1 Introduction
A recent research paper [1] studied the possibility for a random walk (r.w.) ${\textstyle\sum _{i=1}^{t}}{Z_{i}}$ to hit the line $u+2t$, $u\in {\mathbb{N}_{0}}:=\mathbb{N}\cup \{0\}$, $t\in \mathbb{N}$, at least once in time, when r.w. consists of two interchangeably occurring discrete and nonnegative integer-valued random variables, i.e. ${Z_{2i-1}}\stackrel{d}{=}X$ and ${Z_{2i}}\stackrel{d}{=}Y$, $i\in \mathbb{N}$. Here X and Y are independent but not necessarily identically distributed. The described model is called the bi-seasonal discrete time risk model with income rate two.
In this article we define a slightly different model
where:
The present model (1) is called the bi-risk discrete time risk model with income rate two. Its deterministic part $u+2t$ consists of two components: u is deemed as initial wealth or savings in some financial context, and premium rate or income per unit of time, which is just a multiplier of t. The subtracted stochastic part ${\textstyle\sum _{i=1}^{t}}{X_{i}}+{\textstyle\sum _{j=1}^{\lfloor t/2\rfloor }}{Y_{j}}$ is treated as random expenses. In particular, the setup of the random part in (1) is considered in such a way that r.v. X is present at every moment of time, while Y additionally occurs at even moments of time. Of course, there are many other different setups of such type of models as (1). One of the most general models was introduced in [2] and is known as Sparre Andersen collective risk model. Equally, Refs. [9–11, 15, 20] are known as classical works on the subject. The research variety is mainly due to that every model assumption has its impact on the possibility that stochastic part never exceeds deterministic, i.e. savings and earnings are sufficient to cover occurring expenses. There are two objects we deal with investigating the model (1):
(1)
\[ W(t)=u+2t-{\sum \limits_{i=1}^{t}}{X_{i}}-{\sum \limits_{j=1}^{\lfloor t/2\rfloor }}{Y_{j}},\]The probability (2) is called the finite time survival probability while the later one (3) is the ultimate time survival probability and they both deal with the possibility that $W(t)>0$ for all $t\in \{1,\hspace{0.1667em}\dots ,\hspace{0.1667em}T\}$, when T is finite or $T\to \infty $. The ultimate time survival probability (3) heavily depends on the net profit condition, which for the model (1), is defined as
The ultimate time survival probability also depends directly on the minimal value of the sum ${X_{1}}+{X_{2}}+Y$ – see Theorems 2–6.
As mentioned previously, this article is based on the research done in [1] where the bi-seasonal model is studied. The relationship of the ultimate time survival probability expressions between bi-seasonal and bi-risk models is given in Table 1.
Table 1.
Map between bi-seasonal and bi-risk models
| Case | $\min X$ | $\min Y$ | $\min (X+Y)$ | $\min (2X+Y)$ |
| 1 | 0 | 0 | 0 | 0 |
| 2 | 0 | 1 | 1 | 1 |
| 3 | 1 | 0 | 1 | 2 |
| 4 | 1 | 1 | 2 | 3 |
| 5 | 2 | 0 | 2 | 4̸ |
| 6 | 0 | 2 | 2 | 2 |
| 7 | 0 | 3 | 3 | 3 |
| 8 | 1 | 2 | 3 | 4̸ |
| 9 | 2 | 1 | 3 | 5̸ |
| 10 | 3 | 0 | 3 | 6̸ |
Table 1 should be read as follows. The net profit condition for the bi-seasonal risk model with income rate two is $\mathbb{E}X+\mathbb{E}Y<4$. Therefore, the third column $\min (X+Y)$ indicates where the distribution of $X+Y$ may start not violating the net profit condition. That, of course, depends on which minimal value can be attained by X and Y as depicted in the second and the third columns. Turning to the bi-risk model with income rate two, a similar question arises: which minimal value can be attained by ${X_{1}}+{X_{2}}+Y$ so that the net profit condition remain valid? The answer is present in the last column $\min (2X+Y)$, and it shows that four cases are not valid for the ultimate time as they never satisfy $2\mathbb{E}X+\mathbb{E}Y<4$. Moreover, comparing bi-risk model with bi-seasonal one, some of the cases get rearranged due to $\min (2X+Y)\ne \min (X+Y)$ – see cases number 3 and 4 in Table 1.
For more convenient expressions of $\varphi (u,T)$ and $\varphi (u)$, we introduce the following notations. For $u\in {\mathbb{N}_{0}}$, we denote the probability mass functions (PMFs)
\[\begin{aligned}{}& {x_{u}}:=\mathbb{P}(X=u),\hspace{0.1667em}{y_{u}}:=\mathbb{P}(Y=u),\\ {} & {s_{u}}:=\mathbb{P}(X+Y=u),\hspace{0.1667em}{a_{u}}:=\mathbb{P}({X_{1}}+{X_{2}}+Y=u),\end{aligned}\]
the cumulative distribution functions (CDFs)
and tails
The following equality, implied by [1, eq. (2)],
shows where the problem of finding $\varphi (u)$ for all $u\in {\mathbb{N}_{0}}$ stems from. In order to use the recurrence relation (5), we need to know several initial values. How many of them exactly? This depends on the smallest value of ${X_{1}}+{X_{2}}+Y$. For example, if ${a_{0}}>0$ and $u=0$, the formula (5) implies the relation among $\varphi (0)$, …, $\varphi (4)$, if $u=0$ and ${a_{0}}=0$, ${a_{1}}>0$, then we have the relation among $\varphi (0)$, …, $\varphi (3)$, etc. With that said, the next Section 2 is structured as follows: Theorem 1 deals with the finite time survival probability, Theorems 2–5 express the ultimate time survival probability under the net profit condition and the remaining Theorem 6 provides the values of $\varphi (u)$ under the breach of the net profit condition. In Section 3 we present the outputs of these mentioned theorems with some chosen random variables X and Y, while in Section 4 we give a summarized overview and a brief look at more generalized discrete time risk models.
(5)
\[ \varphi (u)={\sum \limits_{k=1}^{u+4}}\varphi (k){a_{u+4-k}}-({x_{u+3}}{s_{0}}+{x_{u+2}}{s_{1}})\varphi (1)-{x_{u+2}}{s_{0}}\varphi (2),\]It is worth mentioning that research related to the present one can also be found in papers [7] and [12]. In general, the research done in this paper might be considered as a study of a certain randomness in attaining some large values. Multiple research papers are written each year on the subject, just few of them are [4–6, 8, 13, 14] and [17]. Also, due to some model’s specifics, only particular distributions of a random part are considered, see [19].
2 Statements and proofs
We start with the statement for finite time survival probability $\varphi (u,T)$ for the bi-risk discrete time risk model with premium two. The model is defined in (1).
Theorem 1.
For any $u\in {\mathbb{N}_{0}}$, the finite time survival probability of the bi-risk discrete time risk model with income rate two, satisfies
\[\begin{aligned}{}\varphi (u,1)& ={F_{X}}(u+1),\\ {} \varphi (u,2)& ={\sum \limits_{k=0}^{u+1}}{x_{k}}{F_{S}}(u+3-k),\\ {} \varphi (u,T)& ={\sum \limits_{k=0}^{u+3}}\varphi (u+4-k,T-2){a_{k}}\\ {} & \hspace{1em}-({x_{u+2}}{s_{1}}+{x_{u+3}}{s_{0}})\varphi (1,T-2)-{x_{u+2}}{s_{0}}\varphi (2,T-2),\hspace{0.1667em}T\geqslant 3.\end{aligned}\]
Proof.
The proof follows by replacing $Y\mapsto X+Y$ and $X+Y\mapsto {X_{1}}+{X_{2}}+Y$ in Theorem 2.1 in [1]. □
Let us observe that Theorem 1 is independent of the net profit condition. We note that [3, Thm. 1] may be adopted for the considered finite time survival probability calculation as well.
Lets turn to the ultimate time. To express the ultimate time survival probability for model (1), when the sum ${X_{1}}+{X_{2}}+Y$ can attain zero with some positive probability, we define four recurrent sequences ${\alpha _{n}}$, ${\beta _{n}}$, ${\gamma _{n}}$ and ${\delta _{n}}$. For $n=0,\hspace{0.1667em}1,\hspace{0.1667em}2,\hspace{0.1667em}3$ we define
and, for $n=4,\hspace{0.1667em}5,\hspace{0.1667em}\dots $,
| n | ${\alpha _{n}}$ | ${\beta _{n}}$ | ${\gamma _{n}}$ | ${\delta _{n}}$ |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 1 | 0 |
| 3 | $-\frac{1}{{a_{0}}}$ | $-\frac{{F_{A}}(2)+{\overline{F}_{X}}(2){s_{0}}+{\overline{F}_{X}}(1){s_{1}}}{{a_{0}}}$ | $-\frac{{F_{A}}(1)+{\overline{F}_{X}}(1){s_{0}}}{{a_{0}}}$ | $\frac{1}{{a_{0}}}$ |
\[\begin{aligned}{}& {\alpha _{n}}=\frac{1}{{a_{0}}}\left({\alpha _{n-4}}-{\sum \limits_{k=1}^{n-1}}{a_{n-k}}{\alpha _{k}}\right),\\ {} & {\beta _{n}}=\frac{1}{{a_{0}}}\left({\beta _{n-4}}-{\sum \limits_{k=1}^{n-1}}{a_{n-k}}{\beta _{k}}+{x_{n-1}}{s_{0}}+{x_{n-2}}{s_{1}}\right),\\ {} & {\gamma _{n}}=\frac{1}{{a_{0}}}\left({\gamma _{n-4}}-{\sum \limits_{k=1}^{n-1}}{a_{n-k}}{\gamma _{k}}+{x_{n-2}}{s_{0}}\right),\hspace{0.1667em}{\delta _{n}}=\frac{1}{{a_{0}}}\left({\delta _{n-4}}-{\sum \limits_{k=1}^{n-1}}{a_{n-k}}{\delta _{k}}\right).\end{aligned}\]
Theorem 2.
Suppose that ${a_{0}}>0$ and $2\mathbb{E}X+\mathbb{E}Y<4$. Then, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two satisfies
(6)
\[\begin{aligned}{}& \left(\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c}{\alpha _{n+1}}-{\alpha _{n}}& {\beta _{n+1}}-{\beta _{n}}& {\gamma _{n+1}}-{\gamma _{n}}\\ {} {\alpha _{n+2}}-{\alpha _{n}}& {\beta _{n+2}}-{\beta _{n}}& {\gamma _{n+2}}-{\gamma _{n}}\\ {} {\alpha _{n+3}}-{\alpha _{n}}& {\beta _{n+3}}-{\beta _{n}}& {\gamma _{n+3}}-{\gamma _{n}}\end{array}\right)\times \left(\begin{array}{c}\varphi (0)\\ {} \varphi (1)\\ {} \varphi (2)\end{array}\right)\\ {} & +\left(\begin{array}{c}{\delta _{n+1}}-{\delta _{n}}\\ {} {\delta _{n+2}}-{\delta _{n}}\\ {} {\delta _{n+3}}-{\delta _{n}}\end{array}\right)\times (4-2\mathbb{E}X-\mathbb{E}Y)=\left(\begin{array}{c}\varphi (n+1)-\varphi (n)\\ {} \varphi (n+2)-\varphi (n)\\ {} \varphi (n+3)-\varphi (n)\end{array}\right),\hspace{0.1667em}n\in {\mathbb{N}_{0}},\end{aligned}\]
\[\begin{aligned}{}\varphi (3)& =\frac{-\varphi (0)-({F_{A}}(2)+{\overline{F}_{X}}(2){s_{0}}+{\overline{F}_{X}}(1){s_{1}})\varphi (1)}{{a_{0}}}\\ {} & \hspace{1em}-\frac{({F_{A}}(1)+{\overline{F}_{X}}(1){s_{0}})\varphi (2)-4+2\mathbb{E}X+\mathbb{E}Y}{{a_{0}}},\\ {} \varphi (u)& =\frac{1}{{a_{0}}}\left(\varphi (u-4)+({x_{u-1}}{s_{0}}+{x_{u-2}}{s_{1}})\varphi (1)+{x_{u-2}}{s_{0}}\varphi (2)-{\sum \limits_{k=1}^{u-1}}{a_{u-k}}\varphi (k)\right),\\ {} & \hspace{1em}u=4,\hspace{0.1667em}5,\hspace{0.1667em}\dots .\end{aligned}\]
Proof.
The proof is implied by Theorem 2.2 in [1] replacing $Y\mapsto X+Y$ and $X+Y\mapsto {X_{1}}+{X_{2}}+Y$ there. □
We now turn to the case when the lowest possible value of ${X_{1}}+{X_{2}}+Y$ (with positive probability) is one. Note that there is just one underlying case satisfying the mentioned condition – case number two in Table 1. Let’s define three recurrent sequences ${\overline{\alpha }_{n}}$, ${\overline{\beta }_{n}}$ and ${\overline{\delta }_{n}}$. For $n=0,\hspace{0.1667em}1,\hspace{0.1667em}2$,
and, for $n=3,\hspace{0.1667em}4,\hspace{0.1667em}\dots $,
| n | ${\overline{\alpha }_{n}}$ | ${\overline{\beta }_{n}}$ | ${\overline{\delta }_{n}}$ |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 2 | $-\frac{1}{{a_{1}}}$ | $-\frac{{F_{A}}(2)+{\overline{F}_{X}}(1){s_{1}}}{{a_{1}}}$ | $\frac{1}{{a_{1}}}$ |
\[\begin{aligned}{}& {\overline{\alpha }_{n}}=\frac{1}{{a_{1}}}\left({\overline{\alpha }_{n-3}}-{\sum \limits_{k=1}^{n-1}}{a_{n+1-k}}{\overline{\alpha }_{k}}\right),\\ {} & {\overline{\beta }_{n}}=\frac{1}{{a_{1}}}\left({\overline{\beta }_{n-3}}-{\sum \limits_{k=1}^{n-1}}{a_{n+1-k}}{\overline{\beta }_{k}}+{x_{n-1}}{s_{1}}\right),\\ {} & {\overline{\delta }_{n}}=\frac{1}{{a_{1}}}\left({\overline{\delta }_{n-3}}-{\sum \limits_{k=1}^{n-1}}{a_{n+1-k}}{\overline{\delta }_{k}}\right).\end{aligned}\]
Theorem 3.
Suppose that ${a_{0}}=0$, ${a_{1}}>0$ and $2\mathbb{E}X+\mathbb{E}Y<4$. Then, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two satisfies
(7)
\[\begin{aligned}{}& \left(\begin{array}{c@{\hskip10.0pt}c}{\overline{\alpha }_{n+1}}-{\overline{\alpha }_{n}}& {\overline{\beta }_{n+1}}-{\overline{\beta }_{n}}\\ {} {\overline{\alpha }_{n+2}}-{\overline{\alpha }_{n}}& {\overline{\beta }_{n+2}}-{\overline{\beta }_{n}}\end{array}\right)\times \left(\begin{array}{c}\varphi (0)\\ {} \varphi (1)\end{array}\right)+\left(\begin{array}{c}{\overline{\delta }_{n+1}}-{\overline{\delta }_{n}}\\ {} {\overline{\delta }_{n+2}}-{\overline{\delta }_{n}}\end{array}\right)\times (4-\mathbb{E}A)\\ {} & \hspace{1em}=\left(\begin{array}{c}\varphi (n+1)-\varphi (n)\\ {} \varphi (n+2)-\varphi (n)\end{array}\right),\hspace{0.1667em}n\in {\mathbb{N}_{0}},\end{aligned}\]
\[\begin{aligned}{}& \varphi (2)=\frac{-\varphi (0)-({F_{A}}(2)+{\overline{F}_{X}}(1){s_{1}})\varphi (1)+4-2\mathbb{E}X-\mathbb{E}Y}{{a_{1}}},\\ {} & \varphi (u)=\frac{1}{{a_{1}}}\left(\varphi (u-3)+{x_{u-1}}{s_{1}}\varphi (1)-{\sum \limits_{k=1}^{u-1}}{a_{u+1-k}}\varphi (k)\right),\hspace{0.1667em}u=3,4,\hspace{0.1667em}\dots \end{aligned}\]
Proof.
The proof follows from Theorem 2.3 in [1] by replacing $Y\mapsto X+Y$, $X+Y\mapsto {X_{1}}+{X_{2}}+Y$ there and observing that ${s_{0}}=0$ because of ${y_{0}}=0$ under the current assumptions. □
We now ask when does the sum ${X_{1}}+{X_{2}}+Y$ attains its minimum value of two with some positive probability? That happens in cases three and six in Table 1. The next theorem requires two recurrent sequences to be defined:
\[\begin{aligned}{}& {\hat{\alpha }_{0}}=1,\hspace{0.1667em}{\hat{\alpha }_{1}}=-\frac{1}{{\overline{F}_{X}}(1){s_{1}}+{a_{2}}},\\ {} & {\hat{\alpha }_{n}}=\frac{1}{{a_{2}}}\left({\hat{\alpha }_{n-2}}-{\sum \limits_{k=1}^{n-1}}{a_{n+2-k}}{\hat{\alpha }_{k}}+{x_{n}}{s_{1}}{\hat{\alpha }_{1}}\right),\hspace{0.1667em}n=2,\hspace{0.1667em}3,\hspace{0.1667em}\dots ,\\ {} & {\hat{\delta }_{0}}=0,\hspace{0.1667em}{\hat{\delta }_{1}}=\frac{1}{{\overline{F}_{X}}(1){s_{1}}+{a_{2}}},\\ {} & {\hat{\delta }_{n}}=\frac{1}{{a_{2}}}\left({\hat{\delta }_{n-2}}-{\sum \limits_{k=1}^{n-1}}{a_{n+2-k}}{\hat{\delta }_{k}}+{x_{n}}{s_{1}}{\hat{\delta }_{1}}\right),\hspace{0.1667em}n=2,\hspace{0.1667em}3,\hspace{0.1667em}\dots .\end{aligned}\]
Theorem 4.
Suppose that ${a_{0}}={a_{1}}=0$, ${a_{2}}>0$ and $2\mathbb{E}X+\mathbb{E}Y<4$. Then, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two satisfies
\[\begin{aligned}{}& ({\hat{\alpha }_{n+1}}-{\hat{\alpha }_{n}})\varphi (0)+({\hat{\delta }_{n+1}}-{\hat{\delta }_{n}})(4-2\mathbb{E}X-\mathbb{E}Y)=\varphi (n+1)-\varphi (n),\hspace{0.1667em}n\in {\mathbb{N}_{0}},\\ {} & \varphi (1)={\hat{\alpha }_{1}}\varphi (0)+{\hat{\delta }_{1}}(4-2\mathbb{E}X-\mathbb{E}Y),\\ {} & \varphi (u)=\frac{1}{{a_{2}}}\left(\varphi (u-2)-{\sum \limits_{k=1}^{u-1}}{a_{u+2-k}}\varphi (k)+{x_{u}}{s_{1}}\varphi (1)\right),\hspace{0.1667em}u=2,\hspace{0.1667em}3,\hspace{0.1667em}\dots \end{aligned}\]
Moreover, ${\hat{\alpha }_{n+1}}-{\hat{\alpha }_{n}}\ne 0$ for all $n\in {\mathbb{N}_{0}}$.
Proof.
The proof is implied by Theorem 2.4 in [1] replacing $Y\mapsto X+Y$, $X+Y\mapsto {X_{1}}+{X_{2}}+Y$ there and observing that ${s_{1}}=0$ under the current assumptions. □
The last case, when the net profit condition can be satisfied, is when the sum ${X_{1}}+{X_{2}}+Y$ can attain its minimum value of three with some positive probability. This is illustrated in cases number four and seven in Table 1 and consequently, the expression of the ultimate time survival probability is straightforward.
Theorem 5.
Suppose that ${a_{0}}={a_{1}}={a_{2}}=0$, ${a_{3}}>0$ and $2\mathbb{E}X+\mathbb{E}Y<4$. Then, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two satisfies
\[\begin{aligned}{}& \varphi (0)=4-2\mathbb{E}X-\mathbb{E}Y,\hspace{0.1667em}\varphi (1)=\varphi (0)/{a_{3}},\\ {} & \varphi (u)=\frac{1}{{a_{3}}}\left(\varphi (u-1)-{\sum \limits_{k=1}^{u-1}}{a_{u+3-k}}\varphi (k)\right),\hspace{0.1667em}u=2,\hspace{0.1667em}3,\hspace{0.1667em}\dots .\end{aligned}\]
Proof.
The proof follows from Theorem 2.5 in [1] by replacing $Y\mapsto X+Y$, $X+Y\mapsto {X_{1}}+{X_{2}}+Y$ there and observing that ${s_{0}}={s_{1}}=0$ under the current assumptions. □
The next theorem shows that survival is impossible, in all but a few trivial cases, if the net profit condition is violated.
Theorem 6.
Suppose that $2\mathbb{E}X+\mathbb{E}Y\geqslant 4$. Then, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two is as follows:
-
(i) $\varphi (u)=0$ for all $u\in {\mathbb{N}_{0}}$, when $2\mathbb{E}X+\mathbb{E}Y>4$,
-
(ii) $\varphi (u)=0$ for all $u\in {\mathbb{N}_{0}}$, when $2\mathbb{E}X+\mathbb{E}Y=4$ and ${a_{4}}<1$,
-
(iii) $\varphi (0)=0$ and $\varphi (u)=1$ for all $u\in \mathbb{N}$, when $2\mathbb{E}X+\mathbb{E}Y=4$ and ${a_{4}}=1$.
Proof.
For the model (1), the equality (20) in [1] implies
where
Due to (8), the condition $2\mathbb{E}X+\mathbb{E}Y>4$ implies $\varphi (\infty )=0$ and consequently $\varphi (u)=0$ for all $u\in {\mathbb{N}_{0}}$ because of $\varphi (u)\leqslant \varphi (u+1)$. Therefore (i) is correct.
(8)
\[\begin{aligned}{}& \varphi (0)+({x_{0}}{s_{2}}+{s_{1}}+{s_{0}})\varphi (1)+({x_{0}}{s_{1}}+{s_{0}})\varphi (2)+{x_{0}}{s_{0}}\varphi (3)\\ {} & =(4-2\mathbb{E}X-\mathbb{E}Y)\cdot \varphi (\infty ),\end{aligned}\]If $2\mathbb{E}X+\mathbb{E}Y=4$ and ${a_{4}}<1$, then, by (8),
In view of (9), we have the following cases:
(9)
\[ \varphi (0)+({x_{0}}{s_{2}}+{s_{1}}+{s_{0}})\varphi (1)+({x_{0}}{s_{1}}+{s_{0}})\varphi (2)+{x_{0}}{s_{0}}\varphi (3)=0.\]
\[\begin{aligned}{}& {a_{0}}>0\hspace{1em}\Rightarrow \hspace{1em}\varphi (0)=\varphi (1)=\varphi (2)=\varphi (3)=0,\\ {} & {a_{0}}=0,\hspace{0.1667em}{a_{1}}>0\hspace{1em}\Rightarrow \hspace{1em}\varphi (0)=\varphi (1)=\varphi (2)=0,\\ {} & {a_{0}}={a_{1}}=0,\hspace{0.1667em}{a_{2}}>0\hspace{1em}\Rightarrow \hspace{1em}\varphi (0)=\varphi (1)=0,\\ {} & {a_{0}}={a_{1}}={a_{2}}=0,\hspace{0.1667em}{a_{3}}>0\hspace{1em}\Rightarrow \hspace{1em}\varphi (0)=0,\end{aligned}\]
while $\varphi (u)=0$ for all $u\in {\mathbb{N}_{0}}$ is implied by (5). Thus, (ii) is correct.Suppose that $2\mathbb{E}X+\mathbb{E}Y=4$ and ${a_{4}}=1$. Then, there are just three degenerated distributions:
\[ X\equiv 0,\hspace{0.1667em}Y\equiv 4\hspace{2.5pt}\text{or}\hspace{2.5pt}X\equiv 2,\hspace{0.1667em}Y\equiv 0\hspace{2.5pt}\text{or}\hspace{2.5pt}X\equiv 1,\hspace{0.1667em}Y\equiv 2,\]
all of which, by model definition (1), imply $\varphi (0)=0$ and $\varphi (u)=1$ for all $u\in \mathbb{N}$. □3 Numerical examples
In this section we verify the formulated Theorems 1–6 with some chosen r.vs. That is performed by choosing particular distributions of X and Y and obtaining $\varphi (u,T)$ and $\varphi (u)$ for some $u\in {\mathbb{N}_{0}}$ and $T\in \mathbb{N}$. Initial wealth u and time T are chosen individually aiming to reflect the dynamics of survival probabilities. The presented survival probabilities are calculated with Python [16] and confirmed with Wolfram Mathematica [18]. Results are rounded up to three decimal places except when the rounding result is 0 or 1. As the initial values of the ultimate time survival probability in Theorems 2–4 depend on $n\in \mathbb{N}$, there $n=100$ is chosen as large enough when applying them. Ideally, we should set $n\to \infty $ and, for example, Theorem 4 would imply
\[ \varphi (0)=(4-2\mathbb{E}X-\mathbb{E}Y)\underset{n\to \infty }{\lim }\frac{{\hat{\delta }_{n}}-{\hat{\delta }_{n+1}}}{{\hat{\alpha }_{n+1}}-{\hat{\alpha }_{n}}}.\]
However, it is not easy to find such limit, especially the corresponding one in Theorems 2 and 3. Therefore, we consider the chosen n as large enough when a relative change of the obtained initial value (or values) is negligible comparing results between n and $n+1$, i.e. $|{L_{n+1}}/{L_{n}}|<\varepsilon $, where ${L_{n}}=({\hat{\delta }_{n}}-{\hat{\delta }_{n+1}})/({\hat{\alpha }_{n+1}}-{\hat{\alpha }_{n}})$.The distribution
is called geometric with parameter $0<p<1$ (denoted $X\sim \mathcal{G}(p)$), while the generalized one
\[ \mathbb{P}(X=k)=\left(\genfrac{}{}{0.0pt}{}{k-1}{r-1}\right){(1-p)^{k-r}}{p^{r}},\hspace{0.1667em}k=r,\hspace{0.1667em}r+1,\hspace{0.1667em}\dots ,\]
is known as negative binomial or Pascal with parameters $0<p<1$ and $r\in \mathbb{N}$ (denoted $X\sim \mathcal{NB}(r,p)$).Table 2.
Survival probability when $X\sim \mathcal{G}(3/4)$ and $Y\sim \mathcal{G}(1/4)$
| T | $u=0$ | $u=1$ | $u=2$ | $u=3$ | $u=4$ | $u=5$ | $u=10$ | $u=25$ | $u=50$ |
| 1 | 0.938 | 0.984 | 0.996 | 0.999 | 1 | 1 | 1 | 1 | 1 |
| 2 | 0.582 | 0.695 | 0.774 | 0.831 | 0.873 | 0.905 | 0.977 | 1 | 1 |
| 3 | 0.580 | 0.693 | 0.772 | 0.830 | 0.872 | 0.904 | 0.977 | 1 | 1 |
| 4 | 0.465 | 0.576 | 0.661 | 0.729 | 0.784 | 0.829 | 0.948 | 0.999 | 1 |
| 5 | 0.465 | 0.575 | 0.660 | 0.728 | 0.784 | 0.828 | 0.948 | 0.999 | 1 |
| 10 | 0.338 | 0.430 | 0.508 | 0.576 | 0.637 | 0.689 | 0.866 | 0.993 | 1 |
| 20 | 0.267 | 0.343 | 0.410 | 0.472 | 0.528 | 0.579 | 0.772 | 0.974 | 1 |
| 30 | 0.235 | 0.303 | 0.364 | 0.420 | 0.472 | 0.521 | 0.713 | 0.953 | 0.999 |
| 40 | 0.216 | 0.279 | 0.336 | 0.389 | 0.438 | 0.484 | 0.672 | 0.933 | 0.997 |
| 50 | 0.203 | 0.263 | 0.317 | 0.367 | 0.414 | 0.458 | 0.642 | 0.914 | 0.996 |
| 100 | 0.172 | 0.224 | 0.270 | 0.314 | 0.355 | 0.395 | 0.562 | 0.850 | 0.982 |
| ∞ | 0.138 | 0.18 | 0.218 | 0.253 | 0.287 | 0.319 | 0.460 | 0.731 | 0.915 |
Example 2.
Assume that $X\sim \mathcal{G}(3/4)$ and $Y\sim \mathcal{NB}(1,1/2)$. Then, $2\mathbb{E}X+\mathbb{E}Y=2\frac{2}{3}<4$ and, according to Theorems 1 and 3, we obtain Table 3.
Table 3.
Survival probability when $X\sim \mathcal{G}(3/4)$ and $Y\sim \mathcal{NB}(1,1/2)$
| T | $u=0$ | $u=1$ | $u=2$ | $u=3$ | $u=4$ | $u=5$ | $u=10$ | $u=15$ |
| 1 | 0.938 | 0.984 | 0.996 | 0.999 | 1 | 1 | 1 | 1 |
| 2 | 0.738 | 0.866 | 0.932 | 0.965 | 0.983 | 0.991 | 1 | 1 |
| 3 | 0.734 | 0.863 | 0.930 | 0.964 | 0.982 | 0.991 | 1 | 1 |
| 4 | 0.686 | 0.823 | 0.902 | 0.946 | 0.970 | 0.984 | 0.999 | 1 |
| 5 | 0.684 | 0.822 | 0.901 | 0.945 | 0.970 | 0.984 | 0.999 | 1 |
| 10 | 0.647 | 0.787 | 0.872 | 0.924 | 0.955 | 0.973 | 0.998 | 1 |
| 20 | 0.638 | 0.778 | 0.864 | 0.917 | 0.949 | 0.969 | 0.997 | 1 |
| 30 | 0.637 | 0.776 | 0.862 | 0.916 | 0.948 | 0.968 | 0.997 | 1 |
| 40 | 0.636 | 0.776 | 0.862 | 0.915 | 0.948 | 0.968 | 0.997 | 1 |
| ∞ | 0.636 | 0.776 | 0.862 | 0.915 | 0.948 | 0.968 | 0.997 | 1 |
Example 3.
Assume that $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{G}(1/2)$. Then, $2\mathbb{E}X+\mathbb{E}Y=3\frac{2}{3}<4$ and, according to Theorems 1 and 4, we obtain Table 4.
Table 4.
Survival probability when $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{G}(1/2)$
| T | $u=0$ | $u=1$ | $u=2$ | $u=3$ | $u=4$ | $u=5$ | $u=10$ | $u=25$ | $u=40$ |
| 1 | 0.750 | 0.938 | 0.984 | 0.996 | 0.999 | 1 | 1 | 1 | 1 |
| 2 | 0.492 | 0.738 | 0.866 | 0.932 | 0.965 | 0.983 | 0.999 | 1 | 1 |
| 3 | 0.475 | 0.722 | 0.854 | 0.925 | 0.962 | 0.981 | 0.999 | 1 | 1 |
| 4 | 0.399 | 0.635 | 0.782 | 0.872 | 0.927 | 0.958 | 0.998 | 1 | 1 |
| 5 | 0.393 | 0.627 | 0.775 | 0.867 | 0.923 | 0.956 | 0.998 | 1 | 1 |
| 10 | 0.307 | 0.511 | 0.658 | 0.765 | 0.841 | 0.894 | 0.989 | 1 | 1 |
| 20 | 0.260 | 0.440 | 0.577 | 0.684 | 0.766 | 0.829 | 0.969 | 1 | 1 |
| 30 | 0.241 | 0.409 | 0.540 | 0.644 | 0.726 | 0.791 | 0.952 | 1 | 1 |
| 40 | 0.230 | 0.392 | 0.518 | 0.620 | 0.702 | 0.768 | 0.938 | 0.999 | 1 |
| 50 | 0.223 | 0.381 | 0.505 | 0.605 | 0.686 | 0.751 | 0.928 | 0.999 | 1 |
| 100 | 0.210 | 0.358 | 0.476 | 0.572 | 0.651 | 0.716 | 0.901 | 0.997 | 1 |
| ∞ | 0.203 | 0.347 | 0.462 | 0.556 | 0.633 | 0.967 | 0.884 | 0.993 | 1 |
Example 4.
Assume that $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{NB}(1,4/5)$. Then, $2\mathbb{E}X+\mathbb{E}Y=3\frac{11}{12}<4$ and, according to Theorems 1 and 5, we obtain Table 5.
Table 5.
Survival probability when $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{NB}(1,4/5)$
| T | $u=0$ | $u=1$ | $u=2$ | $u=3$ | $u=4$ | $u=5$ | $u=10$ | $u=25$ | $u=50$ |
| 1 | 0.750 | 0.938 | 0.984 | 0.996 | 0.999 | 1 | 1 | 1 | 1 |
| 2 | 0.450 | 0.765 | 0.912 | 0.970 | 0.990 | 0.997 | 1 | 1 | 1 |
| 3 | 0.422 | 0.738 | 0.896 | 0.962 | 0.987 | 0.996 | 1 | 1 | 1 |
| 4 | 0.344 | 0.652 | 0.837 | 0.930 | 0.972 | 0.989 | 1 | 1 | 1 |
| 5 | 0.332 | 0.636 | 0.824 | 0.922 | 0.968 | 0.987 | 1 | 1 | 1 |
| 10 | 0.240 | 0.497 | 0.694 | 0.825 | 0.905 | 0.951 | 0.999 | 1 | 1 |
| 20 | 0.185 | 0.398 | 0.579 | 0.717 | 0.817 | 0.885 | 0.993 | 1 | 1 |
| 30 | 0.161 | 0.350 | 0.517 | 0.652 | 0.755 | 0.832 | 0.982 | 1 | 1 |
| 40 | 0.146 | 0.320 | 0.477 | 0.607 | 0.711 | 0.792 | 0.969 | 1 | 1 |
| 50 | 0.137 | 0.300 | 0.449 | 0.575 | 0.678 | 0.760 | 0.956 | 1 | 1 |
| 100 | 0.113 | 0.250 | 0.378 | 0.490 | 0.586 | 0.666 | 0.900 | 0.999 | 1 |
| ∞ | 0.083 | 0.185 | 0.282 | 0.368 | 0.445 | 0.512 | 0.744 | 0.963 | 0.999 |
Example 5.
Assume that $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{NB}(3,3/4)$. Then, the net profit condition is violated, $2\mathbb{E}X+\mathbb{E}Y=6\frac{2}{3}>4$, and, according to Theorems 1 and 6, we obtain Table 6.
Table 6.
Survival probability when $X\sim \mathcal{NB}(1,3/4)$ and $Y\sim \mathcal{NB}(3,3/4)$
| T | $u=0$ | $u=1$ | $u=2$ | $u=3$ | $u=4$ | $u=5$ | $u=10$ | $u=25$ | $u=50$ |
| 1 | 0.750 | 0.938 | 0.984 | 0.996 | 0.999 | 1 | 1 | 1 | 1 |
| 2 | 0 | 0 | 0.237 | 0.534 | 0.756 | 0.886 | 0.999 | 1 | 1 |
| 3 | 0 | 0 | 0.222 | 0.512 | 0.737 | 0.873 | 0.999 | 1 | 1 |
| 4 | 0 | 0 | 0 | 0.056 | 0.197 | 0.391 | 0.960 | 1 | 1 |
| 5 | 0 | 0 | 0 | 0.053 | 0.187 | 0.376 | 0.956 | 1 | 1 |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0.115 | 0.997 | 1 |
| 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.340 | 1 |
| 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.001 | 0.943 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.292 |
| 50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.006 |
| 55 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.001 |
| ∞ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
4 Concluding remarks
Random walks appear in many natural sciences where birth/death, gain/loss and upturn/downturn processes are studied. In this work, we investigated the possibility that the random walk
never hits the line $u+2t,\hspace{0.1667em}u\in {\mathbb{N}_{0}}$, when ${X_{i}}\stackrel{d}{=}X$, ${Y_{j}}\stackrel{d}{=}Y$ for all $i,\hspace{0.1667em}j\in \mathbb{N}$ and X, Y are integer-valued, nonnegative and independent random variables, which may be distributed differently. Here $t\in \{1,\hspace{0.1667em}2,\hspace{0.1667em}\dots ,\hspace{0.1667em}T\}$, $T\in \mathbb{N}$ or $T\to \infty $. The finite time survival probability $\varphi (u,T)$ does not depend on the net profit condition $2\mathbb{E}X+\mathbb{E}Y<4$ and its recurrent expressions are given in Theorem 1. Theorems 2–5 express the ultimate time survival probability $\varphi (u)$ when the net profit condition is satisfied, and show $\varphi (u)$ dependency on $\min (2X+Y)$. The last Theorem 6 states that survival is impossible, in all but few trivial cases, if the net profit condition is violated. In Section 3, there are examples given for particular values of survival probabilities calculated according to Theorems 1–6. Summarizing, the ultimate time survival probability of the bi-risk discrete time risk model with income rate two is expressible via certain recurrent sequences and limit laws, i.e. initial values of φ are determined by the limits of certain recurrent sequences.
As mentioned in the introduction, each distinct setup of random or deterministic part in any discrete time risk model influences an expression of survival probability. This work might be deemed as preparation for studying more generalized discrete time risk models with an arbitrary income rate $\kappa \in \mathbb{N}$ in deterministic part $u+\kappa t$ and/or an arbitrary number of nonidentically distributed random variables generating the random walk ${\textstyle\sum _{i=1}^{t}}{Z_{i}}$. To avoid being buried in too many details, we just mention that such generalized models raise the level of abstraction significantly: a corresponding version of recurrence relation (5) would require more initial values and consequently much more effort in finding them.
