# Improving CT Image Segmentation Accuracy Using
StyleGAN Driven Data Augmentation

Soham Bhosale  Department of Computer Science, Stony Brook University, Stony
Brook, NY, USA  Arjun Krishna  Department of Computer Science, Stony Brook
University, Stony Brook, NY, USA  Ge Wang  Biomedical Imaging Center, School
of Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA  Klaus Mueller
Department of Computer Science, Stony Brook University, Stony Brook, NY, USA

Abstract Medical Image Segmentation is a useful application for medical image
analysis including detecting diseases and abnormalities in imaging modalities
such as MRI, CT etc. Deep learning has proven to be promising for this task
but usually has a low accuracy because of the lack of appropriate publicly
available annotated or segmented medical datasets. In addition, the datasets
that are available may have a different texture because of different dosage
values or scanner properties than the images that need to be segmented. This
paper presents a StyleGAN-driven approach for segmenting publicly available
large medical datasets by using readily available extremely small annotated
datasets in similar modalities. The approach involves augmenting the small
segmented dataset and eliminating texture differences between the two
datasets. The dataset is augmented by being passed through six different
StyleGANs that are trained on six different style images taken from the large
non-annotated dataset we want to segment. Specifically, style transfer is used
to augment the training dataset. The annotations of the training dataset are
hence combined with the textures of the non-annotated dataset to generate new
anatomically sound images. The augmented dataset is then used to train a U-Net
segmentation network which displays a significant improvement in the
segmentation accuracy in segmenting the large non-annotated dataset.
---

##  1 Introduction

CT Image Segmentation is a hallmark of Computer-Aided Diagnosis (CAD). Medical
professionals often use it to isolate specific areas of interest in a medical
image. This allows them to conduct effective medical analyses for detecting
abnormalities such as lung cancer and positioning implants [1]. However,
manual segmentation can become time-consuming and complicated depending on
what part of a body is being analyzed. Deep learning can make this task
significantly more efficient, affordable, and accessible. Nevertheless, this
approach is currently very ineffective due to the low availability of large
labeled and annotated training datasets. The reasons for this shortage include
privacy concerns and the high cost of labeling CT datasets by human experts.

In addition, the added challenge with segmenting a CT dataset is that there is
a lot of variability between the publicly available datasets. For example, the
images frequently have different dosages which is the difference in amount of
x-ray radiation used to synthesize CT images or they could be generated from
several different machines each having different parameters for CT
reconstruction, which for example could mean the thickness of tissue that each
image slice represents may vary. Hence, if we train a segmentation network
with a dataset that has images of different dosage and scanner properties than
the set of images we intend to segment then the deep network will have low
accuracy.

We present a new strategy for augmenting segmented datasets which eliminates
the differences between the segmented dataset we use for training and the
dataset we wish to segment, by using texture learning. This strategy will
create synthetic images which imitate dosage and scanner properties of the
unsegmented dataset and maintain anatomical accuracy within the process.

Figure 1: Flow starts at the top right corner with the two datasets - a small
segmented and a large unsegmented dataset. It illustrates the inputs and
outputs involved in training the StyleGANs and the U-Net to finally segment
the test set taken from the large dataset.

As shown in Figure 1, we build on our previous work of texture learning [2] to
expand our small annotated dataset with textures present in the large
unlabeled dataset. In the first step, we ask an expert, a radiologist in this
case, to manually segment a few CT images from the large non-segmented dataset
which were chosen as "style images" for our StyleGAN training. We then use the
StyleGAN architecture presented in Krishna et al. [2] to create new CT images
based on the texture of a style image and the content, or segmentation maps,
of another image.

We train 6 different StyleGANs with 6 separate texture images. Since these
texture images came from the non-annotated dataset which contains the images
we want to segment, the newly generated segmented images will make good
candidates for training the segmentation network. Next, we take all the
segmentation maps of the small segmented dataset and use it as "content"
inputs to each of 6 trained StyleGAN networks to generate new images. This
technique augments the segmented dataset by 7 folds and creates additional
images with sample textures from the unsegmented dataset. We then train a
U-Net [3] segmentation network with the augmented segmented dataset. The
trained U-Net is then used to segment all the images of the large non-
annotated dataset. Segmentation accuracy of the network is computed on a test
set that consists of around 100 randomly chosen images from the non-annotated
dataset. In the absence of the ground-truth for the test set, annotations are
created manually with the assistance of a few radiologists.

To the best of our knowledge, we are the first to enhance segmentation
accuracy by reducing texture differences between training and testing CT
datasets using StyleGANs.

Figure 2: Our Expanded Dataset. For "m" images in annotated dataset and "n"
style images, expanded dataset size is mx(n+1). Note, that the heart anatomy
isn’t perfect in the generated images, but as long as generated images could
exhibit enough anatomy and texture, it could still help in improving final
segmentation accuracy.

Figure 3: Left: A standard U-Net network which segments a low-dose soft-tissue
CT image into lungs, heart, torso, spinal-cord and esophagus. Right: Top row
shows some well formed segmentation maps of various ct scans, while bottom row
highlights flawed results from the U-Net when trained on a smaller different
dataset with high-dose CT images.

##  2 Materials and Methods

Fig. 1 highlights each step of the process. As discussed, there are three
major steps: training the StyleGAN networks, augmenting the annotated dataset,
and then training the segmentation U-Net with this augmented dataset. We will
briefly describe each step below

###  2.1 Datasets

The top right corner of Fig. 1 highlights the two different datasets (in red
colored boxes) that were involved in this work. The smaller dataset consists
of 512x512 high dose chest CT images along with their segmentation maps of
thirty patients depicting their lungs, heart, spinal cord, esophagus,
surrounding tissue and bones within their torso. The larger dataset consists
of non-annotated low dose chest CT images of similar resolution of around 14k
patients taken with several different scanners at several different locations.
As stated before, for using the datasets together for improving segmentation,
we pre-process and augment the smaller annotated dataset multiple folds [2]
using CT image textures present in the larger dataset.

###  2.2 StyleGAN

We use our encoder-decoder based styleGAN architecture and segment-wise style
loss [2] for learning and generating segment-wise textures of heart, torso and
surrounding tissues present in the larger dataset. For this, we first choose
five or six different "style images" from the large dataset. Since the dataset
contains low-dose CT scans collected over a period of three years from several
different scanners, it contains a few distinct textures over all the CT-scans
corresponding to distinct noise characteristics of these scanners. The "style-
images" are manually selected corresponding to these distinct textures. For
our large dataset six style-images seem enough to cover all the different
texture features present in the dataset. Images in the top row of Figure 2
show three such style images. Note that as per our styleGAN architecture, the
segmentation maps of both "style images", from the large dataset, and "content
images", from the small dataset, are needed for generating CT images having
textures of the large dataset’s images while retaining anatomy of the small
dataset’s "content images". This is maintained by our segment-wise style loss,
Ls and content loss, Ls which is as follows:

$\displaystyle
L_{s}=\sum_{sg{\in}SG}\sum_{l{\in}SL}\frac{1}{4N^{2}_{l}M^{2}_{l}}\lVert
G(R^{l}_{sg})-G(S^{l}_{sg})\rVert_{F}^{2}$ (1)

$\displaystyle
L_{c}=\sum_{sg{\in}SG}\sum_{l{\in}CL}\frac{1}{2N_{l}M_{l}}\lVert
R^{l}_{sg}-O^{l}_{sg}\rVert_{2}^{2}$ (2)

where Ol, Sl, and Rl denote the feature maps extracted from the pre-trained
VGG network at layer l, for the original image xo, the style image xs and the
stylized image output xr respectively; $G(R^{l})$ and $G(S^{l})$ denote the
encoded Graham Matrices [4] of those feature maps; SG is the set of all six
segments including heart, torso, lungs, spinal cord and esophagus in the CT
images and SL/CL is the set of all style/content layers in the VGG network.
Since, the large training dataset’s CT scans are not segmented, we manually
build the segmentation maps of the chosen six style images with the assistance
of a radiologist, as mentioned.

The StyleGAN architecture proposed by Krishna et al. [2] takes the
segmentation map of one image xo and the segments’ texture of another image xs
as input, and outputs a new image xr with the contents (organs) of the
segmentation map xo and style of image xs. Similar to the original work, all
the training images are resized to 512x512 pixels. Each StyleGAN is trained
for 100 epochs. For the loss function, the input images, generated images, and
style images along with their segmentation maps are fed into a VGG-19 network.
The feature maps generated by VGG-19 are used to calculate style loss and
content loss. Both content loss and style loss are calculated separately for
each segmented region and then combined.

###  2.3 Augmenting Segmentation Training Dataset

Next, all the segmentation maps of the smaller CT segmentation training
dataset are passed through each trained StyleGAN networks as inputs generating
new CT images for each segmentation map. Note that we train six style
generators corresponding to each style image hence augmenting the dataset
seven fold. The original 700 images segmented lung CT dataset is now expanded
to around 4900 images. In our case, the generated images have textures
simulating low dosage CT images similar to the images of the large dataset we
intend to segment. Also, all of the generated images have associated
annotations or segmentation maps which are the same as the inputs or the
annotations of the original dataset

Figure 2 shows a few results of the expanded annotated dataset. In the figure
we are showing results for three such generators corresponding to three style
images shown in the top row and three segmentation maps corresponding to three
"content"/training dataset images shown in the right-most column. Since these
generators do not serve as our final segmentation networks, the generated
images do not have to be anatomically prefect as long as they can learn and
exhibit enough anatomy with the large dataset’s texture in generated images
that could help in training a U-Net effectively in our last step. We chose the
soft tissue window of CT scans for texture learning and augmenting the smaller
dataset. Soft tissue images exhibit all the anatomical details necessary for
learning annotations of the smaller dataset which in our case are segmentation
maps having segments depicting heart, lungs, torso esophagus and spinal cord.
These segments are clearly distinguishable in the soft tissue window based on
the pixel values of the images. Having the ability to learn annotations or
segmentation from the augmented dataset will enable the trained U-Net to
recognize the CT images’ segments from the larger dataset having similar
textures.

###  2.4 Training the U-Net

We train a U-Net [3] for creating a segmentation map as output when given a
chest CT image as input. We chose the U-Net architecture due to its excellent
performance for segmenting biomedical images, especially when coupled with
data augmentation techniques like elastic deformation. The architecture of the
network is based on the original work done by the authors where each blue
block in Figure 3a consists of two convolutional layers with a batch
normalization layer in between those layers and a max-pooling layer as the
last layer. In the up-sampling part, max-pooling layers are replaced by the
transposed convolutions layer as the first layer followed by the two
convolutional layers. As shown in the figure the results of each block in the
down-sampling part are concatenated to their corresponding block in the up-
sampling half. Kernels of size 3x3 are used for convolutional layers with a
stride of 2. We use the augmented textured annotated dataset created in the
previous step for training our U-Net. Having similar texture across the two
different datasets helps training a segmentation network on the segmented
dataset to segment the images of the large non-annotated dataset. We use the
trained U-Net to segment randomly chosen 100 CT images of the larger dataset
which forms our test set.

We analyze the improvement of the network trained on the augmented CT image
dataset as compared to the network trained on the small high-dose image
dataset. In the absence of the ground-truth segmentation maps for the test
set, we manually corrected the erroneous predicted segmentation outputs of the
network models with the help of a few radiologists. The corrected segmentation
outputs of the test set images are then treated as the ground truths for the
images. The first row of Figure 3b highlights some of these well formed ground
truth segmentation maps, while the second row shows the erroneous predicted
segmentation maps from the network trained on the smaller high-dose image
dataset. We discuss the improvements and compare the two networks in the next
section. Both are trained for 30 epochs with a batch size of 16.

Figure 4: The above figures display a side-by-side comparison of the test
results of the segmentation network (U-Net 1) trained on the small high-dose
CT dataset versus the network (U-Net 2) trained on the augmented dataset. The
segmentation maps generated by the augmented dataset show greater visual
accuracy than those by the network trained on the small dataset.

##  3 Results, Further Work and Conclusion

Figure 4 highlights some of our results. We observe that the U-Net trained
with our style learning based augmented data (U-Net 2) has a greater
segmentation accuracy than the network trained with the original annotated
dataset (U-Net 1). For instance, the augmented dataset-trained network
classified much more of lung and torso matter correctly than the network
trained with the original dataset. As for metrics, U-Net 2 has a testing
accuracy of 95.7% whereas the U-Net 1 has a testing accuracy of 93.1%.

We also observe that we still have segmentation flaws around smaller segments
like esophagus and spinal cord. Since the small dataset represents limited
anatomy of a few patients, there are errors in the segmentation outputs of the
patients in the larger dataset with more varied anatomy. This becomes more
evident around smaller segments since the segmentation maps of the small
dataset are on an average significantly smaller, i.e., have less spatial
detail, than those of the large dataset which is due to the differences in
positioning of the scanner lens between the two datasets.

One way to resolve this is to explore the PCA space of segments within
segmentation maps as we did in Krishna el al. [2] to generate CT images with
varied anatomy through our styleGANs for augmenting the small annotated
dataset. Also, we propose to further improve our segmentation network accuracy
by training U-Nets and StyleGANs in a cyclic approach. We will use StyleGANs
to augment a dataset, pass it through the segmentation network, and then use
the segmentation maps generated by this network to provide additional data for
the StyleGAN training and then repeat the process. Another option is to use a
combination of all CT window images including lung and bone windows for
segmenting the images. We propose to use a combination of above mentioned
strategies to further improve our segmentation accuracy.

According to these results, augmenting medical datasets using StyleGANs has
proven to be a promising method to resolve texture differences between
different medical datasets paving a way to analyze one dataset based on the
annotations of another. In the segmentation task of lung CT, the StyleGANs
assisted expanded dataset helped a U-Net to improve its accuracy by 2.6% on a
completely different dataset. Hopefully, this new technique can mitigate the
annotated medical data shortage issue that is required for training data
intensive deep learning networks for various applications.

## References

  * [1] Brahim Ait Skourt, Abdelhamid El Hassani and Aicha Majda  “Lung CT Image Segmentation Using Deep Neural Networks”  In _Procedia Computer Science_ Elsevier, 2018, pp. 109–113
  * [2] Arjun Krishna and Klaus Mueller  “Medical (CT) image generation with style”  In _International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear_
  * [3] Olaf Ronneberger, Philipp Fischer and Thomas Brox  “U-Net: Convolutional Networks for Biomedical Image Segmentation”  In _International Conference on Medical image computing and computer-assisted intervention_ , 2015, pp. 234–241
  * [4] Leon Gatys, Alexander S Ecker and Matthias Bethge  “Texture synthesis using convolutional neural networks”  In _Advances in neural information processing systems_ 28, 2015


$\Gamma{\mathrm{M}}$ (red) and $\Gamma{\mathrm{X}}{\mathrm{M}}$ (blue). The
two parities on $\Gamma{\mathrm{M}}$ taken together have a winding number
$-1$, while the positive parities taken together have a winding number 0.

Figure 19:  (Left) Surface spectrum showing a quadratic band touching at the
${\overline{\Gamma}}$ point for the model defined in Eq. (181) with parameters
$t=t^{\prime}=1$ and $m=2.5$. The remaining plots show the surface spectrum
near the ${\overline{\Gamma}}$-point for two copies of this model with
$t^{\prime}=1$ on both copies (center) and $t^{\prime}=\pm 1$ for the two
copies (right). The two copies are coupled by the Hamiltonian defined in Eq.
(187) with $\lambda_{1}=0.3$ and $\lambda_{2}=1.0$. The two cases show eight
and four Dirac nodes in the surface Brillouin zone.

###  G.2 Three dimensions

We consider the 3d Bloch Hamiltonian

$H_{3d}(\mathbf{k})=H(\mathbf{k}_{\perp})+\sigma^{0}\otimes\left[\cos
k_{z}\tau^{1}+\sin k_{z}\tau^{2}\right],$ (181)

where $\mathbf{k}_{\perp}=(k_{x},k_{y})$. For $1<\left|m\right|<3$, the
restriction of $H_{3d}(\mathbf{k})$ to $k_{z}=0$ and $k_{z}=\pi$ are in
different topological phases, so that $H_{3d}(\mathbf{k})$ has nontrivial
level-2 invariants

$\mathrm{C}^{\Gamma{\mathrm{M}}}={\rm sgn}\,\\!(mt),\qquad\mathrm{C}^{+}={\rm
sgn}\,(mt^{\prime})\Theta(-tt^{\prime}).$ (182)

For this model, we only get
$\mathrm{C}^{\Gamma{\mathrm{M}}}=\mathrm{C}^{{\mathrm{M}}{\mathrm{X}}\Gamma}=\pm
1$, so that we expect a quadratic band touching at a surface normal to $k_{z}$
for $t,t^{\prime}\neq 0$ for any $1<\left|m\right|<3$ (see the discussion in
Sec. VIII.3). This is indeed what we see in numerical exact diagonalization,
as shown in Fig. 19(left).

The boundary modes can be computed analytically following the generalized
transfer matrix approach [106]. We follow the notation of Sec. V of Ref. [106]
and identify the hopping and on-site matrices as [see 106, Eq. (124)]

$J=\begin{pmatrix}0&\openone_{2}\\\ 0&0\end{pmatrix},\quad
M(\mathbf{k}_{\perp})=\begin{pmatrix}\mathbf{b}\cdot\mbox{\boldmath$\sigma$}&\mu\sigma^{0}\\\
\mu\sigma^{0}&-\mathbf{b}\cdot\mbox{\boldmath$\sigma$}\end{pmatrix},$ (183)

where $\mu(\mathbf{k}_{\perp})=m-\cos k_{x}-\cos k_{y}$ and

$\mathbf{b}(\mathbf{k}_{\perp})=2(\cos k_{x}+\cos k_{y})\begin{pmatrix}t\sin
k_{x}\sin k_{y}\\\ 0\\\ t^{\prime}(\cos k_{y}-\cos k_{x})\end{pmatrix}.$

The transfer matrix is given by [106, Eq. (131)]

$T(\mathbf{k}_{\perp})=\frac{1}{\mu}\begin{pmatrix}\left(\varepsilon^{2}-b^{2}-\mu^{2}\right)\sigma^{0}&-\varepsilon\sigma^{0}+\mathbf{b}\cdot\mbox{\boldmath$\sigma$}\\\
\varepsilon\sigma^{0}+\mathbf{b}\cdot\mbox{\boldmath$\sigma$}&-\sigma^{0}\end{pmatrix}.$
(184)

The modes localized at the surface satisfy

$\varepsilon_{\text{bdry}}(\mathbf{k}_{\perp})=\pm\left|\mathbf{b}(\mathbf{k}_{\perp})\right|,\qquad\left|\mu(\mathbf{k}_{\perp})\right|<1.$
(185)

The first equation is obtained by imposing the Dirichlet boundary condition at
the surface, while the inequality follows from demanding that the states decay
into the bulk. Near the ${\overline{\Gamma}}$ and ${\overline{\mathrm{M}}}$
points, to lowest order,
$\varepsilon_{\text{bdry}}(\mathbf{k}_{\perp})=2\left|\delta\mathbf{k}_{\perp}\right|^{2}$,
where $\delta\mathbf{k}_{\perp}$ is measured from the gapless point. Since
$\mu=m\mp 2$ for the ${\overline{\Gamma}}$ and ${\overline{\mathrm{M}}}$
points, we get quadratic band touching at ${\overline{\Gamma}}$ for $1<m<3$
and at ${\overline{\mathrm{M}}}$ for $-3<m<-1$.

The nature of the anomalous boundary modes is independent of $\mathrm{C}^{+}$
so far. However, we can see its effect by taking two copies of this model.
From the transfer matrix, the surface Hamiltonian near the
${\overline{\Gamma}}$ point to the quadratic order in $\mathbf{k}_{\perp}$ is
given by

$H_{\text{surf}}(\mathbf{k}_{\perp})\approx
2tk_{x}k_{y}\sigma^{1}-2t^{\prime}(k_{x}^{2}-k_{y}^{2})\sigma^{3}.$ (186)

Assuming $m,t>0$, we get $\mathrm{C}^{\Gamma{\mathrm{M}}}=1$ and
$\mathrm{C}^{+}=\Theta(-t^{\prime})$. We now consider the doubled Hamiltonian
with the symmetry allowed mixing term

$\delta
H(\mathbf{k}_{\perp})=\lambda_{0}\sigma^{0}\rho^{1}+\lambda_{2}k_{x}k_{y}(k_{x}^{2}-k_{y}^{2})\sigma^{2}\rho^{2},$
(187)

where $\rho^{\mu}$ are Pauli matrices in the space of the two copies. For two
identical copies of $H_{\text{surf}}(\mathbf{k}_{\perp})$ with $t^{\prime}=1$,

$\mathrm{C}^{\Gamma{\mathrm{M}}}=2,\;\mathrm{C}^{+}=0\implies\mathrm{C}^{\Gamma{\mathrm{X}}{\mathrm{M}}}=2\mathrm{C}^{+}-\mathrm{C}^{\Gamma{\mathrm{M}}}=-2.$

Thus, we get one Dirac cone on both $\Gamma{\mathrm{M}}$ and
$\Gamma{\mathrm{X}}{\mathrm{M}}$, leading to eight Dirac cones in the surface
Brillouin zone, as shown in Fig. 19(center). On the other hand, for two
distinct copies of $H_{\text{surf}}(\mathbf{k}_{\perp})$ with $t^{\prime}=\pm
1$,

$\mathrm{C}^{\Gamma{\mathrm{M}}}=2,\;\mathrm{C}^{+}=1\implies\mathrm{C}^{\Gamma{\mathrm{X}}{\mathrm{M}}}=2\mathrm{C}^{+}-\mathrm{C}^{\Gamma{\mathrm{M}}}=0.$

Thus, we get four Dirac cones in the surface Brillouin zone, as shown in Fig.
19(right).

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[1]Andrius Grigutis

[1]Institute of Mathematics, Vilnius university, Naugarduko 24, Vilnius,
LT-03225, Lithuania

# On the exact survival probability by setting discrete random variables in E.
Sparre Andersen’s model

<EMAIL_ADDRESS>*

###### Abstract

In this work, we propose a simplification of the Pollaczek–Khinchine formula
for the ultimate time survival (or ruin) probability calculation in exchange
for a few assumptions on the random variables which generate the renewal risk
model. More precisely, we show the expressibility of the distribution function

$\mathbb{P}\left(\sup_{n\geqslant
1}\sum_{i=1}^{n}(X_{i}-c\theta_{i})<u\right),\,u\in\mathbb{N}_{0}$

via the roots of the probability generating function $G_{X-c\theta}(s)=1$, the
expectation $\mathbb{E}(X-c\theta)$, and the probability mass function of
$X-c\theta$. We assume that the random variables $X_{1},\,X_{2},\,\ldots$ and
$c\theta_{1},\,c\theta_{2},\,\ldots$ are independent copies of $X$ and
$c\theta$ respectively, $c>0$, $X$ and $c\theta$ are independent non-negative
and integer-valued, and the support of $\theta$ is finite. We give few
numerical outputs of the proven theoretical statements when the mentioned
random variables admit some particular distributions.

###### keywords:

Ruin theory, renewal theory, random walk, survival probability, generating
function, Pollaczek–Khinchine formula, initial values

###### pacs:

[

MSC Classification]60G50, 60J80, 91G05

##  1 Introduction

In applied probability, the following stochastic process

$\displaystyle W(0):=u,\,W(t):=u+ct-\sum_{i=1}^{N(t)}X_{i},\,t>0,$ (1)

is well known. Here $u\geqslant 0$, $c>0$, the non-negative random variables
$X_{1},\,X_{2},\,\ldots$ are independent copies of $X$, and

$N(t)=\max\\{n\in\mathbb{N}:\,\theta_{1}+\theta_{2}+\ldots+\theta_{n}\leqslant
t\\}$

is the renewal process generated by the non-negative random variables
$\theta_{1},\,\theta_{2},\,\ldots$, which are the independent copies of
$\theta$.

The model (1) is often met in queuing theory arguing that it represents the
G/G/1 queue. The notation G/G/1 means that the queue length in a system with a
single server is described by the interarrival times having an arbitrary
distribution and the service times having some different distribution, see
[1]. In ruin theory and insurance mathematics, the process (1) is known as E.
Sparre Andersen’s model or the renewal risk model. One may assume that $W(t)$
describes the insurer’s wealth in time, where $u\geqslant 0$ denotes the
initial surplus, $c>0$ represents the constant premium amount paid by the
customers per unit of time and the subtracted random sum means payoffs caused
by the random size claims at the random points in time, see for example [2] or
[3] among voluminous other literature regarding the renewal risk models.
Whatever is modeled by $W(t)$, one of the key aspects to know is whether
$W(t)>0$ (or $W(t)\geqslant 0$) for all such moments of time when the counting
process $N(t)$ attains its jumps.

Let us define the ultimate time survival probability

$\displaystyle\varphi(u)$
$\displaystyle:=\mathbb{P}\left(u+ct-\sum_{i=1}^{N(t)}X_{i}>0\textit{ for all
}t>0\right)$
$\displaystyle=\mathbb{P}\left(u+c\sum_{i=1}^{n}\theta_{i}-\sum_{i=1}^{N(\theta_{1}+\theta_{2}+\ldots+\theta_{n})}X_{i}>0\textit{
for all }n\geqslant 1\right)$ $\displaystyle=\mathbb{P}\left(\sup_{n\geqslant
1}\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right)<u\right),\,u\geqslant 0$ (2)

and, by the same argumentation, the finite time survival probability

$\displaystyle\varphi(u,\,T):=\mathbb{P}\left(\sup_{1\leqslant n\leqslant
T}\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right)<u\right),\,u\geqslant
0,\,T\in\mathbb{N}.$ (3)

Clearly, both functions $\varphi(u)$ and $\varphi(u,\,T)$ in (1) and (3)
respectively, serve the purpose to represent the distribution functions of the
underlying random walk

$\displaystyle\left\\{\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right),\,n\in\mathbb{N}\right\\}.$
(4)

Perhaps the probability distribution of
$\sup\limits_{n}\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right)$ is one of the
most natural numerical characteristics which is desired to know as this
information allows us to compute any other characteristics of interest for the
renewal risk models, see [4], [5], [6].

We further assume the existence of the first moments of independent and non-
negative random variables $X$ and $\theta$, i.e. $\mathbb{E}X<\infty$ and
$\mathbb{E}\theta<\infty$, where $X_{1},\,X_{2},\,\ldots$ and
$\theta_{1},\,\theta_{2},\,\ldots$ in (4) are independent copies of $X$ and
$\theta$ respectively.

The computation of $\varphi(u,\,T)$ is far easier than $\varphi(u)$. Let us
explain why. The finite time survival probability (see Proposition 1 in
Section 2 below) is based on the $T$-fold convolution of the distribution
function $\mathbb{P}(X-c\theta<u),\,u\geqslant 0$. As $T\to\infty$, the
general tool for $\varphi(u)$ calculation is the Pollaczek–Khinchine formula
which states that, if $\mathbb{E}(c\theta-X)>0$, then

$\displaystyle\varphi(u)$
$\displaystyle=e^{-A}\left(1+\sum_{n=1}^{\infty}\left(1-e^{-A}\right)^{n}H^{*n}(u)\right),\,u\geqslant
0,$ (5)

where

$\displaystyle
A=\sum_{n=1}^{\infty}\frac{\mathbb{P}(S_{n}>0)}{n},\,H(u)=\frac{F_{+}(u)}{F_{+}(\infty)},\,F_{+}(u)=\mathbb{P}(S_{N^{+}}\leqslant
u),$ $\displaystyle N^{+}=\inf\\{n\geqslant
1:S_{n}>0\\},\,S_{n}=\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right),$

and $H^{*n}$ denotes the $n$-fold convolution of the distribution function
$H$, see [7, eq. (10)]. Despite the formula (5) admitting neat closed-form
expression, its practical usage is complicated due to the convolution’s
$H^{*n}(u)$ calculation.

In this work, we demonstrate the closed-form expressiveness of the ultimate
time survival probability $\varphi(u)$ via the roots of the probability-
generating function of the random variable $X-c\theta$ where some other
numerical characteristics of $X-c\theta$ are involved under certain
assumptions of $X$ and $c\theta$. To formulate more precisely what is being
done, let us first define the auxiliary notations and formulate assumptions.

Let $s\in\mathbb{C}$ be the complex number and define the probability-
generating function for some arbitrary non-negative and integer-valued random
variable $X$

$\displaystyle
G_{X}(s):=\mathbb{P}(X=0)+s\mathbb{P}(X=1)+s^{2}\mathbb{P}(X=2)+\ldots,\,|s|\leqslant
1.$ (6)

The condition $|s|\leqslant 1$ in (6) ensures the convergence of the provided
power series in general. However, in many examples, $G_{X}(s)$ exists outside
of the unit circle too. We now aim to define the probability-generating
function of the random variable $X-c\theta$. This requires a couple of
assumptions:
(a) The independent random variables $X$ and $c\theta$ are non-negative and
integer-valued,
(b) The random variable $c\theta$ has the finite support, i.e.
$\mathbb{P}(c\theta\leqslant m)=1$ for some $m\in\mathbb{N}$.

So, under assumptions (a) and (b), the probability generating function of the
random variable $X-c\theta$, where $X$ and $\theta$ are independent, is

$G_{X-c\theta}(s)=G_{X}(s)G_{c\theta}\left(1/s\right),\,0<|s|\leqslant 1.$

Consequently, the expression of $\varphi(u)$ in this work is given under
assumptions (a) and (b) too. In other words, we assume that the random walk
(4) runs over the integers $\geqslant-m$, $m\in\mathbb{N}$. The assumption (a)
implies that it is sufficient to consider $\varphi(u)$ for
$u\in\mathbb{N}_{0}$ only.

Let us comment on that assumption (a) serves the purpose of the neater
expressions of $\varphi(u)$. The assumption (a) may be considered as not too
restrictive since every continuous or semi-continuous process can be
discretized, i.e. ”we can always calculate the money in cents”. The assumption
(b) excludes the opportunity ”to live happily ever after” and the statement of
(b) in this work is unavoidable for a number of reasons:

  * •

we require the existence of $G_{X-c\theta}(s)$ in general,

  * •

we require $m-1$ roots of $G_{X-c\theta}(s)=1$ in $0<|s|\leqslant 1$, $s\neq
1$, see Lemma 3 below,

  * •

we cannot get the information on all of $\varphi(u),\,u\in\mathbb{N}_{0}$ at
once, see eq. (12) below.

Again, assumption (b) might be considered as not too restrictive from a
practical perspective. The number $m$ in (b) can be arbitrarily large and,
aiming to incorporate some inter-occurrence time $\theta$ of infinite support,
we can define a new random variable $\theta_{m}$ whose distribution is

$\displaystyle\begin{cases}&\mathbb{P}\left(c\theta_{m}=k\right)=\mathbb{P}\left(c\theta=k\right),\,k\in\\{0,\,1,\,\ldots,\,m-1\\}\\\
&\mathbb{P}\left(c\theta_{m}=m\right)=\mathbb{P}\left(c\theta\geqslant
m\right)\end{cases}.$ (7)

Then, when $\theta^{m}_{1},\,\theta^{m}_{2},\,\ldots$ are independent copies
of $\theta_{m}$,

$\mathbb{P}\left(\sup_{n\geqslant
1}\sum_{i=1}^{n}\left(X_{i}-c\theta^{m}_{i}\right)<u\right)\leqslant\mathbb{P}\left(\sup_{n\geqslant
1}\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right)<u\right)$

for all $u\in\mathbb{N}_{0}$, and this can be considered as a more
conservative approach to the modeled process since the ultimate time survival
probability does not increase switching inter-occurrence time of infinite
support to the finite one, see Example 4 in Section 5 below. Moreover, let’s
say that the model (1) is generated by the non-negative integer-valued and
independent random variables $X$ and $c\theta$, where the inter-occurrence
time $\theta$ is of infinite support. Then, switching $c\theta$ to
$c\theta_{m}$ of the finite support as provided in (7), we can modify the
probabilities of $X$, i.e. define a new random variable $X_{m}$, in a way that
certain numerical characteristics of the random variables $X-c\theta$ and
$X_{m}-c\theta_{m}$ are aligned. For instance, having some $X$, regardless of
its support, we can take some of its value $l\neq 0$ and define the random
variable $X_{m}$

$\displaystyle\begin{cases}&\mathbb{P}(X_{m}=0)=\mathbb{P}(X=0)+\frac{1}{l}\sum\limits_{i=1}^{\infty}i\mathbb{P}(c\theta=m+i)\\\
&\mathbb{P}(X_{m}=k)=\mathbb{P}(X=k),\,k\in\mathbb{N},\,k\neq l\\\
&\mathbb{P}(X_{m}=l)=\mathbb{P}(X=l)-\frac{1}{l}\sum\limits_{i=1}^{\infty}i\mathbb{P}(c\theta=m+i)\end{cases},$

achieving $\mathbb{E}(X_{m}-c\theta_{m})=\mathbb{E}(X-c\theta)$. In other
words, we can balance between the ”softer punishment more often” and the
”harsher punishment less often”. In addition, in view of
$\varphi(u)\leqslant\varphi(u+1)$ for all $u\in\mathbb{N}_{0}$ and
$\varphi(u)\to 1$ as $u\to\infty$ under the net profit condition (see eq. (14)
below), eq. (12) implies the estimates

$\displaystyle\inf_{i\geqslant
m+1}\varphi(i)\mathbb{P}(X-c\theta\geqslant-m-1)\leqslant\varphi(0)-\sum_{i=1}^{m}\varphi(i)f(-i)\leqslant\mathbb{P}(X-c\theta\geqslant-m-1),$
(8)

where $f(j)=\mathbb{P}(X-c\theta=j)$ for any integer $j$. This suggests that
the error in estimating $\varphi(0)$ (the other survival probabilities too)
when truncating the inter-occurrence time $\theta$ of infinite support is
controlled by the probability tail mass of $\theta$.

Let us recall that the expectation $\mathbb{E}(X-c\theta)$ is called the net
profit condition and it is said that the net profit condition holds if
$G^{\prime}_{X-c\theta}(1)=\mathbb{E}(X-c\theta)<0$, i.e. on average the
claims’ sizes are less than collected premiums. It is well known that
$\varphi(u)=0$ for all $u\geqslant 0$ if $\mathbb{E}(X-c\theta)\geqslant 0$
except some cases when $\mathbb{E}(X-c\theta)=0$ and
$\mathbb{P}(X-c\theta=0)=1$, see [8].

As mentioned, the calculation of the survival (or ruin $\psi=1-\varphi$)
probabilities is nothing but the convolution’s calculation of the distribution
function $\mathbb{P}(X-c\Theta<u)$, and of course, the finite convolution is
far simpler than the ultimate one. Applying the total probability formula for
the ultimate time survival probability under assumptions (a) and (b) we obtain
the following recurrence relation (see Lemma 1 and its proof below)

$\displaystyle\varphi(u)=\sum_{i=1}^{u+m}\varphi(i)f(u-i),\,u\in\mathbb{N}_{0},$
(9)

where, as previuosly, $f(j)=\mathbb{P}(X-c\theta=j)$, $j=-m,\,-m+1,\,\ldots$
is the probability mass function. By setting $u=0,\,1,\,\ldots$ in formula (9)
we get

$\varphi(0)=\sum_{i=1}^{m}\varphi(i)f(-i),\,\varphi(1)=\sum_{i=1}^{m+1}\varphi(i)f(1-i),\,\ldots,$

which means that willing to get $\varphi(m),\,\varphi(m+1),\,\ldots$ we must
know $\varphi(0),\,\varphi(1),\,\ldots,\,\varphi(m-1)$. Thus, the calculation
of $\varphi(u),\,u\in\mathbb{N}_{0}$ turns into the finding of the initial
values $\varphi(0),\,\varphi(1),\,\ldots,\,\varphi(m-1)$ for the recurrence
(9). Let

$\displaystyle\mathcal{M}=\left(\sup_{n\geqslant
1}\sum_{i=1}^{n}\left(X_{i}-c\theta_{i}\right)\right)^{+},$ (10)

where $a^{+}=\max\\{0,\,a\\}$, $a\in\mathbb{R}$ and
$\pi_{i}=\mathbb{P}(\mathcal{M}=i),\,i\in\mathbb{N}_{0}$, be the probability
mass function of the newly defined random variable $\mathcal{M}$. Then,
according to (1) above,

$\displaystyle\varphi(u+1)=\mathbb{P}(\mathcal{M}\leqslant
u)=\sum_{i=0}^{u}\pi_{i},\,u\in\mathbb{N}_{0}$ (11)

and the search of the initial values of
$\varphi(0),\,\varphi(1),\,\ldots,\,\varphi(m-1)$ turns into the search of the
initial values of the probability mass function of $\mathcal{M}$. However, the
technique given in this work is applicable when finding the first initial
values of $\pi_{0},\,\pi_{1},\,\ldots,\,\pi_{m-1}$ only. If $u=0$ and
$m\to\infty$, the recurrence (9) implies

$\displaystyle\varphi(0)=\sum_{i=1}^{\infty}\varphi(i)f(-i)$ (12)

and there is nothing better than the Pollaczek–Khinchine formula (eq. (5)
above) to compute the exact values of $\varphi(u)$ out of this ultimate
recurrence (12).

Let us explain how we obtain the desired initial values of
$\pi_{0},\,\pi_{1},\,\ldots,\,\pi_{m-1}$. The defined random variable
$\mathcal{M}$ (eq. (10) above) admits the following distribution property

$\displaystyle\mathcal{M}\buildrel d\over{=}(\mathcal{M}+X-c\theta)^{+},$ (13)

see [9, p. 198] or [10, Lem. 4.1]. The net profit condition
$\mathbb{E}(X-c\theta)<0$ ensures

$\displaystyle\lim_{u\to\infty}\varphi(u)=\mathbb{P}(\mathcal{M}<\infty)=1,$
(14)

this can be proved by following the proof of [11, Lem.1 ] or [12, Lem. 4].
Distributions’ equality (13) implies the equality of the corresponding
probability-generating functions

$\displaystyle\mathbb{E}s^{\mathcal{M}}=\mathbb{E}s^{(\mathcal{M}+X-c\theta)^{+}}=\mathbb{E}\left(\mathbb{E}s^{(\mathcal{M}+X-c\theta)^{+}}|\mathcal{M}\right),\,|s|\leqslant
1.$ (15)

Replicating (15) over the roots of

$\displaystyle G_{X-c\theta}(s)=1,\,0<|s|\leqslant 1,\,s\neq 1$ (16)

and letting $s\to 1^{-}$ in derivative (with respect to $s$) of both sides of
(15), we get a system of linear equations (see (22) below) whose solution is
the wanted probabilities $\pi_{0},\,\pi_{1},\,\ldots,\,\pi_{m-1}$, see Theorem
3 below. Note that if $\mathbb{P}(c\theta\leqslant m)=1$ and
$\mathbb{P}(X=0)>0$ there are exactly $m-1$ roots, counted with their
multiplicities, of (16) in $0<|s|\leqslant 1,\,s\neq 1$, see Lemma 3 below,
and it is not clear in general what happens to the roots of (16) in the unit
circle or the entire complex plane if $m\to\infty$.

On top of the described setup of the ultimate time survival probability
$\varphi(u),\,u\in\mathbb{N}_{0}$ we can construct the ultimate time survival
probability generating function $\Xi(s)$ too. Indeed,

$\Xi(s)=\sum_{i=0}^{\infty}\varphi(i+1)s^{i}=\sum_{i=0}^{\infty}\sum_{j=0}^{i}\pi_{j}s^{i}=\frac{G_{\mathcal{M}}(s)}{1-s},\,|s|<1,$

where the probability-generating function $G_{\mathcal{M}}(s)$ can be solved
out of (15), see eq. (19) below.

The majority of the work in setting and solving the described system (22)
remains similar as in [10], where the same discretized model (1) with
$\theta\equiv 1$ and $c\in\mathbb{N}$ is considered. So, the essence of this
work can be described as following the reference [10] carefully and adjusting
the corresponding formulas where needed.

Let us mention that the described initial values
$\varphi(0),\,\varphi(1),\,\ldots,\,\varphi(m-1)$ can be found not using
neither the generating function $G_{X-c\theta}(s)$ nor its roots, but rather
calculating the approximate limits of certain recurrent sequences, see [13],
[14] and related works. However, the method of recurrent sequences is more
extensive and fails either when $m\to\infty$.

##  2 Main results

In this section, we formulate the main statements for the finite time
$\varphi(u,\,T)$ and the ultimate time $\varphi(u)$ survival probabilities
calculation. The listed statements are proved in Section 4 below.

######  Proposition 1.

Let $\varphi(u,\,T)$ be the finite time survival probability of the stochastic
process (1). Then, for all $T\in\mathbb{N}$,

$\displaystyle\varphi(u,\,T)=\int\limits_{(-\infty,\,u)}\tilde{F}^{*(T-1)}(u-x)\,d\,\tilde{F}(x),\,u\geqslant
0,$

where $\tilde{F}^{*T}$ denotes the $T$-fold convolution of the distribution
function $\tilde{F}(u)=\mathbb{P}(X-c\theta<u)$ and $\tilde{F}^{*0}(u):=1$.

Note that Proposition 1 holds regardless of the assumptions (a) and (b)
formulated in the previous section. Of course, the calculation of
$\varphi(u,\,T)$ is more convenient assuming (a) and (b). This implies the
following corollary of Proposition 1.

######  Corollary 1.

Assume that the stochastic process (1) is generated by the non-negative,
independent, and integer-valued random variables $X$ and $c\theta$, where
$\mathbb{P}(c\theta\leqslant m)=1$ for some $m\in\mathbb{N}$. Then, for all
$u\in\mathbb{N}_{0}$,

$\displaystyle\varphi(u,\,1)=F(u-1),\,\varphi(u,\,T)=\sum_{k=-m}^{u+1}\varphi(u-k,\,T-1)f(k),\,T\in\\{2,\,3,\,\ldots\\},$

where $f(j)=\mathbb{P}(X-c\theta=j)$ and $F(j)=\mathbb{P}(X-c\theta\leqslant
j)$ for any integer $j$.

Let us now turn to the ultimate time survival probability $\varphi(u)$. The
next statement provides the relation of probabilities
$\pi_{0},\,\pi_{1},\,\ldots,\,\pi_{m-1}$, probability-generating functions of
$\mathcal{M}$ and $X-c\theta$, and the ultimate time survival probability
generating function $\Xi(s)$.

######  Theorem 2.

Assume that the stochastic process (1) is generated by the non-negative,
independent, and integer-valued random variables $X$ and $c\theta$, where
$\mathbb{P}(c\theta\leqslant m)=1$ for some $m\in\mathbb{N}$. Then, under the
net profit condition $c\mathbb{E}\theta-\mathbb{E}X>0$, the following
equalities hold:

$\displaystyle\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}(1-s^{i-j})f(-j)=G_{\mathcal{M}}(s)(1-G_{X-c\theta}(s)),\,|s|\leqslant
1,$ (17)
$\displaystyle\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}(j-i)f(-j)=c\mathbb{E}\theta-\mathbb{E}X,$
(18)
$\displaystyle\Xi(s)=\frac{1}{s^{m}(G_{X-c\theta}(s)-1)}\sum_{i=0}^{m-1}\pi_{i}\sum_{j=0}^{m-i-1}s^{j+i}F(-m+j),\,|s|<1,$
(19)

where, as previously, $f(j)=\mathbb{P}(X-c\theta=j)$,
$F(j)=\mathbb{P}(X-c\theta\leqslant j)$ for any integer $j$, and
$s\in\mathbb{C}$ such that $s^{m}(1-G_{X-c\theta}(s))\neq 0$ in eq. (19).

Let us mention that if $m=c=1$ and $\theta\equiv 1$, the equations (18) and
(9) imply

$\pi_{0}=\varphi(1)=\frac{1-\mathbb{E}X}{\mathbb{P}(X=0)},\,\varphi(u)=\sum_{i=1}^{u+1}\varphi(i)\mathbb{P}(X=u+1-i),\,u\in\mathbb{N}_{0},$

where $\mathbb{P}(X=0)>0$ because of the net profit condition. This is the
well-known result for the homogeneous discrete-time risk model, see [15],
[16], [17], and many other sources.

Under assumptions (a), (b), and the net profit condition, eq. (17) in Theorem
2 implies the following corollary.

######  Corollary 2.

Let $|\alpha|\leqslant 1$, $\alpha\neq\\{0,\,1\\}$ be the root of

$G_{X}(s)G_{c\theta}\left(\frac{1}{s}\right)=1$

and recall that $F(j)=\mathbb{P}(X-c\theta\leqslant j)$ for any integer $j$.
Then

$\displaystyle\sum_{i=0}^{m-1}\pi_{i}\sum_{j=0}^{m-i-1}\alpha^{j+i}F(-m+j)=0.$
(20)

Moreover, the $n$’th derivative

$\displaystyle\frac{d^{n}}{ds^{n}}\left(\sum_{i=0}^{m-1}\pi_{i}\sum_{j=0}^{m-i-1}s^{j+j}F(-m+j)\right)\Bigg{|}_{s=\alpha}=0$
(21)

for all $n\in\\{0,\,1,\,\ldots,\,r-1\\}$, where
$r\in\\{1,\,2,\,\ldots,\,m-1\\}$ denotes the multiplicity of the root
$\alpha$.

We recall that there are exactly $m-1$ roots of
$G_{X}(s)G_{c\theta}\left(1/s\right)=1$ in $|s|\leqslant 1$,
$s\neq\\{0,\,1\\}$ counted with their multiplicities, see Lemma 3 below.

Suppose all the roots
$\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{m-1}\neq\\{0,\,1\\}$ of
$G_{X}(s)G_{c\theta}\left(1/s\right)=1$ in $|s|\leqslant 1$ are simple. Then,
replicating the equation (20) over these roots and including the equality (18)
we set up the following system

$\displaystyle\begin{pmatrix}&\sum\limits_{j=0}^{m-1}\alpha_{1}^{j}F(-m+j)&\sum\limits_{j=0}^{m-2}\alpha_{1}^{j+1}F(-m+j)&\ldots&\alpha_{1}^{m-1}f(-m)\\\
&\sum\limits_{j=0}^{m-1}\alpha_{2}^{j}F(-m+j)&\sum\limits_{j=0}^{m-2}\alpha_{2}^{j+1}F(-m+j)&\ldots&\alpha_{2}^{m-1}f(-m)\\\
&\vdots&\vdots&\ddots&\vdots\\\
&\sum\limits_{j=0}^{m-1}\alpha_{m-1}^{j}F(-m+j)&\sum\limits_{j=0}^{m-2}\alpha_{m-1}^{j+1}F(-m+j)&\ldots&\alpha_{m-1}^{m-1}f(-m)\\\
&\sum_{j=1}^{m}jf(-j)&\sum_{j=2}^{m}(j-1)f(-j)&\ldots&f(-m)\end{pmatrix}\begin{pmatrix}&\pi_{0}\\\
&\pi_{1}\\\ &\vdots\\\ &\pi_{m-1}\end{pmatrix}$ $\displaystyle\hskip
256.0748pt=\begin{pmatrix}&0\\\ &0\\\ &\vdots\\\ &0\\\
&c\mathbb{E}\theta-\mathbb{E}X\end{pmatrix}$ (22)

Let us denote the system (22) as $A{\bm{\pi}}=B$. The matrix $A$ is
Vandermonde-like and its determinant $|A|$ admits the following representation

$\displaystyle|A|=(-1)^{m-1}f^{m}(-m)\prod_{j=1}^{m-1}(\alpha_{j}-1)\prod_{1\leqslant
i<j\leqslant m-1}\left(\alpha_{j}-\alpha_{i}\right),$

see Lemma 2 below. Thus, $|A|\neq 0$ as long as
$\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{m-1}\neq\\{0,\,1\\}$ are the
simple roots of $G_{X}(s)G_{c\theta}\left(1/s\right)=1$ in $|s|\leqslant 1$
and $f(-m)>0$. In the next theorem, we provide the solution of (22) and the
exact values of $\varphi(u),\,u\in\mathbb{N}_{0}$.

######  Theorem 3.

Suppose $\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{m-1}\neq\\{0,\,1\\}$ are
the simple roots of $G_{X}(s)G_{c\theta}\left(1/s\right)=1$ in $|s|\leqslant
1$, and recall that $f(-m)=\mathbb{P}(X-c\theta=-m)>0$,
$F(j)=\mathbb{P}(X-c\theta\leqslant j)$ for any integer $j$. Then

$\displaystyle\tilde{\pi}_{0}$
$\displaystyle=\frac{1}{f(-m)}\prod_{j=1}^{m-1}\frac{\alpha_{j}}{\alpha_{j}-1},$
$\displaystyle\tilde{\pi}_{1}$ $\displaystyle=-\frac{\sum\limits_{1\leqslant
j_{1}<\ldots<j_{m-2}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-2}}}{f(-m)\prod\limits_{j=1}^{m-1}(\alpha_{j}-1)}-\frac{F(-m+1)}{f(-m)}\tilde{\pi}_{0},$
$\displaystyle\tilde{\pi}_{2}$ $\displaystyle=\frac{\sum\limits_{1\leqslant
j_{1}<\ldots<j_{m-3}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-3}}}{f(-m)\prod\limits_{j=1}^{m-1}(\alpha_{j}-1)}-\frac{F(-m+2)}{f(-m)}\tilde{\pi}_{0}-\frac{F(-m+1)}{f(-m)}\tilde{\pi}_{1},$
$\displaystyle\,\,\,\vdots$ $\displaystyle\tilde{\pi}_{m-1}$
$\displaystyle=\frac{(-1)^{m+1}}{f(-m)}\prod\limits_{j=1}^{m-1}\frac{1}{\alpha_{j}-1}-\frac{1}{f(-m)}\sum_{i=0}^{m-2}\tilde{\pi}_{i}F(-1-i),$

where $\tilde{\pi}_{i}=\pi_{i}/(c\mathbb{E}\theta-\mathbb{E}X)$. Moreover, if
$\tilde{\varphi}(u)=\varphi(u)/(c\mathbb{E}\theta-\mathbb{E}X)$, then

$\displaystyle\tilde{\varphi}(0)$
$\displaystyle=(-1)^{m+1}\prod_{j=1}^{m-1}\frac{1}{\alpha_{j}-1},\,\tilde{\varphi}(1)=\frac{1}{f(-m)}\prod_{j=1}^{m-1}\frac{\alpha_{j}}{\alpha_{j}-1},$
$\displaystyle\tilde{\varphi}(2)$
$\displaystyle=-\frac{F(-m+1)}{f(-m)}\tilde{\varphi}(1)$ $\displaystyle\hskip
14.22636pt+\prod_{j=1}^{m-1}\frac{1/f(-m)}{\alpha_{j}-1}\left(\prod_{j=1}^{m-1}\alpha_{j}-\sum\limits_{1\leqslant
j_{1}<\ldots<j_{m-2}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-2}}\right),$
$\displaystyle\tilde{\varphi}(3)$
$\displaystyle=-\frac{F(-m+1)}{f(-m)}\tilde{\varphi}(2)-\frac{F(-m+2)}{f(-m)}\tilde{\varphi}(1)+\prod_{j=1}^{m-1}\frac{1/f(-m)}{\alpha_{j}-1}$
$\displaystyle\times\left(\prod_{j=1}^{m-1}\alpha_{j}-\sum_{1\leqslant
j_{1}<\ldots<j_{m-2}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-2}}+\sum_{1\leqslant
j_{1}<\ldots<j_{m-3}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-3}}\right),$ $\displaystyle\,\,\vdots$
$\displaystyle\tilde{\varphi}(m)$
$\displaystyle=-\frac{1}{f(-m)}\sum_{i=1}^{m-1}F(-i)\tilde{\varphi}(i)+\prod_{j=1}^{m-1}\frac{1/f(-m)}{\alpha_{j}-1}\times\Bigg{(}\prod_{j=1}^{m-1}\alpha_{j}-$
$\displaystyle\sum_{1\leqslant j_{1}<\ldots<j_{m-2}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-2}}+\sum_{1\leqslant
j_{1}<\ldots<j_{m-3}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-3}}+\ldots+(-1)^{m+1}\Bigg{)}$

and

$\displaystyle\varphi(u)=\frac{1}{f(-m)}\left(\varphi(u-m)-\sum_{i=1}^{u-1}\varphi(i)f(u-m-i)\right),\,u=m,\,m+1,\,\ldots$
(23)

If $f(-m)>0$ and there are multiple roots among
$\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{m-1}\neq\\{0,\,1\\}$, in order to
avoid identical lines, we then shall modify the system (22) by replacing its
line (or lines) by derivatives, as provided by the equality (21), see Example
3 below. Such a modified system remains non-singular too because, roughly, the
derivative represents the linear mapping, see [10, Lem 4.3] for more detailed
proof.

The system (22) may require modifications not only due to the multiple roots.
If $\mathbb{P}(X-c\theta>i)=1$ for some $i>-m$ under assumption
$\mathbb{P}(c\theta=m)>0$, i.e. the values of the random variable $X$ (with
positive probability) start by some value which is greater than zero, we then
shall rebuild the system (22) according to the equations (20), including (21)
in case there are multiple roots, and (18). In other words, if
$\mathbb{P}(c\theta=m)>0$ and $\mathbb{P}(X\geqslant i)=1$ for some
$i\in\mathbb{N}$, the modification is necessary in order to avoid the zero-
columns in system’s matrix in (22). Let us observe that (if
$\mathbb{P}(c\theta=m)>0$) $\mathbb{P}(X\geqslant m)=1$ violates the net
profit condition.

##  3 Lemmas

In this section, we formulate and prove several auxiliary statements that
Section 2 is based on.

######  Lemma 1.

The ultimate time survival probability (1) satisfies the following recurrence
relation

$\varphi(u)=\sum_{i=1}^{u+m}\varphi(i)f(u-i),\,u\in\mathbb{N}_{0}.$

###### Proof.

Since the random variables $X_{1}-c\theta_{1},\,X_{2}-c\theta_{2},\,\ldots$
are independent and identically distributed, and
$\mathbb{P}(X-c\theta\geqslant-m)=1$, then

$\displaystyle\varphi(u)$
$\displaystyle=\mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\\{\sum_{i=1}^{n}(X_{i}-c\theta_{i})<u\right\\}\right)$
$\displaystyle=\mathbb{P}\left(\bigcap_{n=2}^{\infty}\left\\{X_{1}-c\theta_{1}+\sum_{i=2}^{n}(X_{i}-c\theta_{i})<u\right\\},\,X_{1}-c\theta_{1}<u\right)$
$\displaystyle=\sum_{i=-m}^{u-1}\varphi(u-i)f(i)=\sum_{i=1}^{u+m}\varphi(i)f(u-i).$

∎

######  Lemma 2.

Let $A$ be the system’s matrix from (22) and denote
$f^{m}(-m)=\left(\mathbb{P}(X-c\theta=-m)\right)^{m}$. Then, the determinant
of $A$ is

$|A|=(-1)^{m-1}f^{m}(-m)\prod_{j=1}^{m-1}(\alpha_{j}-1)\prod_{1\leqslant
i<j\leqslant m-1}\left(\alpha_{j}-\alpha_{i}\right).$

Moreover, the minors of the last line of $A$ are:

$\displaystyle M_{m,\,1}$
$\displaystyle=f^{m-1}(-m)\prod_{i=1}^{m-1}\alpha_{i}\prod_{1\leqslant
i<j\leqslant m-1}(\alpha_{j}-\alpha_{i}),$ $\displaystyle M_{m,\,2}$
$\displaystyle=f(-m)^{m-1}\prod_{1\leqslant i<j\leqslant
m-1}(\alpha_{j}-\alpha_{i})\sum_{1\leqslant j_{1}<\ldots<j_{m-2}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-2}}$
$\displaystyle+\frac{F(-m+1)}{f(-m)}M_{\kappa,\,1},$ $\displaystyle M_{m,\,3}$
$\displaystyle=f(-m)^{m-1}\prod_{1\leqslant i<j\leqslant
m-1}(\alpha_{j}-\alpha_{i})\sum_{1\leqslant j_{1}<\ldots<j_{m-3}\leqslant
m-1}\alpha_{j_{1}}\cdots\alpha_{j_{m-3}},$
$\displaystyle-\frac{F(-m+2)}{f(-m)}M_{\kappa,\,1}+\frac{F(-m+1)}{f(-m)}M_{\kappa,\,2},$
$\displaystyle\vdots$ $\displaystyle M_{m,\,m}$
$\displaystyle=f(-m)^{m-1}\prod_{1\leqslant i<j\leqslant
m-1}(\alpha_{j}-\alpha_{i})+(-1)^{m}\frac{F(-1)}{f(-m)}M_{m,\,1}$
$\displaystyle+(-1)^{m+1}\frac{F(-2)}{f(-m)}M_{m,\,2}+\ldots+(-1)^{2m-1}\frac{F(-m+2)}{f(-m)}M_{\kappa,\,\kappa-2}$
$\displaystyle+\frac{F(-m+1)}{f(-m)}M_{m,\,m-1}.$

###### Proof.

The proof is pretty technical and is based on elementary calculations of the
determinants, see the reference [10, Lem 4.2 and pp. 5192-5193] where the same
determinants with $\theta\equiv 1$ and $c\in\mathbb{N}$ are calculated more
extensively. ∎

######  Lemma 3.

Let $s\in\mathbb{C}$. For the non-negative, independent and integer-valued
random variables $X$ and $c\theta$ assume $\mathbb{P}(c\theta\leqslant m)=1$
and $\mathbb{P}(X-c\theta=-m)>0$, i.e. $\mathbb{P}(X=0)>0$ and
$\mathbb{P}(c\theta=m)>0$. Then, there are exactly $m-1$ roots, counted with
their multiplicities, of

$\displaystyle G_{X-c\theta}(s)=1$ (24)

in $|s|\leqslant 1$, $s\neq\\{0,\,1\\}$.

###### Proof.

We shall follow the reference [14, Sec. 4]. Let us temporarily assume
$\mathbb{P}(c\theta=m)=1$. Then, the equation (24) becomes

$G_{X}(s)=s^{m}.$

Since $\mathbb{P}(X=0)>0$, the latter equation has exactly $m-1$ roots in
$0<|s|\leqslant 1$, $s\neq 1$, counted with their multiplicities. This fact is
implied by (i) the fundamental theorem of algebra, which states that (without
loosing the generality) the equation $s^{m}=1$ has $m$ roots counted with
their multiplicities (or $m-1$ not counting $s=1$), (ii) the estimate

$|G_{X}(s)|\leqslant 1<|\lambda s^{m}|,\,\lambda>1,$

and (iii) Rouché’s theorem, see [18]. Then, declining the requirement
$\mathbb{P}(c\theta=m)=1$, we can rewrite the equation (24) as follows

$\displaystyle\left(\mathbb{P}(X=0)+\mathbb{P}(X=0)s+\ldots\right)\left(\mathbb{P}(c\theta=0)+\frac{\mathbb{P}(c\theta=1)}{s}+\ldots+\frac{\mathbb{P}(c\theta=m)}{s^{m}}\right)=1.$

Multiplying both sides of the last equation by $s^{m}$, we obtain

$\displaystyle\left(\mathbb{P}(X=0)+\mathbb{P}(X=0)s+\ldots\right)\left(\mathbb{P}(c\theta=0)s^{m}+\mathbb{P}(c\theta=1)s^{m-1}+\ldots+\mathbb{P}(c\theta=m)\right)=s^{m},$

where the left-hand side of the last equation is the probability-generating
function of the non-negative discrete and integer-valued distribution and has
the same number of roots in $|s|\leqslant 1$, $s\neq\\{0,\,1\\}$ as in the
previous case when $\mathbb{P}(c\theta=m)=1$ was assumed. ∎

##  4 Proofs of the main results

In this section, we prove all of the statements formulated in Section 2.

###### Proof of Proposition 1.

The proof is pretty evident. If $T=1$, then

$\varphi(u,\,1)=\int\limits_{(-\infty,\,u)}\,d\,\tilde{F}(x)=\mathbb{P}(X-c\theta<u).$

If $T=2$, then

$\displaystyle\varphi(u,\,2)$
$\displaystyle=\mathbb{P}\left(\\{X_{1}-c\theta_{1}<u\\}\cap\\{X_{1}-c\theta_{1}+X_{2}-c\theta_{2}<u\\}\right)$
$\displaystyle=\int\limits_{(-\infty,\,u)}\mathbb{P}(X-c\theta<u-x)\,d\,\mathbb{P}(X-c\theta<x)=\int\limits_{(-\infty,\,u)}\tilde{F}^{*1}(u-x)\,d\,\tilde{F}(x)$

and, by mathematical induction,

$\displaystyle\varphi(u,\,T)=\int\limits_{(-\infty,\,u)}\tilde{F}^{*(T-1)}(u-x)\,d\,\tilde{F}(x).$

∎

###### Proof of Corollary 1.

The expression of $\varphi(u,\,1)$ is evident. The formula of $\varphi(u,\,T)$
for $T\in\\{2,\,3,\,\ldots\\}$ is implied as follows

$\displaystyle\varphi(u,\,T)$
$\displaystyle=\mathbb{P}\left(\bigcap\limits_{n=1}^{T}\left\\{\sum_{i=1}^{n}(X_{i}-c\theta_{i})<u\right\\}\right)$
$\displaystyle=\mathbb{P}\left(\bigcap\limits_{n=2}^{T}\left\\{\sum_{i=1}^{n}(X_{i}-c\theta_{i})<u\right\\}\cap\\{X_{1}-c\theta_{1}<u\\}\right)$
$\displaystyle=\sum_{k=-m}^{u-1}\mathbb{P}\left(\bigcap\limits_{n=1}^{T-1}\left\\{\sum_{i=1}^{n}(X_{i}-c\theta_{i})<u-(X_{1}-c\theta_{1})\right\\}\right)\mathbb{P}(X-c\theta=k)$
$\displaystyle=\sum_{k=-m}^{u-1}\varphi(u-k,\,T-1)f(k).$

∎

###### Proof of Theorem 2.

The equality $\mathcal{M}\buildrel
d\over{=}\left(\mathcal{M}+X-c\theta\right)^{+}$ (see eq. (13) above) implies
the equality of the probability-generating functions of the underlying
distributions. Then, applying the rule of total expectation when having in
mind that $\mathbb{P}(X-c\theta\geqslant-m)=1$ and
$\mathbb{P}(\mathcal{M}<\infty)=1$ due to the net profit condition (see eq.
(14) above), we obtain

$\displaystyle\mathbb{E}s^{\mathcal{M}}$
$\displaystyle=\mathbb{E}s^{\left(\mathcal{M}+X-c\theta\right)^{+}}=\mathbb{E}\left(\mathbb{E}\left(s^{\left(\mathcal{M}+X-c\theta\right)^{+}}|\mathcal{M}\right)\right)$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\mathbb{E}s^{(i+X-c\theta)^{+}}+\sum_{i=m}^{\infty}\pi_{i}\mathbb{E}s^{i+X-c\theta}$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\left(\sum_{j=-m}^{-i}f(j)+\sum_{j=-i+1}^{\infty}s^{i+j}f(j)\right)+\mathbb{E}s^{X-c\theta}\sum_{i=m}^{\infty}\pi_{i}s^{i}$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\left(\sum_{j=-m}^{-i}f(j)+\sum_{j=-i+1}^{\infty}s^{i+j}f(j)\right)+G_{X-c\theta}(s)\left(G_{\mathcal{M}}(s)-\sum_{i=0}^{m-1}\pi_{i}s^{i}\right)$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\left(\sum_{j=-m}^{-i}f(j)+s^{i}\sum_{j=-i+1}^{\infty}s^{j}f(j)-s^{i}\sum_{j=-m}^{\infty}s^{j}f(j)\right)+G_{X-c\theta}(s)G_{\mathcal{M}}(s)$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\sum_{j=-m}^{-i-1}f(j)(1-s^{i+j})+G_{\mathcal{M}}(s)G_{X}(s)G_{c\theta}(1/s),$

which implies

$\displaystyle\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}f(-j)(1-s^{i-j})=G_{\mathcal{M}}(s)(1-G_{X}(s)G_{c\theta}(1/s))$
(25)

and the equality (17) is proved. The second Theorem’s equality (18) is implied
by the proven equality (25) by taking the derivative of both sides with
respect to $s$ and letting $s\to 1^{-}$, see [12, Lem. 9] and [10, p. 5191]. ∎

###### Proof of Corollary 2.

If $|\alpha|\leqslant 1$, $\alpha\neq\\{0,\,1\\}$ is the root of

$\displaystyle G_{X}(s)G_{c\theta}\left(\frac{1}{s}\right)=1,$

the equality (17) implies

$\displaystyle\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}f(-j)(1-s^{i-j})=0.$ (26)

Since

$(1-s^{i-j})=\left(1+s^{-1}+\ldots+s^{i-j+1}\right)(1-s^{-1}),$

we multiply the both sides of (26) by $1-s^{-1}$, $s\neq 0$ and obtain

$\displaystyle(1-s^{-1})\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}f(-j)(1-s^{i-j})=\sum_{i=0}^{m-1}\pi_{i}\sum_{j=i+1}^{m}f(-j)\sum_{l=0}^{-i+j-1}s^{-j}$
$\displaystyle=\sum_{i=0}^{m-1}\pi_{i}\sum_{l=0}^{m-i-1}s^{-l}\sum_{j=i+1+l}^{m}f(-j)=\sum_{i=0}^{m-1}\pi_{i}\sum_{l=0}^{m-i-1}s^{-l}F(-i-1-l)$
$\displaystyle=s^{-m+1}\sum_{i=0}^{m-1}\pi_{i}\sum_{j=0}^{m-1-i}s^{j+i}F(-m+j)=0.$

∎

###### Proof of Theorem 3.

As the roots $\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{m-1}$ are assumed to
be simple and $f(-m)>0$, the system (22) has the solution
$(\pi_{0},\,\pi_{1},\,\ldots,\,\pi_{m-1})$ where

$\pi_{i}=\frac{(-1)^{m+i+1}M_{m,\,i+1}}{|A|}(\mathbb{E}(c\theta-X)),\,i=0,\,1,\,\ldots,\,m-1$

and the minors $M_{m,\,1},\,M_{m,\,2},\,\ldots,\,M_{m,\,m}$, including
determinant $|A|$, are given in Lemma 2 above. Then

$\varphi(i+1)=\sum_{j=0}^{i}\pi_{i},\,i=0,\,1,\,\ldots,\,m-1,$

due to eq. (11). The formula of $\varphi(0)$ is implied by the recurrence (9)
and the expression of $\pi_{m-1}$. Indeed,

$\displaystyle\varphi(0)$
$\displaystyle=\sum_{i=1}^{m}\varphi(i)f(-i)=\sum_{i=1}^{m}f(-i)\sum_{j=0}^{i-1}\pi_{j}=\sum_{j=0}^{m-1}\pi_{j}\sum_{i=j+1}^{m}f(-i)=\sum_{j=0}^{m-1}\pi_{j}F(-j-1)$
$\displaystyle=(-1)^{m+1}\,\mathbb{E}(c\theta-X)\,\prod_{j=1}^{m-1}\frac{1}{\alpha_{j}-1}.$

The formula (23) is the rearranged version of (9).

∎

##  5 Numerical examples

In this section, we demonstrate some numerical outputs of statements
formulated in the previous Section 2 when $X$ and $c\theta$ admit some chosen
distributions. All the necessary calculations and visualizations are performed
using the software [19].

######  Example 1.

Let $\mathbb{P}(X=0)=\mathbb{P}(X=1)=1/2$, $X\buildrel d\over{=}\theta$ and
$c=2$. We give the exact values of $\varphi(0),\,\varphi(1),\,\varphi(2)$ and
$\varphi(3)$ and provide an algorithm for arbitrary $\varphi(u)$
$u\in\mathbb{N}_{0}$ calculation.

Let us observe the validity of the net profit condition
$\mathbb{E}(2\theta-X)=1/2>0$. Then

$G_{X-2\theta}(s)=\left(\frac{1}{2}+\frac{s}{2}\right)\left(\frac{1}{2}+\frac{1}{2s^{2}}\right)=1\quad\Rightarrow\quad
s=1-\sqrt{2}=:\alpha$

and, according to Theorem 3,

$\displaystyle\varphi(0)=\frac{\sqrt{2}}{4}\approx 0.354,$
$\displaystyle\varphi(1)=2-\sqrt{2}\approx 0.586,$
$\displaystyle\varphi(2)=2(\sqrt{2}-1)\approx 0.828,$
$\displaystyle\varphi(3)=8-5\sqrt{2}\approx 0.929.$

We can proceed using the recurrence (23)

$\varphi(u)=4\left(\varphi(u-2)-\sum_{i=1}^{u-1}\varphi(i)f(u-2-i)\right),\,u=2,\,3,\,\ldots,$

or employ the survival probability-generating function

$\Xi(s)=\frac{2-\sqrt{2}+\sqrt{2}s}{1+s-3s^{2}+s^{3}},\,s\neq\alpha,$

see Theorem 2, i.e.

$\varphi(u+1)=\frac{1}{u!}\lim_{s\to
0}\frac{d^{u}}{ds^{u}}\Xi(s),\,u\in\mathbb{N}_{0}.$

######  Example 2.

Let $p=1/2$,

$\mathbb{P}(X=k)=(1-p)^{k}p,\,k=0,\,1,\,\ldots$

and

$\mathbb{P}(c\theta=k)=\binom{4}{k}p^{k}(1-p)^{4-k},\,k=0,\,1,\,\ldots,\,4,$

i.e. we assume the claim amount $X$ is geometric and the inter-occurrence time
$\theta$ multiplied by premium $c$ is binomial distributed with the provided
parameters. We give the exact values of
$\varphi(0),\,\varphi(1),\,\varphi(2)$, $\varphi(3)$, $\varphi(4)$ and provide
an algorithm for arbitrary $\varphi(u)$ $u\in\mathbb{N}_{0}$ calculation.

Let us observe that the net profit condition is valid
$\mathbb{E}(c\theta-X)=1>0$. Then, the equation

$\displaystyle
G_{X-c\theta}(s)=\frac{p}{1-(1-p)s}\left(1-p+\frac{p}{s}\right)^{4}=1,\,p=\frac{1}{2},$

has three roots inside the unit circle:

$\displaystyle\alpha_{1}$ $\displaystyle=-0.15434+0.342115i,$
$\displaystyle\alpha_{2}$ $\displaystyle=-0.15434-0.342115i,$
$\displaystyle\alpha_{3}$ $\displaystyle=-0.289014.$

Note that the complex roots always appear in conjugate pairs, see [12, Rem.
10].

According to Theorem 3,

$\displaystyle\varphi(0)$
$\displaystyle=-\prod_{j=1}^{3}\frac{1}{\alpha_{j}-1}=0.535194,\,\varphi(1)=32\prod_{j=1}^{3}\frac{\alpha_{j}}{\alpha_{j}-1}=0.697233,$
$\displaystyle\varphi(2)$
$\displaystyle=32\left(-\frac{11}{64}\varphi(1)+\frac{\alpha_{1}\alpha_{2}\alpha_{3}-\alpha_{1}\alpha_{2}-\alpha_{1}\alpha_{3}-\alpha_{2}\alpha_{3}}{(\alpha_{1}-1)(\alpha_{2}-1)(\alpha_{3}-1)}\right)=0.802783,$
$\displaystyle\varphi(3)$
$\displaystyle=32\left(-\frac{11}{64}\varphi(2)-\frac{55}{128}\varphi(1)+\frac{\alpha_{1}\alpha_{2}\alpha_{3}-\alpha_{1}\alpha_{2}-\alpha_{1}\alpha_{3}-\alpha_{2}\alpha_{3}+\alpha_{1}+\alpha_{2}+\alpha_{3}}{(\alpha_{1}-1)(\alpha_{2}-1)(\alpha_{3}-1)}\right)$
$\displaystyle=0.871536$

We can proceed using the recurrence (23)

$\varphi(u)=32\left(\varphi(u-4)-\sum_{i=1}^{u-1}\varphi(i)f(u-4-i)\right),\,u=4,\,5,\,\ldots,$

which gives

$\varphi(4)=32\left(\varphi(0)-\varphi(1)\cdot\frac{65}{256}-\varphi(2)\cdot\frac{33}{128}-\varphi(3)\cdot\frac{9}{64}\right)=0.916321,$

or employ the survival probability-generating function (see Theorem 2)

$\Xi(s)=\frac{0.0435771+0.224482s+0.516629s^{2}+0.750506s^{3}-0.535194s^{4}}{0.0625+0.25s+0.375s^{2}+0.25s^{3}-1.9375s^{4}+s^{5}},\,s\neq\alpha_{1},\,\alpha_{2},\,\alpha_{3},$

where $\pi_{0},\,\pi_{1},\,\pi_{2},\,\pi_{3}$ can be obtained from
$\varphi(0),\,\varphi(1),\,\varphi(2)$, $\varphi(3)$, $\varphi(4)$ or directly
by Theorem 3.

######  Example 3.

Let $\mathbb{P}(c\theta=1)=p=1-\mathbb{P}(c\theta=3)$ and

$\mathbb{P}(X=0)=\frac{-1+p+\sqrt{1-p}}{2p}=1-\mathbb{P}(X=1),\,0<p\leqslant
1.$

We find $\varphi(u),\,u\in\mathbb{N}_{0}$ and the survival probability
generating function $\Xi(s)$, $|s|<1$.

It is easy to see that the net profit condition holds for all $0<p<1$. The
equation

$\left(\frac{-1+p+\sqrt{1-p}}{2p}+\frac{1+p-\sqrt{1+p}}{2p}s\right)\left(\frac{p}{s}+\frac{1-p}{s^{3}}\right)=1$

has a double root

$s=-\frac{1-p}{1-p+\sqrt{1-p}}:=\alpha,\,0<p<1.$

inside the unit circle. We now setup the modified system (22) by replacing its
second line with derivative

$\displaystyle\begin{pmatrix}F(-3)+\alpha
F(-2)+\alpha^{2}F(-1)&F(-3)\alpha+F(-2)\alpha^{2}&\alpha^{2}f(-3)\\\
F(-2)+2\alpha F(-1)&F(-3)+2\alpha F(-2)&2\alpha f(-3)\\\
f(-1)+2f(-2)+3f(-3)&f(-2)+2f(-3)&f(-3)\end{pmatrix}\begin{pmatrix}\pi_{0}\\\
\pi_{1}\\\ \pi_{2}\end{pmatrix}=\begin{pmatrix}&0\\\ &0\\\
&\mathbb{E}(c\theta-X)\end{pmatrix},$

where the involved probability mass function and expectation are described by
the assumed distribution of $X-c\theta$. The provided system has a solution
$(\pi_{0},\,\pi_{1},\,\pi_{2})=(1,\,0,\,0)$, which implies $\varphi(u)=1$ for
all $u\in\mathbb{N}$ and $\Xi(s)=1/(1-s)$, $|s|<1$, while the recurrence (23)
gives

$\varphi(0)=\sum_{i=1}^{3}\varphi(i)f(-i)=f(-1)+f(-2)+f(-3)=1-f(0)=\frac{1-p+\sqrt{1-p}}{2}$

and that is natural because under the assumed distributions the ruin can
appear when $X=1$, $c\theta=1$ and $u=0$ only.

######  Example 4.

Let $X\sim\mathcal{P}(1)$ and $c\theta\sim{P}(1.01)$ be Poisson distributed
with parameters $\lambda=1$ and $\lambda=1.01$, i.e.

$\mathbb{P}(X=k)=\frac{e^{-1}}{k!},\,\mathbb{P}(c\theta=k)=e^{-1.01}\frac{1.01^{k}}{k!},\,k=0,\,1,\,\ldots$

We truncate $c\theta$ to have the finite support and calculate the exact
survival probabilities $\varphi(0),\,\varphi(1),\,\ldots,\,\varphi(10)$.

As the proposed algorithm for the ultimate time survival probability
calculation does not allow the inter-occurrence time of infinite support, for
the illustrative purposes we define two new random variables $c\theta_{10}$
and $c\theta_{15}$:

$\displaystyle\mathbb{P}(c\theta_{10}=k)=\mathbb{P}(c\theta=k),\,k=0,\,1,\,\ldots,\,9,\,\mathbb{P}(c\theta_{10}=10)=\mathbb{P}(c\theta\geqslant
10),$ (27)
$\displaystyle\mathbb{P}(c\theta_{15}=k)=\mathbb{P}(c\theta=k),\,k=0,\,1,\,\ldots,\,14,\,\mathbb{P}(c\theta_{15}=15)=\mathbb{P}(c\theta\geqslant
15).$ (28)

It is easy to observe $\mathbb{E}(X-c\theta)=-0.01<0$. The net profit
condition remains valid for the truncated distributions $c\theta_{10}$ and
$c\theta_{15}$ too:

$\displaystyle\mathbb{E}(X-c\theta_{10})$ $\displaystyle=-0.009999988,$
$\displaystyle\mathbb{E}(X-c\theta_{15})$ $\displaystyle=-0.00999999999998.$

Then, the equation

$\displaystyle
e^{s-1}\left(\sum_{k=0}^{9}\frac{\mathbb{P}(c\theta=k)}{s^{k}}+\frac{\mathbb{P}(c\theta\geqslant
10)}{s^{10}}\right)=1$ (29)

has nine roots inside the unit circle, the red points in Figure 1 below, while

$\displaystyle
e^{s-1}\left(\sum_{k=0}^{14}\frac{\mathbb{P}(c\theta=k)}{s^{k}}+\frac{\mathbb{P}(c\theta\geqslant
15)}{s^{15}}\right)=1$ (30)

has fourteen roots, the blue points in Figure 1 below.

Figure 1: Roots of (29) (red) and (30) (blue) inside the unit circle.

Let us denote $\varphi_{10}(u)$ and $\varphi_{15}(u)$ the survival
probabilities when the process (1) is generated by $\\{X,\,c\theta_{10}\\}$
and $\\{X,\,c\theta_{15}\\}$ respectively. We then multiply the roots depicted
in Figure 1 as provided in Theorem 3 and obtain the survival probabilities
$\varphi_{10}(u)$ and $\varphi_{15}(u)$ when $u=0,\,1,\,\ldots,\,10$, see
Table 1 below.

$u$ | $\varphi_{10}(u)$ | $\varphi_{15}(u)$ | $\varphi_{15}(u)-\varphi_{10}(u)$
---|---|---|---
$0$ | $0.0067795743$ | $0.0067795818$ | $7.5E-09$
$1$ | $0.0145425921$ | $0.0145456080$ | $1.6E-08$
$2$ | $0.0238700927$ | $0.0238701187$ | $2.6E-08$
$3$ | $0.0334952018$ | $0.0334952381$ | $3.6E-08$
$4$ | $0.0430669381$ | $0.0430669845$ | $4.6E-08$
$5$ | $0.0525424876$ | $0.0525425439$ | $5.6E-08$
$6$ | $0.0619232839$ | $0.0619233499$ | $6.6E-08$
$7$ | $0.0712111444$ | $0.0712112199$ | $7.6E-08$
$8$ | $0.0804070612$ | $0.0804071458$ | $8.5E-08$
$9$ | $0.0895119320$ | $0.0895120511$ | $1.2E-07$
$10$ | $0.0985266555$ | $0.0985268429$ | $1.9E-07$
Table 1: The survival probabilities of $\varphi_{10}(u)$ and
$\varphi_{15}(u)$.

The obtained values of $\varphi_{10}(u)$ and $\varphi_{15}(u)$ in Table 1
indicate the negligible effect truncating the distribution of $c\theta$ up to
$10$ in comparison to truncation up to $15$, see (27) and (28) and recall the
estimates (8).

##  6 Acknowledgements

The author is grateful to Professor Jonas Šiaulys for carefully reading the
first draft of the manuscript and raising many inaccuracies.

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