices as a point-separating sub-algebra and is of $\psi$-moderate growth,
even for $\xi>1$. ∎

##  Appendix B Proofs of Section 4

The following lemma is needed in the proof of Theorem 4.3.

######  Lemma B.1.

Let $D$ be a (subset of a) metric space. Consider a matrix-valued function
$M:D\ni x\mapsto M(x)$ whose coefficients are continuous functions from $D$ to
$\mathbb{R}$ and which is non-singular for all $x\in D$, then the coefficients
of its matrix-inverse $M^{-1}$ are again continuous functions.

###### Proof.

Let us prove the statement by induction:

  * •

Let $M:D\rightarrow\mathbb{R}^{2\times 2}$. Then the inverse of

$M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

is given by

$M^{-1}=\frac{1}{(ad-bc)}\begin{pmatrix}d&-b\\\ -c&a\end{pmatrix}$

By assumption the determinant $(ad-bc)$ is non-zero, which implies continuity
of the coefficients of $M^{-1}$.

  * •

Assume the statement holds for an arbitrary but fixed $n\geq 2$.

  * •

Let $M:D\rightarrow\mathbb{R}^{(n+1)\times(n+1)}$. Then we can write $M$ in
block matrix form

$M=\begin{pmatrix}\tilde{M}&m^{1}\\\ m^{2}&{m^{3}}\end{pmatrix},$

where $\tilde{M}$ takes values in $\mathbb{R}^{n\times n}$,
$m^{1},(m^{2})^{\top}$ in $\mathbb{R}^{n}$ and $m^{3}$ in $\mathbb{R}$. The
inverse is then computed via

$M^{-1}=\begin{pmatrix}\tilde{M}^{-1}+\frac{1}{\Delta}\tilde{M}^{-1}m^{1}\
m^{2}\tilde{M}^{-1}&-\frac{1}{\Delta}\tilde{M}^{-1}m^{1}\\\
\frac{1}{\Delta}m^{2}\tilde{M}^{-1}&\frac{1}{\Delta}\end{pmatrix}$

where $\Delta=m^{3}-m^{2}\tilde{M}^{-1}m^{1}$ with values in
$\mathbb{R}\setminus\\{0\\}$ by assumption on invertability. As (the
components of) $\tilde{M}^{-1},m^{1},m^{2}$ and $m^{3}$ are all continuous
functions, continuity of the components of $M^{-1}$ follows as well.

∎

##  Appendix C Proofs of Section 5

###### Proof of Lemma 5.14.

By Equation (5.2), we know that

$\displaystyle\log\left(\frac{W^{\pi}_{t}}{W^{{\mathcal{U}}}_{t}}\right)=\sum_{i\in\mathcal{U}}\int_{t_{0}}^{t}\gamma_{s}^{i}f^{i}(s,{X}_{[0,s]})d\mu_{s}^{\mathcal{U},i}$
$\displaystyle-\frac{1}{2}\sum_{i,j\in\mathcal{U}}\int_{t_{0}}^{t}\gamma_{s}^{i}\gamma_{s}^{j}f^{i}(s,{X}_{[0,s]})f^{j}(s,{X}_{[0,s]})d[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s}$
$\displaystyle+\sum_{i\in\mathcal{U}}\int_{t_{0}}^{t}(1-\sum_{k=1}^{d}\lambda_{s}^{k}f^{k}(s,{X}_{[0,s]}))d\mu_{s}^{\mathcal{U},i}$
(C.1)
$\displaystyle-\frac{1}{2}\sum_{i,j\in\mathcal{U}}\int_{t_{0}}^{t}(1-\sum_{k=1}^{d}\lambda_{s}^{k}f^{k}(s,{X}_{[0,s]}))(1-\sum_{m=1}^{d}\lambda_{s}^{m}f^{m}(s,{X}_{[0,s]}))d[\mu^{\mathcal{U},i},\mu^{\mathcal{U},j}]_{s},$
(C.2)

for

$\lambda_{s}^{k}=\begin{cases}\mu_{s}^{\mathcal{U},k}&\textrm{ if $\pi$ is of
type $I$}\\\ 1&\textrm{ if $\pi$ is of type $II$}\end{cases}.$

Note that the terms in (C.1) and (C.2) are both zero due to
$\sum_{i\in\mathcal{U}}\mu_{t}^{\mathcal{U},i}\equiv 1$.

∎

##  Appendix D Proofs of Section 6

###### Proof of Proposition 6.1.

  1. 1.

This case always holds for long-only portfolios and is discussed in [59]. The
argument in the long-only case is that $L(1)<R(1)$ and $L(0)>R(0)$, while both
sides are obviously continuous and monotone. While the former statement still
holds in the case with short-selling, the monotonicity is no longer true.
However, one can show by a tedious case-by-case study that where
$R^{\prime}(\alpha)$ exists it is piece-wise constant and increasing.
Therefore a unique solution for $\alpha^{*}\in[0,1]$ is still assured in this
case.

  2. 2.

Here the above argument does not hold any more. This is because now $L(0)\leq
R(0)$, while it still holds that $L(1)<R(1)$. Let us give three examples, one
for each of the cases a) no solution b) no solution in $[0,1]$, c) no unique
solution in $[0,1]$.

    1. (a)

_no solution_ : consider a market of two stocks and $c=0.05$, i.e. 5%
transaction costs. For the portfolio weights

$\pi_{t^{-}}=(13.1,-12.1)^{\mathsf{T}},\hskip
28.45274pt\pi_{t}=(13,-12)^{\mathsf{T}}$

there does not exist a solution for $\alpha$.

    2. (b)

_no solution in $[0,1]$_: again for $c=0.05$ and a market of two stocks. For
the portfolio weights

$\pi_{t^{-}}=(11,-10)^{\mathsf{T}},\hskip
28.45274pt\pi_{t}=(10,-9)^{\mathsf{T}}$

the solution is $\alpha=-1$.

    3. (c)

_no unique solution in $[0,1]$_: consider $c=0.05$ and a market of three
stocks with portfolio weights

$\pi_{t^{-}}=(5,6,-10)^{\mathsf{T}},\hskip
28.45274pt\pi_{t}=(5.5,6.5,-11)^{\mathsf{T}}$

then the solutions are $\alpha=0.3333$ and $\alpha=0.9535$ (numbers are
rounded).

∎


# A noetherian criterion for sequences of modules

Wee Liang Gan  Department of Mathematics, University of California, Riverside,
CA 92521, USA<EMAIL_ADDRESS>and  Khoa Ta  Department of Mathematics,
University of California, Riverside, CA 92521, USA<EMAIL_ADDRESS>
###### Abstract.

Patzt defined linear categories associated to certain families of diagram
algebras. We prove that finitely generated modules over these linear
categories are noetherian.

Gan and Li proved a noetherian criterion for infinite EI categories in [gl].
In this note, we adapt their proof to a slightly more general setting and
apply it to the linear categories studied by Patzt in [patzt].

We work over a field $k$. Denote by ${\mathbb{N}}$ the set of non-negative
integers. Let $A$ be a sequence $\\{A_{i}\\}_{i\in{\mathbb{N}}}$ where $A_{i}$
is a $k$-algebra for each $i\in{\mathbb{N}}$.

######  Definition 1.

An _$A$ -module_ $M$ is a sequence of pairs
$\\{(M_{i},\phi^{M}_{i})\\}_{i\in{\mathbb{N}}}$ where, for each
$i\in{\mathbb{N}}$,

  * •

$M_{i}$ is an $A_{i}$-module;

  * •

$\phi^{M}_{i}:M_{i}\to M_{i+1}$ is a $k$-linear map.

######  Definition 2.

Let $M$ and $N$ be $A$-modules. A _morphism_ $f:M\to N$ of $A$-modules is a
sequence $\\{f_{i}\\}_{i\in{\mathbb{N}}}$ where, for each $i\in{\mathbb{N}}$,

  * •

$f_{i}:M_{i}\to N_{i}$ is a homomorphism of $A_{i}$-modules;

  * •

$\phi^{N}_{i}\circ f_{i}=f_{i+1}\circ\phi^{M}_{i}$.

######  Definition 3.

Let $M$ be an $A$-module.

  1. (i)

An $A$-module $N$ is an _$A$ -submodule_ of $M$ if each $N_{i}$ is a subset of
$M_{i}$ and the sequence of inclusion maps $\\{N_{i}\hookrightarrow
M_{i}\\}_{i\in{\mathbb{N}}}$ is a morphism of $A$-modules (called the
_inclusion morphism_).

  2. (ii)

Let $E\subseteq\bigsqcup_{i\in{\mathbb{N}}}M_{i}$. The $A$-submodule of $M$
_generated by_ $E$ is the smallest $A$-submodule $N$ of $M$ such that
$E\subseteq\bigsqcup_{i\in{\mathbb{N}}}N_{i}$.

######  Definition 4.

  1. (i)

An $A$-module is _finitely generated_ if it can be generated by a finite set.

  2. (ii)

An $A$-module is _noetherian_ if each of its $A$-submodules is finitely
generated.

######  Notation 5.

  1. (i)

Denote by $\mathrm{Mod}(k)$ the category of $k$-modules.

  2. (ii)

Let $M$ be an $A$-module. Denote by $\mathrm{Sub}(M)$ the category whose
objects are the $A$-submodules of $M$ and whose morphisms are the inclusion
morphisms. For each $i\in{\mathbb{N}}$, denote by $F^{M}_{i}$ the functor

$F^{M}_{i}:\mathrm{Sub}(M)\to\mathrm{Mod}(k),\quad
N\mapsto\mathrm{Hom}_{A_{i}}(M_{i},N_{i}).$

(An inclusion morphism $N^{\prime}\hookrightarrow N$ between $A$-submodules of
$M$ induces an inclusion map $F^{M}_{i}(N^{\prime})\hookrightarrow
F^{M}_{i}(N)$.)

######  Lemma 6.

Let $M$ be an $A$-module, let $N$ be an $A$-submodule of $M$, and let
$N^{\prime}$ be an $A$-submodule of $N$. Let $i\in{\mathbb{N}}$ and assume
that:

  * •

$M_{i}$ is a semisimple $A_{i}$-module;

  * •

$N^{\prime}_{i}\subsetneqq N_{i}$.

Then $F^{M}_{i}(N^{\prime})\subsetneqq F^{M}_{i}(N)$.

###### Proof.

Since $N^{\prime}_{i}\subseteq N_{i}\subseteq M_{i}$, and $M_{i}$ is a
semisimple $A_{i}$-module, there exist $A_{i}$-submodules $Q$ and $Q^{\prime}$
of $M_{i}$ such that

$M_{i}=N_{i}\oplus Q\quad\mbox{ and }\quad N_{i}=N^{\prime}_{i}\oplus
Q^{\prime}.$

Since $N^{\prime}_{i}\neq N_{i}$, we have $Q^{\prime}\neq 0$. Now
$M_{i}=N^{\prime}_{i}\oplus Q^{\prime}\oplus Q$. The projection map from
$M_{i}$ onto $Q^{\prime}$ along $N^{\prime}_{i}\oplus Q$ belongs to
$F^{M}_{i}(N)$ but not $F^{M}_{i}(N^{\prime})$. ∎

Our noetherian criterion is:

######  Theorem 7.

Let $M$ be an $A$-module such that each $M_{i}$ is finite dimensional. Let
$d\in{\mathbb{N}}$. Assume that for each integer $i\geqslant d$,

  1. (7.1)

$M_{i}$ is a semisimple $A_{i}$-module;

  2. (7.2)

there exists a morphism of functors $\nu_{i}:F^{M}_{i}\to F^{M}_{i+1}$ such
that

$\nu_{i}(M):F^{M}_{i}(M)\to F^{M}_{i+1}(M)$

is a bijection.

Then $M$ is noetherian.

###### Proof.

We follow the proof of [gl, Proposition 4.10].

Assume on the contrary that there exists an $A$-submodule $N$ of $M$ which is
not finitely generated. For each $\ell\in{\mathbb{N}}$, let $N^{(\ell)}$ be
the $A$-submodule of $N$ generated by $\bigsqcup_{i=0}^{\ell}N_{i}$. Observe
that if $i\in\\{0,\ldots,\ell\\}$, then $N^{(\ell)}_{i}=N_{i}$.

Since each $N_{i}$ is finite dimensional, the $A$-module $N^{(\ell)}$ is
finitely generated; but $N$ is not finitely generated, so
$N^{(\ell)}\subsetneqq N$. It follows that there exists an integer
$d_{\ell}>\ell$ such that $N^{(\ell)}_{d_{\ell}}\subsetneqq N_{d_{\ell}}$. We
have $N^{(\ell)}_{d_{\ell}}\subsetneqq N^{(d_{\ell})}_{d_{\ell}}$.

Define a sequence of integers $\\{\ell_{i}\\}_{i\in{\mathbb{N}}}$ recursively
by $\ell_{0}=d$ and

$\ell_{i+1}=d_{\ell_{i}}\quad\mbox{ for each }i\in{\mathbb{N}}.$

We have $d=\ell_{0}<\ell_{1}<\ell_{2}<\cdots$. For each $i\in{\mathbb{N}}$,

$N^{(\ell_{i})}_{\ell_{i+1}}\subsetneqq N^{(\ell_{i+1})}_{\ell_{i+1}}.$

Hence, by Lemma 6 and (7.1),

$F^{M}_{\ell_{i+1}}(N^{(\ell_{i})})\subsetneqq
F^{M}_{\ell_{i+1}}(N^{(\ell_{i+1})}).$

In particular,

(8) $\dim F^{M}_{\ell_{i+1}}(N^{(\ell_{i})})<\dim
F^{M}_{\ell_{i+1}}(N^{(\ell_{i+1})}).$

By (7.2), we also have, for each $i\in{\mathbb{N}}$, a commuting diagram

$\textstyle{F^{M}_{d}(N^{(\ell_{i})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d}(N^{(\ell_{i})})}$$\textstyle{F^{M}_{d+1}(N^{(\ell_{i})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d+1}(N^{(\ell_{i})})}$$\textstyle{F^{M}_{d+2}(N^{(\ell_{i})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d+2}(N^{(\ell_{i})})}$$\textstyle{\cdots}$$\textstyle{F^{M}_{d}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d}(M)}$$\textstyle{F^{M}_{d+1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d+1}(M)}$$\textstyle{F^{M}_{d+2}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{d+2}(M)}$$\textstyle{\cdots}$

In the above diagram, the maps in the second row are bijective, hence the maps
in the first row are injective. We deduce that, for each $i\in{\mathbb{N}}$,

(9) $\dim F^{M}_{\ell_{i}}(N^{(\ell_{i})})\leqslant\dim
F^{M}_{\ell_{i+1}}(N^{(\ell_{i})})$

and

(10) $\dim F^{M}_{\ell_{i}}(N^{(\ell_{i})})\leqslant\dim F^{M}_{d}(M).$

From (8) and (9), we obtain, for each $i\in{\mathbb{N}}$,

$\dim F^{M}_{\ell_{i}}(N^{(\ell_{i})})<\dim
F^{M}_{\ell_{i+1}}(N^{(\ell_{i+1})}).$

Hence,

$\dim F^{M}_{\ell_{0}}(N^{(\ell_{0})})<\dim
F^{M}_{\ell_{1}}(N^{(\ell_{1})})<\dim
F^{M}_{\ell_{2}}(N^{(\ell_{2})})<\cdots,$

a contradiction to (10). ∎

######  Example 11.

Let $k=\mathbb{C}$. Let $\\{A_{i}\\}_{i\in{\mathbb{N}}}$ be one of the three
sequences of $\mathbb{C}$-algebras $\\{TL_{i}\\}_{i\in{\mathbb{N}}}$,
$\\{Br_{i}\\}_{i\in{\mathbb{N}}}$, or $\\{P_{i}\\}_{i\in{\mathbb{N}}}$
associated to a parameter $\delta\in\mathbb{C}$ (see [patzt] for definitions);
assume that $\delta$ is chosen so that each $A_{i}$ is a semisimple
$\mathbb{C}$-algebra. Let $A$ be the sequence
$\\{A_{i}\\}_{i\in{\mathbb{N}}}$. Let $C_{A}$ be the linear category defined
in [patzt, Theorem 2.1]. We claim that every finitely generated $C_{A}$-module
is noetherian. By a standard argument111See, for example, [gl, §4.2], it
suffices to prove that for each $m\in{\mathbb{N}}$, the $C_{A}$-module $M(m)$
defined in [patzt, Theorem 3.15] is noetherian.

Let $m\in{\mathbb{N}}$. Define an $A$-module $M$ as follows: for each
$i\in{\mathbb{N}}$, let $M_{i}=M(m)_{i}$ and let $\phi^{M}_{i}:M_{i}\to
M_{i+1}$ be the map induced by the element $1\otimes 1\in
A_{i+1}\otimes_{A_{1}}\mathbb{C}$ (see [patzt, Lemma 2.8]). Then each $M_{i}$
is a finite dimensional semisimple $A_{i}$-module. To apply Theorem 7, it
remains to verify that there exists $d\in{\mathbb{N}}$ such that for each
$i\geqslant d$, condition (7.2) holds.

For each integer $i\geqslant m$, define a functor $F^{\prime}_{i}$ by

$F^{\prime}_{i}:\mathrm{Sub}(M)\to\mathrm{Mod}(\mathbb{C}),\quad
N\mapsto\mathbb{C}\otimes_{A_{i-m}}N_{i}.$

Claim 11.1. For each integer $i\geqslant m$, the functors $F^{M}_{i}$ and
$F^{\prime}_{i}$ are isomorphic.

###### Proof.

We have:

$\displaystyle F^{M}_{i}(N)$
$\displaystyle=\mathrm{Hom}_{A_{i}}(A_{i}\otimes_{A_{i-m}}\mathbb{C},N_{i})$
$\displaystyle\cong\mathrm{Hom}_{A_{i-m}}(\mathbb{C},N_{i}).$

Since $N_{i}$ is a semisimple $A_{i-m}$-module, it is a direct sum of isotypic
components. Let $N_{i}^{\mathrm{triv}}$ be the isotypic component of $N_{i}$
spanned by the submodules isomorphic to the trivial $A_{i-m}$-module
$\mathbb{C}$. Then we have:

$\displaystyle F^{M}_{i}(N)$ $\displaystyle\cong N_{i}^{\mathrm{triv}}$
$\displaystyle\cong\mathrm{Hom}_{\mathbb{C}}(\mathrm{Hom}_{\mathbb{C}}(N_{i}^{\mathrm{triv}},\mathbb{C}),\mathbb{C})$
$\displaystyle\cong\mathrm{Hom}_{\mathbb{C}}(\mathrm{Hom}_{A_{i-m}}(N_{i},\mathbb{C}),\mathbb{C})$
$\displaystyle\cong\mathrm{Hom}_{\mathbb{C}}(\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}\otimes_{A_{i-m}}N_{i},\mathbb{C}),\mathbb{C})$
$\displaystyle\cong\mathbb{C}\otimes_{A_{i-m}}N_{i}$
$\displaystyle=F^{\prime}_{i}(N).$

∎

For each integer $i\geqslant m$ and $R$-submodule $N$ of $M$, the map
$\phi^{N}_{i}:N_{i}\to N_{i+1}$ induces a map

$\nu^{\prime}_{i}(N):\mathbb{C}\otimes_{A_{i-m}}N_{i}\to\mathbb{C}\otimes_{A_{i+1-m}}N_{i+1}.$

This defines a functor $\nu^{\prime}_{i}:F^{\prime}_{i}\to F^{\prime}_{i+1}$.
By [patzt, Theorem 3.15], if $i\geqslant 3m$, the map

$\nu^{\prime}_{i}(M):F^{\prime}_{i}(M)\to F^{\prime}_{i+1}(M)$

is a bijection. Hence, by Claim 11.1, if $i\geqslant 3m$, condition (7.2)
holds.

We conclude by Theorem 7 that the $A$-module $M$ (and hence the $C_{A}$-module
$M(m)$) is noetherian.

######  Remark 12.

Similarly to Example 11, we can deduce [gl, Theorem 3.7] from Theorem 7.

## References