Consider first a simple version of the recursion (
4), i.e.
\(W_{n+1}=[V_{n}W_{n}+B_{n}-A_{n}]^{+}\), where now
\(P(V_{n}=a)=p_{1}\),
\(P(V_{n}\in [0,1))=p_{2}\), and
\(P(V_{n}<0)=1-p_{1}-p_{2}\), with
\(a\in (0,1)\),
\(0\le p_{1}\le 1\),
\(0\le p_{2}\le 1\),
\(p_{1}+p_{2}\le 1\). (The general version of (
4) will be considered in Remark
11.) Note that the case where
\(a=1\) was analysed in [
12]. In the following, we fill the gap in the literature, by analysing the case where
\(a\in (0,1)\), which we call
mixed-autoregressive, in the sense that in the obtained functional equation we will have the terms:
\(Z_{w}(r,as)\), and
\(\int _{[0,1)}Z_{w}(r,sy)P(V\in \textrm{d}y)\). Assume that
\(V^{+}{\mathop {=}\limits ^\textrm{def}}(V|V\in [0,1))\),
\(V^{-}{\mathop {=}\limits ^\textrm{def}}(V|V<0)\). Then, for
\(Re(s)=0\),
\(r\in [0,1)\) we have
$$\begin{aligned} Z_{w}(r,s)-e^{-sw}= & {} rp_{1}\phi _{Y}(s)Z_{w}(r,as)+rp_{2}\phi _{Y}(s)\int _{[0,1)}Z_{w}(r,sy)P(V^{+}\in \textrm{d}y) \nonumber \\{} & {} +r(1-p_{1}-p_{2})\phi _{Y}(s)\int _{(-\infty ,0)}Z_{w}(r,sy)P(V^{-}\in \textrm{d}y)\nonumber \\{} & {} +r\left( \frac{1}{1-r}-J^{-}(r,s)\right) , \end{aligned}$$
(109)
where
\(\{Y_{n}=B_{n}-A_{n}\}_{n\in \mathbb {N}_{0}}\) are i.i.d. random variables with LST
\(\phi _{Y}(s):=\frac{N_{Y}(s)}{D_{Y}(s)}\), with
\(D_{Y}(s):=\prod _{i=1}^{L}(s-t_{i})\prod _{j=1}^{M}(s-s_{j})\). Without loss of generality, we assume that
\(Re(t_{i})>0\),
\(i=1,\ldots ,L\),
\(Re(s_{j})<0\),
\(j=1,\ldots ,M\). Thus, (
109) becomes
$$\begin{aligned} \begin{array}{l} D_{Y}(s)(Z_{w}(r,s)-e^{-sw})-rp_{1}N_{Y}(s)Z_{w}(r,as)-rp_{2}N_{Y}(s)\int _{[0,1)}Z_{w}(r,sy)P(V^{+}\in \textrm{d}y)\\ =r(1-p_{1}-p_{2})N_{Y}(s)\int _{(-\infty ,0)}Z_{w}(r,sy)P(V^{-}\in \textrm{d}y)+rD_{Y}(s)\left( \frac{1}{1-r}-J^{-}(r,s)\right) .\end{array} \nonumber \\ \end{aligned}$$
(110)
It is readily seen that:
Thus, Liouville’s theorem [
14, Theorem 10.52] implies that for
\(Re(s)\ge 0\),
$$\begin{aligned}{} & {} D_{Y}(s)(Z_{w}(r,s)-e^{-sw})-rp_{1}N_{Y}(s)Z_{w}(r,as)-rp_{2}N_{Y}(s)\int _{[0,1)}\nonumber \\{} & {} Z_{w}(r,sy)P(V^{+}\in \textrm{d}y)=\sum _{l=0}^{M+L}C_{l}(r)s^{l}, \end{aligned}$$
(111)
and for
\(Re(s)\le 0\),
$$\begin{aligned}{} & {} r(1-p_{1}-p_{2})N(s)\int _{(-\infty ,0)}Z_{w}(r,sy)P(V^{-}\in \textrm{d}y)+rD_{Y}(s)(\frac{1}{1-r}-J^{-}(r,s))\nonumber \\{} & {} \quad =\sum _{l=0}^{M+L}C_{l}(r)s^{l}. \end{aligned}$$
(112)
By using either (
111) or (
112) for
\(s=0\), we obtain,
$$\begin{aligned} C_{0}(r)=\frac{r(1-p_{1}-p_{2})}{1-r}\prod _{i=1}^{L}t_{i}\prod _{j=1}^{M}s_{j}. \end{aligned}$$
Denoting by
\(\mu \) the probability measure on [0, 1) induced by
\(V^{+}\), the expression (
111) is written as
$$\begin{aligned} Z_{w}(r,s)=p_{1}K(r,s)Z_{w}(r,as)+p_{2}K(r,s) \int _{[0,1)}Z_{w}(r,sy_{1})\mu (\textrm{d}y_{1})+L_{w}(r,s), \nonumber \\ \end{aligned}$$
(113)
where
$$\begin{aligned} K(r,s):=r\phi _{Y}(s),\,\,L_{w}(r,s):=e^{-sw}+\frac{\sum _{l=0}^{M+L}C_{l}(r)s^{l}}{D_{Y}(s)}. \end{aligned}$$
Our aim is to solve (
113), which combines the model in [
8], with those in [
5,
12], i.e. in the functional equation the unknown function
\(Z_{w}(r,s)\) arises also as
\(Z_{w}(r,as)\) as well as in
\(\int _{[0,1)}Z_{w}(r,sy)\mu (\textrm{d}y)\). Let for
\(i,j=0,1,\ldots ,\)$$\begin{aligned} f_{i,j}(s):=&a^{i}\prod _{k=1}^{j}y_{k}s,\,y_{k}\in [0,1),k=1,\ldots ,j, \\ F(r,f_{i,j}(s)):=&\left\{ \begin{array}{ll} Z_{w}(r,a^{i}s),&{}j=0, \\ \int \ldots \int _{[0,1)^{j}}Z_{w}(r,f_{i,j}(s))\mu (\textrm{d}y_{1})\ldots \mu (\textrm{d}y_{j}),&{}j\ge 1, \end{array}\right. \end{aligned}$$
where
\(f_{i,0}(s)=a^{i}s\) (i.e.
\(\prod _{k=1}^{0}y_{k}:=1\)). Moreover,
\(f_{i,j}(f_{k,l}(s))=f_{i+k,j+l}(s)=f_{k,l}(f_{i,j}(s))\). Then, (
113) becomes
$$\begin{aligned} F(r,s)=p_{1}K(r,s)F(r,f_{1,0}(s))+p_{2}K(r,s)F(r,f_{0,1}(s))+L_{w}(r,s), \end{aligned}$$
(114)
where
\(F(r,s)=F(r,f_{0,0}(s))=Z_{w}(r,s)\). Iterating (
114)
\(n-1\) times yields,
$$\begin{aligned} F(r,s)= & {} \sum _{k=0}^{n}p_{1}^{k}p_{2}^{n-k}G_{k,n-k}(s)F(r,f_{k,n-k}(s))\nonumber \\{} & {} \quad +\sum _{k=0}^{n-1}\sum _{m=0}^{k}p_{1}^{m}p_{2}^{k-m}G_{m,k-m}(s)\tilde{L}(r,f_{m,k-m}(s)), \end{aligned}$$
(115)
where
\(G_{k,n-k}(r,s)\) are recursively defined as follows (with
\(G_{-1,.}(s) = G_{.,-1}(s) \equiv 0\),
\(G_{0,0}(s)=1\)):
$$\begin{aligned} G_{1,0}(s):=&K(r,s),\,\,\,\, G_{0,1}(s):=K(r,s), \\ G_{k+1,n-k}(s)=&G_{k,n-k}(s) \tilde{K}(r,f_{k,n-k}(s))+G_{k+1,n-1-k}(s) \tilde{K}(r,f_{k+1,n-1-k}(s)),\\ G_{k,n+1-k}(s)=&G_{k-1,n+1-k}(s) \tilde{K}(r,f_{k-1,n+1-k}(s))+G_{k,n-k}(s) \tilde{K}(r,f_{k,n-k}(s)), \end{aligned}$$
where also
$$\begin{aligned} \tilde{K}(r,f_{i,j}(s)):=&\left\{ \begin{array}{ll} K(r,a^{i}s),&{}j=0, \\ \int \ldots \int _{[0,1)^{j}}K(r,f_{i,j}(s))\mu (\textrm{d}y_{1})\ldots \mu (\textrm{d}y_{j}),&{}j\ge 1, \end{array}\right. \end{aligned}$$
and
$$\begin{aligned} \begin{array}{rl} \tilde{L}(r,f_{i,j}(s)):=&{}\left\{ \begin{array}{ll} L_{w}(r,a^{i}s),&{}j=0, \\ \int \ldots \int _{[0,1)^{j}}L_{w}(r,f_{i,j}(s))\mu (\textrm{d}y_{1})\ldots \mu (\textrm{d}y_{j}),&{}j\ge 1. \end{array}\right. \end{array} \end{aligned}$$
It can be easily verified that
\(G_{k,n-k}(r,s)\) is a sum of
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) terms, and each of them is a product of
n terms of values of
\(\tilde{K}(r,f_{.,.}(.))\), which are related to the LST
\(\phi _{Y}(.)\). We have to mention that our framework is related to the one developed in [
1] with the difference that the functions
\(f_{i,j}(s)\) (for
\(j>0\)) are more complicated compared to the corresponding
\(a_{i}(z)\) in [
1], and inherit difficulties in solving (
114).
In what follows, we will let
\(n\rightarrow \infty \) in (
115) so as to obtain an expression for
F(
r,
s). In doing that, we have to verify the convergence of the summation in the second term in the right-hand side of (
115), as well as to estimate the limit of the corresponding first term in the right-hand side of (
115). The key ingredient is to show that
\(G_{k,n-k}(s)\) is bounded. Similarly to [
1, p. 8],
\(G_{k,n-k}(s)\) can be interpreted as the total weight of all
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) paths from (0, 0) to
\((k,n-k)\). Let
\(C_{k,n-k}\) the set of all paths leading from (0, 0) to
\((k,n-k)\), where a path from (0, 0) to
\((k,n-k)\) is defined as a sequence of grid points starting from (0, 0) and ending to
\((k,n-k)\) by only taking unit steps (1, 0), (0, 1). Then, a typical term (one of the
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) terms) of
\(G_{k,n-k}(s)\) should be the following:
$$\begin{aligned} \int \ldots \int _{[0,1)^{m}}\prod _{(l,m)\in C_{k,n-k}}\tilde{K}(r,a^{l}y_{1}\ldots y_{m}s)\mu (\textrm{d}y_{1})\ldots \mu (\textrm{d}y_{n-k}), \end{aligned}$$
for
\(m=0,\ldots , n-k\), and
\(l=0,\ldots ,k\) with
\((l,m)\ne (k,n-k)\). For
\(Re(s)\ge 0\),
\(M_{1}(r,s):=\sup _{y\in [0,1]}|K(r,sy)|<\infty \),
\(M_{2}(r,s):=\sup _{y\in [0,1]}|L(r,sy)|<\infty \), and
\(|K(r,s)|\le r<1\). Then, for
\(a\in (0,1)\),
\(M_{l}(r,a^{i}s)<M_{l}(r,s)\),
\(i\ge 1\),
\(l=1,2\). Following [
12],
$$\begin{aligned}&\left| \int \ldots \int _{[0,1)^{m}}\prod _{(l,m)\in C_{k,n-k}}\tilde{K}(r,a^{l}y_{1}\ldots y_{m}s)\mu (\textrm{d}y_{1})\ldots \mu (\textrm{d}y_{n-k})\right| \\&\quad \le E\left[ \prod _{(l,m)\in C_{k,n-k}}|\tilde{K}(r,a^{l}Z_{1}\ldots Z_{m}s)|\right] , \end{aligned}$$
where
\(Z_{1},Z_{2},\ldots \) is a sequence of i.i.d. random variables with the same distribution as
\(V^{+}\). Following the same procedure as in [
12, pp. 8-9], we can show that each of the weights of the path is bounded, implying that
\(G_{k,n-k}(s)\) is also bounded. This result will imply as
\(n\rightarrow \infty \) that the first term in the right-hand side of (
115) vanishes. Thus,
$$\begin{aligned} F(r,s)=\sum _{k=0}^{\infty }\sum _{m=0}^{k}p_{1}^{m}p_{2}^{k-m}G_{m,k-m}(s)\tilde{L}(r,f_{m,k-m}(s)). \end{aligned}$$
(116)
We are now ready to obtain the coefficients
\(C_{l}(r)\),
\(l=1,\ldots ,M+L\). For
\(s=t_{i}\),
\(i=1,\ldots ,L\), in (
111), we have
$$\begin{aligned} -rp_{1}N_{Y}(t_{i})Z_{w}(r,at_{i})-rp_{2}N_{Y}(t_{i})\int _{[0,1)}Z_{w}(r,t_{i}y)\mu (\textrm{d}y)=\sum _{l=0}^{M+L}C_{l}(r)t_{i}^{l}. \nonumber \\ \end{aligned}$$
(117)
Setting
\(s=s_{j}\),
\(j=1,\ldots ,M\), in (
112) yields
$$\begin{aligned} r(1-p_{1}-p_{2})N(s_{j})\int _{(-\infty ,0)}Z_{w}(r,s_{j}y)P(V^{-}\in \textrm{d}y)=\sum _{l=0}^{M+L}C_{l}(r)s_{j}^{l}. \end{aligned}$$
(118)
Equations (
117), (
118) constitute a system of equations to obtain the unknown coefficients
\(C_{l}(r)\),
\(l=1,\ldots ,M+L\).