# Schrödinger Risk Diversification Portfolio

Yusuke Uchiyama
MAZIN, Inc.
Amazing Building 3F, 3-29-14 Nishiasakusa, Taito-ku, Tokyo 111-0035, Japan
<EMAIL_ADDRESS>
&Kei Nakagawa
NOMURA Asset Management Co. Ltd,
2-2-1 Toyosu, Koto-ku, Tokyo 135-0061
<EMAIL_ADDRESS>

###### Abstract

The mean-variance portfolio that considers the trade-off between expected
return and risk has been widely used in the problem of asset allocation for
multi-asset portfolios. However, since it is difficult to estimate the
expected return and the out-of-sample performance of mean-variance portfolio
is poor, risk-based portfolio construction methods focusing only on risk have
been proposed, and are attracting attentions mainly in practice. In terms of
risk, asset fluctuations that make up the portfolio are thought to have common
factors behind them, and principal component analysis, which is a dimension
reduction method, is applied to extract the factors. In this study, we propose
the Schrödinger risk diversification portfolio as a factor risk diversifying
portfolio using Schrödinger principal component analysis that applies the
Schrödinger equation in quantum mechanics. The Schrödinger principal component
analysis can accurately estimate the factors even if the sample points are
unequally spaced or in a small number, thus we can make efficient risk
diversification. The proposed method was verified to outperform the
conventional risk parity and other risk diversification portfolio
constructions.

_K_ eywords portfolio management $\cdot$ risk diversification $\cdot$
Schrödinger principal component analysis

##  1 Introduction

The mean-variance portfolio, which takes into account the trade-off between
expected return and risk, has been used in the problem of asset allocation for
portfolios consisting of multiple assets. Since its proposal by Markowitz [1],
the mean-variance portfolio has long been an important framework in investment
decision making. The reason is that the mean-variance portfolio is easy to
handle in practice because the optimization problem can be expressed as a
quadratic programming problem. The mean-variance portfolio requires the
expected return, the variance, and the correlation among the assets and uses
them as inputs to determine the asset allocation by considering the trade-off
between the expected return and the variance of the portfolio under various
investment constraints.

However, the mean-variance portfolio has several drawbacks. The most
frequently pointed out problem is that it is quite difficult to predict the
exact value of the expected returns as an input in the first place, thus the
out-of-sample performance is too poor [2]. Market uncertainty is increasing
year by year, as exemplified by the Lehman shock and the recent Covid-19
shock, and the estimating expected returns is becoming more and more difficult
under such market conditions. Another problem is that the mean-variance
portfolio obtained as a result of optimization differs greatly for small
changes in expected returns [3].

Therefore, risk-based portfolio construction methods that focus only on risk
without expected returns have been proposed and are attracting attention
mainly in practical applications. The minimum risk portfolio [4, 5], the risk
parity portfolio [6, 7], and the maximum diversification portfolio [8] are
typical examples, and risk-based portfolios that are further developed from
them are proposed [9, 10, 11, 12, 11]. In fact, in various empirical studies
and backtests on equity portfolios and asset allocations, these risk-based
portfolio construction methods perform better than traditional mean-variance
portfolios and capitalization-weighted portfolios [13]. Also, unlike the mean-
variance portfolio, the performance of the risk-based portfolio is less
affected by the estimation accuracy of the risk (covariance matrix) [14].

On the other hand, asset fluctuations that make up a portfolio are thought to
have common factors behind them. Principal component analysis, which is a
dimension reduction method, is applied to extract these factors. Based on this
idea, the factor-based risk parity portfolio, which allocates risk equally to
factors extracted by principal component analysis, has been proposed as a
risk-based portfolio construction method [15]. In contrast to the usual risk
parity portfolio, which allocates risk equally on assets, the factor-based
risk parity portfolio is called a risk diversification portfolio.

In addition, for the purpose of extracting dynamic risk factors, a complex-
valued risk diversification portfolio is proposed, which allocates risk
equally to the factors extracted by Hilbert principal component analysis [9],
and a quaternion risk diversification portfolio is further developed [16].

In this study, we propose the Schrödinger Risk Diversification Portfolio as a
novel risk diversification portfolio using Schrödinger principal component
analysis [17], which is an application of the Schrödinger equation in quantum
mechanics. The Schrödinger principal component analysis can estimate the
principal components and cross-correlations with high accuracy even when the
data are unequally sampled or a small number. Therefore, we expect to estimate
factors accurately and enable efficient risk diversification. We compare the
proposed method with existing risk diversification portfolios to verify its
effectiveness using real asset data.

##  2 Preliminary

###  2.1 Gaussian Filed and Schrödinger Equation

Suppose, given a scalar field ${\phi}:\mathbb{R}^{n}\to{\mathbb{R}}$ for
spatial coordinates $\bm{x}\in{\mathbb{R}^{d}}$.

When the quantity of the field
${\phi}(\bm{x}_{1}),{\phi}(\bm{x}_{2}),{\cdots},{\phi}(\bm{x}_{N})$
corresponding to any $N$ coordinate vectors
$\bm{x}_{1},\bm{x}_{2},{\cdots},\bm{x}_{N}$ is a random variable following an
$N$-dimensional Gaussian distribution, ${\phi}({\cdot})$ is called a Gaussian
process.

In particular, ${\phi}({\cdot})$ is also called a Gaussian field because it is
a field quantity with respect to the spatial coordinate $\bm{x}$. By
subtracting the mean value $\mathbb{E}[{\phi}(\bm{x})]$ from the Gaussian
field, we can assume to set $\mathbb{E}[{\phi}(\bm{x})]=0$ without loss of
generality.

Then, the positive definite symmetric function given as follows is called the
covariance function or kernel function.

$k(\bm{x},\bm{x}^{\prime})=\mathbb{E}[{\phi}(\bm{x}){\phi}(\bm{x}^{\prime})]$
(1)

Thus, for any $N$ coordinate vector
$\bm{x}_{1},\bm{x}_{2},{\cdots},\bm{x}_{N}$, the kernel function gives the
covariance matrix of the $N$-dimensional Gaussian distribution.

$K=\left[\begin{array}[]{cccc}k(\bm{x}_{1},\bm{x}_{1})&k(\bm{x}_{1},\bm{x}_{2})&\ldots&k(\bm{x}_{1},\bm{x}_{N})\\\
k(\bm{x}_{2},\bm{x}_{1})&k(\bm{x}_{2},\bm{x}_{2})&\ldots&k(\bm{x}_{2},\bm{x}_{N})\\\
\vdots&\vdots&\ddots&\vdots\\\
k(\bm{x}_{N},\bm{x}_{1})&k(\bm{x}_{N},\bm{x}_{2})&\ldots&k(\bm{x}_{N},\bm{x}_{N})\end{array}\right]$
(2)

For the covariance function of a Gaussian field, there exists a real function
$A({\cdot})$ and a positive definite symmetric matrix ${\Sigma}$ that
satisfies

$\displaystyle k(\bm{x},\bm{x}^{\prime})$
$\displaystyle=\sqrt{A(\bm{x})A(\bm{x}^{\prime})}$
$\displaystyle{\times}\exp{\left[-\frac{1}{2}(\bm{x}-\bm{x}^{\prime})^{T}{\Sigma}^{-1}(\bm{x}-\bm{x}^{\prime})\right]}.$
(3)

On the other hand, for an integral operator with a kernel function as its
integral core, there exist the following eigenvalue problem

$\int_{{\Omega}_{\bm{x}}}k(\bm{x},\bm{x}^{\prime}){\varphi}_{i}(\bm{x}^{\prime})d\bm{x}^{\prime}={\lambda}_{i}{\varphi}_{i}(\bm{x}^{\prime}),$
(4)

where ${\lambda}_{i}$ and ${\varphi}_{i}$ are $i$-th eigenvalue and
eigenfunction, respectively, which satisfy the orthogonality condition

$\int_{{\Omega}_{\bm{x}}}{\varphi}_{i}(\bm{x}){\varphi}_{j}(\bm{x})d\bm{x}={\delta}_{i,j}$
(5)

where ${\delta}_{i,j}$ is the Kronecker delta. Due to the positive symmetry of
the kernel function, the eigenvalues are positive real numbers. The subscripts
$i$ and $j$ of the eigenvalues and eigenfunctions are taken to satisfy
${\lambda}_{1}\geq{{\lambda}_{2}}\geq{{\cdots}}\geq{0}$. Since the Eq. (4)
corresponds to the infinite-dimensional vector of the eigenvalue problem of
the covariance matrix, the eigenfunction expansion censored by an appropriate
finite number of eigenfunctions corresponds to the principal component
analysis.

The eigenvalue problem of the Schrödinger equation in quantum mechanics is
given as follows:

$-\frac{1}{2}{\Delta}_{{\Sigma}_{\bm{m}}}{\psi}(\bm{x})+V(\bm{x}){\psi}(\bm{x})=E{\psi}(\bm{x}).$
(6)

Here, ${\psi}({\cdot})$ is the wave function and
${\Delta}_{{\Sigma}_{\bm{m}}}$ is the second-order partial differential
operator weighted by the positive definite symmetric matrix
${\Sigma}_{\bm{m}}$.

${\Delta}_{{\Sigma}_{\bm{m}}}=\sum_{i,j}\frac{{\partial}}{{\partial}x_{i}}({\Sigma}_{\bm{m}})_{i,j}\frac{{\partial}}{{\partial}x_{j}}$
(7)

The real function $V({\cdot})$ is a potential function and $E$ is an energy
eigenvalue. The linear operator appearing on the left-hand side of Eq. (6) is
also called the Hamilton operator or Hamiltonian.

$H=-\frac{1}{2}{\Delta}_{{\Sigma}_{\bm{m}}}+V(\bm{x})$ (8)

Since the Hamiltonian operator is a Hermite operator, the energy eigenvalues
are non-negative real values.

###  2.2 Schrödinger PCA

The integral equation in Eq. (4) and the Schrödinger equation expressed in Eq.
(6) agree to the second order of accuracy according to the following
correspondence:

$\displaystyle{\phi}(\bm{x})\;$
$\displaystyle{\Leftrightarrow}\;{\psi}(\bm{x})$ (9)
$\displaystyle-A(\bm{x})\;$ $\displaystyle{\Leftrightarrow}\;V(\bm{x})$ (10)
$\displaystyle A(\bm{x}){\Sigma}\;$
$\displaystyle{\Leftrightarrow}\;{\Sigma}_{\bm{m}}.$ (11)

Therefore, we can perform principal component analysis by expanding the
original Gaussian field by the eigenfunctions of the Schrödinger equation and
then terminating the expansion at an appropriate point. This is called
Schrödinger principal component analysis (PCA)[17]. Since ordinary principal
component analysis can be performed using a finite data set, it is known that
the estimation accuracy deteriorates when only a few samples are available. On
the other hand, since Schrödinger PCA does not depend on the number of data
samples, we can obtain stable estimation results within the range of
approximation accuracy up to the second order.

##  3 Method:Schrödinger Risk Diversification Portfolio

Schrödinger PCA considers the quantity of a field, that is, the quantity that
fluctuates at each point of spatial coordinates. On the other hand, the asset
allocation problem considers a system in which the prices of individual assets
fluctuate, which cannot be mapped to spatial coordinates. Therefore, we apply
Schrödinger PCA to multivariate time series consisting of multiple assets
price fluctuations by conveniently mapping the $i$th asset to the spatial
coordinate $x_{i}$. We formulate the risk diversification portfolio
constructed using the results of Schrödinger PCA as the Schrödinger risk
diversification portfolio.

The procedure for constructing a Schrödinger risk diversification portfolio is
as follows:

  1. 1.

For the series of $T$-period returns of $N$ assets,
$\\{r_{t,i}\\}(i=1,2,{\cdots},N;t=1,2,{\cdots},T)$, calculate

$\displaystyle{\mu}_{i}=\frac{1}{T}\sum_{t=1}^{T}r_{t,i},$ (12)
$\displaystyle{\sigma}_{i}^{2}=\frac{1}{T-1}\sum_{t=1}^{T}(r_{t,i}-{\mu}_{r,i})^{2}.$
(13)

  2. 2.

Assume the functional form of the potential function, we perform the
substitution $i{\mapsto}j(i)$ such that $V(x_{j(i)})={\sigma}_{i}^{2}$ and
estimate $x_{0}$ and ${\Delta}x$ in $x_{j}=x_{0}+j{\Delta}x$ and the
parameters of the potential function.

  3. 3.

By solving the eigenvalue problem of the Schrödinger equation in one spatial
dimension,

$-\frac{1}{2}\frac{d^{2}}{dx^{2}}{\psi}(x)+V(x){\psi}(x)=E{\psi}(x)$ (14)

we obtain the pair of eigenvalues and eigenfunctions
$\\{E_{l},{\psi}_{l}\\}(l=1,2,{\cdots},L)$. Here, $L$ corresponds to the
censored order of Schrödinger PCA.

  4. 4.

Generate a following matrix from the eigenfunctions.

${\Psi}_{l,i}={\psi}_{l}(x_{i})\;(i=1,2,{\cdots},N;l=1,2,{\cdots},L)$ (15)

For the portfolio weights $w_{i}\;(i=1,2,{\cdots},N)$, we calculate

$\tilde{w}_{l}=\sum_{i=1}^{N}{\Psi}_{l,i}w_{i}$ (16)

  5. 5.

For each $l$, we obtain $v_{l}$ from the eigenvalues $E_{l}$ and
$\tilde{w}_{l}$.

$v_{l}=\frac{E_{l}\tilde{w}_{l}^{2}}{\sum_{l=1}^{L}E_{l}\tilde{w}_{l}^{2}}$
(17)

  6. 6.

We find the optimal weights by maximizing the entropy
$\mathcal{H}=-\sum_{l=1}^{L}v_{l}\log{v_{l}}$ subject to constraints i.e.,
$\sum_{i=1}^{N}w_{i}=1$.

In the case of one-dimensional space, it is known that the Schrödinger
equation under appropriate potential functions has an analytic solution.

For example, in the case of a harmonic oscillator whose potential function is
given by

$V(x)=-\frac{1}{2}kx^{2}.$ (18)

Then, $y=x/\sqrt{k}$, ${\lambda}=2E/\sqrt{k}$, we have a second-order ordinary
differential equation

$\frac{d^{2}{\psi}}{dy^{2}}+({\lambda}-y^{2}){\psi}=0$ (19)

Solving this equation, we obtain

$\displaystyle{\lambda}_{l}=2l+1,$ (20)
$\displaystyle{\psi}_{l}(y)=H_{l}(y)e^{-y^{2}/2}$ (21)

Here, the function $H_{l}({\cdot})$ is a Hermitian polynomial. This gives us
the eigenvalues and eigenfunctions we wish to obtain. There are other
potential functions for which analytical solutions can be obtained, and we can
use them according to our problem.

##  4 Experiment

In this section, we test the performance of the purposed portfolio
construction by comparing with the conventional risk diversifying portfolio
constructions, the PCA and HPCA portfolio constructions. Data description and
the results of performance test are given in the following subsections.

###  4.1 Dataset

To evaluate the performance of each portfolio, we selected six bond futures
and six equity futures, as shown in Table LABEL:tabl:statistics.

All of the daily historical data were corrected during May 2000 to April 2017.
We obtained all of these data from Bloomberg terminals. The descriptive
statistics of the returns of the assets are shown in Table
LABEL:tabl:statistics.

Table 1: List of Assets. Ticker | Type | Description
---|---|---
TX1 | Bond Future | 10 Year T-Note Futures
XM1 | Bond Future | Australian 10 Year Bond
CN1 | Bond Future | Canadian Government 10 Year Note
RX1 | Bond Future | Eurex Euro Bund
G1 | Bond Future | Gilt UK
JB1 | Bond Future | Japan 10 Year Bond Futures
SP1 | Equity Future | S&P500
XP1 | Equity Future | S&P/ASX 200(Austraria)
PT1 | Equity Future | S&P/TSX 60 Index(Canada)
GX1 | Equity Future | DAX(German)
Z1 | Equity Future | FTSE100(UK)
NK1 | Equity Future | Nikkei225(Japan)
Table 2: Descriptive statistics of the dataset. | Mean [%] | Standard Deviation [%] | Skewness | Kurtosis
---|---|---|---|---
TY1 | 0.006763 | 0.395198 | -0.28289 | 4.197417
XM1 | 0.00094 | 0.065218 | -0.13666 | 1.812167
CN1 | 0.008609 | 0.363551 | -0.26973 | 2.366885
RX1 | 0.010441 | 0.352686 | -0.54011 | 3.825943
G 1 | 0.003761 | 0.442155 | -5.72552 | 150.2678
JB1 | 0.003133 | 0.203517 | -0.52795 | 5.763622
SP1 | 0.018462 | 1.207641 | 0.202656 | 12.4827
XP1 | 0.019369 | 1.012393 | -0.22977 | 4.928742
PT1 | 0.017505 | 1.172459 | -0.4545 | 9.33117
GX1 | 0.022293 | 1.489021 | 0.245713 | 6.613814
Z 1 | 0.009463 | 1.184334 | -0.01916 | 6.814881
NK1 | 0.012986 | 1.555291 | 0.023165 | 12.68528

###  4.2 Experimental Settings

We compare the proposed method Schrodinger Risk diversification Portfolio
(SPCA) with the Equal Weight Portfolio (EW), Principal Component risk
diversification Portfolio (PCA), and Complex Principal Component Risk Parity
Portfolio (HPCA). Each portfolio re-estimates its parameters and rebalances on
the 20 days (monthly).

EW:

"EW" stands for equally-weighted portfolio [18].

PCA:

"PCA" stands for principal component risk diversification portfolio that
invests evenly in principal component risk by performing principal component
analysis on return data for the past 250 days [15].

HPCA:

"HPCA" stands for Hilbert transformed principal component risk diversification
portfolio that invests evenly in principal component risk by performing
principal component analysis on Hilbert transformed return data for the past
250 days [9].

SPCA:

"SPCA" stands for our proposed portfolio by performing Schrodinger principal
component analysis on return data for the past 250 days.

###  4.3 Performance Measures

In evaluating the portfolio strategy, we used the following measures that are
widely used in the field of finance [11].

The portfolio return at time $t$ is defined as

$R_{t}=\sum_{i=1}^{n}r_{it}w_{it-1}$ (22)

where $r_{it}$ is the return of $i$-th asset at time $t$, $w_{it-1}$ is the
weight of $i$-th asset in the portfolio at time $t-1$, and $n$ is the number
of assets.

We evaluated the portfolio strategy using its annualized return (AR), the risk
as the standard deviation of return (RISK), and the risk/return (R/R) (return
divided by risk) as the portfolio strategy. R/R is the most important
performance measure.

$\displaystyle{\bf AR}$ $\displaystyle=\frac{250}{T}\sum_{t=1}^{T}R_{t}$ (23)
$\displaystyle{\bf RISK}$
$\displaystyle=\sqrt{\frac{250}{T-1}\times\sum_{t=1}^{T}(R_{t}-\mu)^{2}}$ (24)
$\displaystyle{\bf R/R}$ $\displaystyle={\bf AR}/{\bf RISK}$ (25)

Here, $\mu=(1/T)\sum_{t=1}^{T}R_{t}$ is the average return of the portfolio.

We also evaluated the maximum drawdown (MaxDD), which is another widely used
risk measure [19], for the portfolio strategy. In particular, MaxDD is defined
as the largest drop from an extremum:

$\displaystyle{\bf MaxDD}$
$\displaystyle=\min_{k\in[1,T]}\left(0,\frac{W_{k}^{\mathrm{Port}}}{\max_{j\in[1,k]}W_{j}^{\mathrm{Port}}}-1\right)$
(26) $\displaystyle W_{k}^{\mathrm{Port}}$
$\displaystyle=\sum_{i=1}^{k}(1+R_{t}).$ (27)

###  4.4 Results

Table 3 summarizes the performance measures of each portfolio. We can see that
SPCA has the highest AR and R/R while HPCA has the lowest Risk. Therefore, the
proposed method can efficiently earn returns and can also control risks by
appropriately diversifying the factor risks.

Table 3: Annual return (AR), risk (RISK) and, risk/return (R/R), maximum drawdown(MaxDD) of the EW, PCA, HPCA and SPCA portfolio. | EW | PCA | HPCA | SPCA
---|---|---|---|---
AR [%] | 3.16 | 1.58 | 3.20 | 3.84
RISK [%] | 6.67 | 6.31 | 5.36 | 5.59
R/R | 0.47 | 0.25 | 0.60 | 0.69
MaxDD [%] | 32.35 | 26.76 | 21.53 | 34.66

In fact, as is shown in Figure 4, the cumulative return of the SPCA is stable.

Table 4 shows the time averaged weight coefficients of the portfolios in terms
of assets and type of assets, respectively. The corresponding sequences of the
weights of the PCA, HPCA and SPCA portfolios are shown in Figures 1-3. We can
see that the weights of equity futures are higher in the order of PCA, SPCA,
and HPCA.

Figure 1: The allocation of the assets in the PCA portfolio. Figure 2: The allocation of the assets in the HPCA portfolio. Figure 3: The allocation of the assets in the SPCA portfolio. Figure 4: The cumulative returns of the EW (blue), PCA (orange), HPCA (green), and SPCA (red) portfolio. Table 4: Averaged weights of the portfolios. | PCA [%] | HPCA [%] | SPCA [%]
---|---|---|---
TY1 | 17.04 | 6.40 | 11.63
XM1 | 23.19 | 10.88 | 31.22
CN1 | 14.49 | 7.37 | 6.96
RX1 | 9.14 | 6.17 | 8.76
G 1 | 8.69 | 7.28 | 3.87
JB1 | 6.57 | 25.32 | 9.52
SP1 | 5.57 | 10.06 | 5.75
XP1 | 4.17 | 9.32 | 2.32
PT1 | 4.00 | 5.78 | 9.37
GX1 | 3.23 | 4.87 | 3.26
Z 1 | 1.98 | 2.09 | 3.82
NK1 | 1.92 | 4.45 | 3.52
Bond | 79.12 | 63.42 | 71.96
Equity | 20.88 | 36.58 | 28.04

##  5 Conclusion

We proposed the Schrödinger risk diversification portfolio which applies
Schrödinger PCA to extract risk factors. We perform experiments on multi-asset
market data to evaluate the proposed portfolio. Compared with various
portfolios, the proposed portfolio demonstrates a higher risk-adjusted
returns.

In the empirical analysis, we used the harmonic oscillator potential as the
potential function and assumed the spatial arrangement among the assets are
evenly spaced. In general, when we conduct Schrödinger PCA, there is room for
selecting the potential function and making the spatial arrangement
unreasonable. Also, we can extend our proposed method to complexifying the
time series of returns of each asset by Hilbert transform as in [9]. The
future task is to verify the effectiveness of the proposed method developed in
these directions.

## References

  * [1] Harry Markowitz.  Portfolio selection.  The Journal of Finance, 1952.
  * [2] Robert C Merton.  On estimating the expected return on the market: An exploratory investigation.  Journal of financial economics, 8(4):323–361, 1980.
  * [3] Richard O Michaud.  The markowitz optimization enigma: Is 窶椀ptimized窶冩ptimal?  Financial analysts journal, 45(1):31–42, 1989.
  * [4] Roger Clarke, Harindra De Silva, and Steven Thorley.  Minimum-variance portfolio composition.  The Journal of Portfolio Management, 37(2):31–45, 2011.
  * [5] R Tyrrell Rockafellar, Stanislav Uryasev, et al.  Optimization of conditional value-at-risk.  Journal of risk, 2:21–42, 2000.
  * [6] Edward Qian.  Risk parity portfolios: Efficient portfolios through true diversification.  Panagora Asset Management, 2005.
  * [7] Sébastien Maillard, Thierry Roncalli, and Jérôme Teïletche.  The properties of equally weighted risk contribution portfolios.  The Journal of Portfolio Management, 36(4):60–70, 2010.
  * [8] Yves Choueifaty and Yves Coignard.  Toward maximum diversification.  The Journal of Portfolio Management, 35(1):40–51, 2008.
  * [9] Yusuke Uchiyama, Takanori Kadoya, and Kei Nakagawa.  Complex valued risk diversification.  Entropy, 21(2):119, 2019.
  * [10] Kei Nakagawa and Yusuke Uchiyama.  Go-gjrsk model with application to higher order risk-based portfolio.  Mathematics, 8(11):1990, 2020.
  * [11] Kei Nakagawa, Shuhei Noma, and Masaya Abe.  Rm-cvar: Regularized multiple $\beta$-cvar portfolio.  In IJCAI, pages 4562–4568, 2020.  Special Track on AI in FinTech.
  * [12] Yusuke Uchiyama and Kei Nakagawa.  Tplvm: Portfolio construction by student窶冱 t-process latent variable model.  Mathematics, 8(3):449, 2020.
  * [13] Thorsten Poddig and Albina Unger.  On the robustness of risk-based asset allocations.  Financial Markets and Portfolio Management, 26(3):369–401, 2012\.
  * [14] Kei Nakagawa, Mitsuyoshi Imamura, and Kenichi Yoshida.  Risk-based portfolios with large dynamic covariance matrices.  International Journal of Financial Studies, 6(2):52, 2018.
  * [15] Attilio Meucci.  Managing diversification.  Risk, 22(5):74, 2009.
  * [16] Seisuke Sugitomo and Keiichi Maeta.  Quaternion valued risk diversification.  Entropy, 22(4):390, 2020.
  * [17] Ziming Liu, Sitian Qian, Yixuan Wang, Yuxuan Yan, and Tianyi Yang.  Schrödinger principal-component analysis: On the duality between principal-component analysis and the schrödinger equation.  Phys. Rev. E, 104:025307, Aug 2021.
  * [18] Victor DeMiguel, Lorenzo Garlappi, and Raman Uppal.  Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?  The review of Financial studies, 22(5):1915–1953, 2009.
  * [19] Malik Magdon-Ismail and Amir F Atiya.  Maximum drawdown.  Risk Magazine, 17(10):99–102, 2004.


11institutetext:  22institutetext: Laboratoire de Mathématiques, Université
Bourgogne Franche-Comté, Besançon, France, and Lomonosov Moscow State
University, Moscow, Russia
22email<EMAIL_ADDRESS>
33institutetext: Laboratoire de Mathématiques Raphaël Salem, Université de
Rouen, France,
and National Research Tomsk State University, Russia
33email<EMAIL_ADDRESS>
# On ruin probabilities with investments in a risky asset with a switching
regime price

Yuri Kabanov  Serguei Pergamenshchikov

(Received: date / Accepted: date)

###### Abstract

We investigate the asymptotic of ruin probabilities when the company invests
its reserve in a risky asset with a switching regime price. We assume that the
asset price is a conditional geometric Brownian motion with parameters
modulated by a Markov process with a finite number of states. Using the
technique of the implicit renewal theory we obtain the rate of convergence to
zero of the ruin probabilities as the initial capital tends to infinity.

###### Keywords:

Ruin probabilities Risky investments Stochastic volatility Hidden Markov model
Regime switching Implicit renewal theory

###### MSC:

60G44

JEL Classification G22 $\cdot$ G23

##  1 Introduction

Models, where an insurance company invests its reserve (or a part of it) in a
risky asset, constitutes an important class currently under an extensive
study. Considering a single risky asset is justified by the common practice of
investing in a market portfolio or in an index (a fund which simulates an
index like the DAX or the S&P500) which is an economically reasonable
strategy. Since the insurance contacts usually are of a long duration and the
return may depend, e.g., on the business cycles of economy, models with regime
switching are now more and more popular. The main question is the rate of
decay of the ruin probability as the initial reserve tends to infinity.

In this note we extend the recent result of Ellanskaya and Kabanov, EK ,
established for a model with characteristics of the asset price depending on a
telegraph process, i.e. on a Markov process with two states, $0$ and $1$. Here
we study the case where the characteristics depend on the ergodic Markov
process $\theta$ with a finite number of states. When $\theta_{t}=k$, the
asset price evolves as a geometric Brownian motion with drift $a_{k}$ and
volatility $\sigma_{k}$. It is well known, see, e.g. FrKP ; KP ; KPukh , that
in the case of a single regime, i.e. when the price process is the classic gBm
with drift $a$ and volatility $\sigma$, the ruin probability decreases to zero
as the initial capital $u$ tends to infinity with the rate
$\beta:=2a/\sigma-1$ provided that $\beta>0$. In EK  is was shown that in the
model with two regimes, $0$ and $1$, the ruin probabilities decrease with a
rate $\beta$ where $\beta$ is a number between the values
$\beta_{k}:=2a_{k}/\sigma_{k}-1$, $k=0,1$, assumed to be strictly positive.
This $\beta$ is the root of an algebraic equation of third order and does not
depend on the initial value of $\theta$.

In the present paper we extend this result to the case where the number of
states of the hidden Markov process $\theta$ is $K\geq 2$. It happens that,
provided all $\beta_{k}>0$, $k=0,\dots,K-1$, the rate of convergence to zero
of the ruin probabilities, depending, in general, on the initial state $i$, is
a root of the cumulant generating function of the value of log price process
at the first return time of $\theta$ to the state $i$. It is worth to note
that the switching by telegraph signal is rather specific: the latter returns
to the initial state after the second jump while for a general Markov process
the return may happen after arbitrary number of jumps.

Though the main idea is again based on the implicit renewal theory, it happens
that the analysis of the considered model is much more complicated and the
calculation of the rate parameter, depending, in general, on the initial
state, is not so straightforward. We hope that the result of this paper
elucidate challenging problems of estimation of ruin probabilities for other
stochastic volatility models.

##  2 The model

Let $(\Omega,{\cal F},{\bf F}=({\cal F}_{t})_{t\in{\mathbb{R}}_{+}},{\bf P})$
be a stochastic basis with a Wiener process $W=(W_{t})$, a Poisson random
measure $\pi(dt,dx)$ on ${\mathbb{R}}_{+}\times{\mathbb{R}}$ with the mean
$\tilde{\pi}(dt,dx)=\Pi(dx)dt$, and a piecewise constant right-continuous
Markov process $\theta=(\theta_{t})$. For the latter we assume that it takes
values in the finite set $\\{0,1,...,K-1\\}$, has the $K\times K$ transition
intensity matrix $\Lambda=(\lambda_{jk})$ with communicating states, and the
initial value $\theta_{0}=i$ (so that $\theta=\theta^{i})$. The
$\sigma$-algebras generated by $W$, $\pi$, and $\theta$ are independent.

Recall that $\lambda_{jj}=-\sum_{k\neq j}\lambda_{jk}$ and
$\lambda_{i}:=-\lambda_{ii}>0$ for each $i$.

Let $T_{n}$ be the successive jumps of the Poisson process $N=(N_{t})$ with
$N_{t}:=\pi([0,t],{\mathbb{R}})$ and let $\tau_{n}$ be the successive jumps of
$\theta$ with the convention $T_{0}=0$ and $\tau_{0}=0$.

Recall that the lengths of the intervals between the consecutive jumps of
$\theta$ are independent exponentially distributed random variables.

The reserve $X=X^{u}$ of an insurance company evolves not only due to the
business activity part, described as in the classical Cramér–Lundberg model,
but also due to the stochastic interest rate. We assume that the reserve is
fully invested in a risky asset whose price $S$ is a conditional geometric
Brownian motion given the Markov process $\theta$. That is, $S$ is given by a
so-called hidden Markov model with

$dS_{t}=S_{t}(a_{\theta_{t}}dt+\sigma_{\theta_{t}}dW_{t}),\qquad S_{0}=1,$

where $a_{k}\in{\mathbb{R}}$, $\sigma_{k}>0$, $k=0,...,K-1$. In this case, $X$
is of the form

$X_{t}=u+\int_{0}^{t}X_{s}dR_{s}+dP_{t}$ (1)

where $dR_{t}=a_{\theta_{t}}dt+\sigma_{\theta_{t}}dW_{t}=dS_{t}/S_{t}$, that
is $R$ is the relative price process, and

$P_{t}=ct+\int_{0}^{t}\int x\pi(dt,dx)=ct+x*\pi_{t}.$ (2)

So, the reserve evolution is described by the process
$(X^{u},\theta)=(X^{u,i},\theta^{i})$ where $u>0$ is the initial capital and
$i$ is the initial regime, i.e. the initial value of $\theta$.

We assume that $P$ is not an increasing process: otherwise the probability of
ruin is zero.

We also assume that $\Pi({\mathbb{R}})<\infty$, that is
$\Pi(dx)=\alpha_{1}F_{1}(dx)+\alpha_{2}F_{2}(dx)$ where $F_{1}(dx)$ is a
probability distribution on $]-\infty,0[$ and $F_{2}(dx)$ is a probability
distribution on $]0,\infty[$. In this case the integral with respect to the
jump measure is simply a difference of two independent compound Poisson
processes with intensities $\alpha_{1}$, $\alpha_{2}$ of jumps downwards and
upwards and whose absolute values have the distributions $F_{1}(dx)$ and
$F_{2}(dx)$, respectively.

The solution of the linear equation (1) can be represented as

$X_{t}^{u}={\cal E}_{t}(R)(u-Y_{t})=e^{V_{t}}(u-Y_{t})$ (3)

where

$Y_{t}:=-\int_{[0,t]}{\cal
E}^{-1}_{s}(R)dP_{s}=-\int_{[0,t]}e^{-V_{s}}dP_{s}=-e^{-V}\cdot P_{t},$ (4)

the stochastic exponential ${\cal E}_{t}(R)$ is equal to ${S_{t}}$, and the
log price process $V=\ln{\cal E}(R)$ admits the stochastic differential

$dV_{s}=\sigma_{\theta_{s}}dW_{s}+(a_{\theta_{s}}-(1/2)\sigma_{\theta_{s}}^{2})ds,\quad
V_{0}=0.$

Of course, $S$, $R$, $Y$, and $V$ depend on $i$ (we omitted the superscript
$i$ in the above formulae).

Let $\tau^{u,i}:=\inf\\{t>0:\ X^{u,i}_{t}\leq 0\\}$ be the instant of ruin
corresponding to the initial capital $u$ and the initial regime $i$. Then
$\Psi_{i}(u):={\bf P}[\tau^{u,i}<\infty]$ is the ruin probability and
$\Phi_{i}(u):=1-\Psi_{i}(u)$ is the survival probability. It is clear that
$\tau^{u,i}=\inf\\{t\geq 0:\ Y^{i}_{t}\geq u\\}$.

Recall that the constant parameter values $a=0$, $\sigma=0$, correspond to the
Cramér–Lundberg model for which the process $X^{u,i}=u+P_{t}$. In the
actuarial literature the compound Poisson process $P$ is usually written in
the form

$P_{t}=ct-\sum_{k=1}^{N_{t}}\xi_{k}$ (5)

where either $\xi_{k}\geq 0$, $c>0$ (i.e. $F_{2}=0$ — jumps only downwards —
the case of non-life insurance) or $\xi_{k}\leq 0$, $c<0$ (i.e. $F_{1}=0$ —
jumps only upwards — the case of life insurance or annuity payments). Models
with both kinds of jumps are also considered in the literature, see, e.g., AGY
and references therein. For the classical models with a positive average trend
and $F$ having a “non-heavy” tail, the Lundberg inequality asserts that the
ruin probability decreases exponentially as the initial capital $u$ increases
to infinity. For the exponentially distributed claims the ruin probability
admits an explicit expression, see Asm , Ch. IV.3b, or Gr , Section 1.1.

For the models with investment in a risky asset the situation is completely
different. For example, for the model with exponentially distributed jumps and
price following a geometric Brownian motion with the drift coefficient $a$ and
the volatility $\sigma>0$ in the case where $2a/\sigma^{2}-1>0$, the ruin
probability as a function of the initial capital $u$, decreases as
$Cu^{1-2a/\sigma^{2}}$. If $2a/\sigma^{2}-1\leq 0$ the ruin happens with
probability one, see FrKP , KP , PZ , KPukh .

To formulate our result for the model where the volatility and drift are
modulated by a finite-state Markov process we assume throughout this note that
except Section 7 that

$\beta_{j}:=2a_{j}/\sigma_{j}^{2}-1>0,\quad j=0,...,K-1,$ (6)

(in other words, $\beta_{*}:=\min_{j}\beta_{j}>0$).

Let ${\upsilon}^{i}_{1}:=\inf\\{t>0\colon\theta^{i}_{t-}\neq i,\
\theta^{i}_{t}=i\\}$ be the first return time of the (continuous-time) Markov
process $\theta=\theta^{i}$ to its initial state $i$. We consider further the
consequent return times defined recursively:

${\upsilon}^{i}_{k}:=\inf\\{t>{\upsilon}^{i}_{k-1}\colon\theta^{i}_{t-}\neq
i,\ \theta^{i}_{t}=i\\},\quad k=2,...$

We introduce the random variable $M_{i1}:=e^{-V_{{\upsilon}^{i}_{1}}}$ and
define the moment generating function
${\Upsilon}_{i}:{\mathbb{R}}_{+}\to\bar{\mathbb{R}}_{+}:={\mathbb{R}}_{+}\cup\\{\infty\\}$
with

${\Upsilon}_{i}(q):={\bf E}[M^{q}_{i1}].$

###### Proposition 1

The function ${\Upsilon}_{i}$ is strictly convex, continuous, and there is
unique $\gamma_{i}>0$ such that ${\Upsilon}_{i}(\gamma_{i})=1$.

Note that one can characterize $\gamma_{i}$ also as the strictly positive
roots of the cumulant generating functions $H_{i}(q):=\ln{\bf
E}e^{-V_{{\upsilon}^{i}_{1}}}$, strictly convex and continuous.

Postponing the proof of Proposition 1 to the next section we formulate our
main result:

###### Theorem 2.1

Suppose that $\Pi(|x|^{\gamma_{i}}):=\int|x|^{\gamma_{i}}\Pi(dx)<\infty$. Then

$0<\liminf_{u\to\infty}u^{\gamma_{i}}\Psi_{i}(u)\leq\limsup_{u\to\infty}u^{\gamma_{i}}\Psi_{i}(u)<\infty.$

Important remark. In the case where $\theta$ is a telegraph signal, i.e. a
two-state Markov process, the values $\gamma_{0}$ and $\gamma_{1}$ coincide
(see EK ). In the considered general case, $\gamma_{i}$ may depend on the
initial value $i$. To alleviate formulae, we fix the initial value $i=0$ and
omit the index $i=0$ when this does not lead to ambiguity.

The proof of Theorem 2.1 is based on the implicit renewal theory. To apply it,
we verify that the random variables $Q=Q_{1}:=-e^{-V}\cdot P_{{\upsilon}_{1}}$
belong to $L^{\gamma}(\Omega)$, the process $Y$ has at infinity a finite limit
$Y_{\infty}$ and show that $Y_{\infty}$ is a random variable unbounded from
above with the same law as $Q+MY_{\infty}$ where $M=M_{1}$ and ${\bf
E}[M_{1}^{\gamma}]=1$. Also ${\bf E}[M_{1}^{\gamma+{\varepsilon}}]<\infty$ for
some ${\varepsilon}>0$. Clearly, the law of the random variable $\ln M_{1}$ is
not arithmetic. We establish the bounds $\bar{G}(u)\leq\Psi_{i}(u)\leq
C\bar{G}(u)$ where $\bar{G}_{i}(u)={\bf P}[Y_{\infty}>u]$ and a constant
$C>0$, Lemma 6.

With these facts Theorem 2.1 follows from Theorem 2.2 below which is the
Kesten–Goldie theorem, see Th. 4.1 in Go91 , combined with a statement on
strict positivity of $C_{+}$ due to Guivarc’h and Le Page, BD  (for a simpler
proof of the latter see Buraczewski and Damek, BD , and an extended discussion
in Kabanov and Pergamenshchikov, KP2020 ).

###### Theorem 2.2

Let $Y_{\infty}$ has the same law as $Q+MY_{\infty}$ where $M>0$. Suppose that
$(M,Q)$ is such that the law of $\ln\,M$ is non-arithmetic and, for some
$\gamma>0$,

$\displaystyle{\bf E}[M^{\gamma}]=1,\ \ \ {\bf
E}[M^{\gamma}\,(\ln\,M)^{+}]<\infty,\ \ \ {\bf E}[|Q|^{\gamma}]<\infty.$ (7)

Then

$\displaystyle\lim_{u\to\infty}\,u^{\gamma}\,{\bf
P}[Y_{\infty}>u]=C_{+}<\infty,$
$\displaystyle\lim_{u\to\infty}\,u^{\gamma}\,{\bf
P}[Y_{\infty}<-u)]=C_{-}<\infty,$

where $C_{+}+C_{-}>0$.

If the random variable $Y_{\infty}$ is unbounded from above, then $C_{+}>0$.

##  3 Properties of the moment generating function: the proof of Proposition 1

Recall that $\tau_{n}$ are the moments of consecutive jumps of $\theta$, that
is, $\tau_{0}:=0$,

$\tau_{n}:=\inf\\{t>\tau_{n-1}\colon\theta_{t-}\neq\theta_{t}\\},\quad n\geq
1.$

We introduce the imbedded Markov chain $\vartheta_{n}:=\theta_{\tau_{n}}$,
$n=0,1,...$ with transition probabilities $P_{kl}=\lambda_{kl}/\lambda_{k}$,
$k\neq l$, and $P_{kk}=0$. Then $\varpi:=\inf\\{j\geq
2\colon\vartheta_{j}=0\\}$ is the first return time of the (discrete-time)
Markov chain $\vartheta$ to the starting point $0$ and
${\upsilon}_{1}=\tau_{\varpi}$.

Put

$Z^{j}_{t}:=\sigma_{j}W_{t}+(a_{j}-\sigma_{j}^{2}/2)t=\sigma_{j}W_{t}+(1/2)\sigma_{j}^{2}\beta_{j}t.$

The random variable $M_{1}$ admits the representation

$M_{1}=\sum_{k\geq 2}\sum_{i_{1}\neq 0,i_{2}\neq
0,...,i_{k}=0}I_{\\{\vartheta_{1}=i_{1},\vartheta_{2}=i_{2},...,\vartheta_{k}=i_{k}\\}}e^{-\zeta^{0}_{1}}e^{-\zeta^{i_{1}}_{2}}...e^{-\zeta^{i_{k-1}}_{k}}$

where

$\zeta^{0}_{1}:=Z^{0}_{\tau_{1}}-Z^{0}_{\tau_{0}},\quad\zeta^{i_{1}}_{2}:=Z^{i_{1}}_{\tau_{2}}-Z^{i_{1}}_{\tau_{1}},\quad....,\quad\zeta^{i_{k-1}}_{k}:=Z^{i_{k-1}}_{\tau_{k}}-Z^{i_{k-1}}_{\tau_{k-1}}.$

The conditional law of random variables
$\zeta^{0}_{1}$,…,$\zeta^{i_{k-1}}_{k}$ given
$\vartheta_{1}=i_{1},\vartheta_{2}=i_{2},...,\vartheta_{k}=i_{k}$ is the same
as the unconditional law of independent random variables
$\tilde{\zeta}^{0}_{1}$,…,$\tilde{\zeta}^{i_{k-1}}_{k}$. For any $m$ the law
${\cal L}(\tilde{\zeta}^{j}_{m})={\cal
L}(\sigma_{j}W_{\tau}+(1/2)\sigma_{j}^{2}\beta_{j}\tau)$ where an exponential
random variable $\tau$ with parameter $\lambda_{j}$ is independent on the
Wiener process $W$.

It follows that

${\Upsilon}(q):={\bf E}[M^{q}_{1}]=\sum_{k\geq 2}\sum_{i_{1}\neq 0,i_{2}\neq
0,...,i_{k}=0}P_{0i_{1}}P_{i_{1}i_{2}}...P_{i_{k-1}0}f_{0}(q)f_{i_{1}}(q)...f_{i_{k-1}}(q),$
(8)

where

$f_{j}(q)={\bf E}[e^{-q\tilde{\zeta}^{j}_{m}}]=\lambda_{j}{\bf
E}\left[\int_{0}^{\infty}e^{-q(\sigma_{j}W_{t}+(1/2)\sigma_{j}^{2}\beta_{j}t)}e^{-\lambda_{j}t}dt\right]=\frac{\lambda_{j}}{\lambda_{j}+(1/2)\sigma^{2}_{j}q(\beta_{j}-q)},$

if the denominator is positive, and $f_{j}(q)=\infty$ otherwise.

Clearly, $f_{j}(q)<\infty$, if $q\in[0,r_{j}[$, $f_{j}(q)=\infty$, if
$q\in[r_{j},\infty[$, and $f_{j}(r_{j}-)=\infty$, where $r_{j}$ is the
positive root of the equation

$q^{2}-\beta_{j}q-2\lambda_{j}\sigma_{j}^{-2}=0,$

that is, $r_{j}=r(\lambda_{j},\beta_{j},\sigma_{j})$,

$r(\lambda,\beta,\sigma):=\beta/2+\sqrt{\beta^{2}/4+2\lambda\sigma^{-2}}.$ (9)

Note that the formula (8) can be written in a shorter form

${\Upsilon}(q)={\bf
E}\left[\sum_{k=2}^{\infty}f_{0}(q)f_{\vartheta_{1}}(q),...,f_{\vartheta_{k-1}}(q)I_{\\{\varpi\geq
k,\,\vartheta_{k}=0\\}}\right].$ (10)

If $q\leq\beta_{*}:=\min_{j}\beta_{j}$, then all $f_{j}(q)\leq 1$ and
${\Upsilon}(q)$ is dominated the probability of return of $\theta$ to the
initial state (equal to unit), that is, $[0,\beta_{*}]\subseteq{\rm
dom}\,{\Upsilon}$. Also, $f_{j}(\beta_{*}/2)<1$ for all $j$ and, therefore,
${\Upsilon}(\beta_{*}/2)<1$. Since we assume that any state of $\theta$ can be
reached from any other state, ${\rm dom}\,{\Upsilon}\subseteq[0,r_{*}[$ where
$r_{*}:=\min_{j}r_{j}$.

More precise information gives the following lemma.

###### Lemma 1

We have ${\rm dom}\,{\Upsilon}=[0,r_{*}[$ and $\lim_{q\uparrow
r^{*}}{\Upsilon}(q)=\infty$.

Proof. To explain the idea let us consider first the case $K=3$. Regrouping
terms in the formula (8) according to 4 pairs “exit from $0$ to $l$, return
back from $m$” we get the representation

$\displaystyle{\Upsilon}(q)$ $\displaystyle=$
$\displaystyle\big{[}P_{0,1}f_{0}(q)P_{1,0}f_{1}(q)+P_{0,2}f_{0}(q)P_{2,0}f_{2}(q)+P_{0,1}P_{1,2}P_{2,0}f_{0}(q)f_{1}(q)f_{2}(q)$
$\displaystyle+P_{0,2}P_{2,1}P_{1,0}f_{0}(q)f_{2}(q)f_{1}(q)\big{]}\sum_{k=0}^{\infty}(P_{1,2}f_{1}(q)P_{2,1}f_{2}(q))^{k}.$

Note that if $P_{1,2}f_{1}(q)P_{2,1}f_{2}(q)<1$, then

$\sum_{k=0}^{\infty}(P_{1,2}f_{1}(q)P_{2,1}f_{2}(q))^{k}=\frac{1}{1-P_{1,2}f_{1}(q)P_{2,1}f_{2}(q)},$

otherwise the above sum is equal to infinity. Thus, ${\Upsilon}$ is a product
of two continuous functions with values in $\bar{\mathbb{R}}_{+}$, hence, has
the same property and the result follows.

For a model with an arbitrary $K$ we get the continuity of ${\Upsilon}$ from
the continuity result for more general functions.

Let us consider a subset $A\subset\\{0,1,...,K-1\\}$. For $i,k\notin A$ we
denote by $\Gamma^{A}_{ik}$ the set of vectors
$(i,i_{1},i_{2},\dots,i_{m},k)$, $i_{j}\in A$, $j=1,\dots,m$,
$m\in{\mathbb{N}}$. The elements of $\Gamma^{A}_{ik}$ are interpreted as cuts
of sample paths of the Markov chain entering to $A$ from the state $i$,
evolving in $A$, and living $A$ to the state $k$.

Putting $h_{i,j}(q)=P_{i,j}f_{i}(q)$ with the natural convention
$0\times\infty=0$ we associate with elements of $\Gamma^{A}_{ik}$ the
continuous functions

$q\mapsto h_{i,i_{1}}(q)h_{i_{1},i_{2}}(q)\dots
h_{i_{m-1},i_{m}}(q)h_{i_{m},k}(q)$

with values in $\bar{\mathbb{R}}_{+}$ and consider the sum of all these
functions

$U^{A}_{ik}\colon q\mapsto\sum h_{i,i_{1}}(q)h_{i_{1},i_{2}}(q)\dots
h_{i_{m-1},i_{m}}(q)h_{i_{m},k}(q).$

Since $f_{j}<1$ on the interval $]0,\beta_{*}[$, also $U^{A}_{ik}<1$ on this
interval.

We show by induction that
$U^{A}_{ik}\colon{\mathbb{R}}_{+}\to\bar{\mathbb{R}}_{+}$ is a continuous
function with $U^{A}_{ik}(0)\leq 1$. Since
${\Upsilon}=U^{A\setminus\\{0\\}}_{00}$, this gives the assertion of the
lemma.

The idea of the proof consists in representing $U^{A}_{ik}$ as a sum of finite
number of positive continuous functions using an appropriate partition of
$\Gamma^{A}_{ik}$. Namely, for $i_{1}\in A$ and $n\geq 0$ we define the set

$\Delta_{ik}^{i_{1},n}:=\\{i\\}\times\Gamma^{A\setminus\\{i_{1}\\}}_{i_{1},i_{1}}\times\dots\times\Gamma^{A\setminus\\{i_{1}\\}}_{i_{1},i_{1}}\times\Gamma^{A\setminus\\{i_{1}\\}}_{i_{1},k},$

composed by the vectors with the first component $i$, followed by $n\geq 0$
blocks formed by vectors from $\Gamma^{A\setminus\\{i_{1}\\}}_{i_{1},i_{1}}$,
and completed by vectors from $\Gamma^{A\setminus\\{i_{1}\\}}_{i_{1},k}$.
Clearly, the countable family $\Delta_{ik}^{i_{1},n}$, $i_{1}\in A$, $n\geq
0$, is a partition of $\Gamma^{A}_{ik}$ and

$U^{A}_{ik}(q)=\sum_{i_{1}\in
A}h_{i,i_{1}}(q)U^{A\setminus\\{i_{1}\\}}_{i_{1}k}(q)\sum_{n=0}^{\infty}\left[U^{A\setminus\\{i_{1}\\}}_{i_{1}i_{1}}(q)\right]^{n}.$
(11)

The result is obvious when $A$ is a singleton, i.e. $|A|=1$. Supposing that
the assertion is already proven for the case where $|A|=K_{1}-1$ we consider
the case where $|A|=K_{1}$. By the induction hypothesis
$U^{A\setminus{i_{1}}}_{i_{1}m}\colon{\mathbb{R}}_{+}\to\bar{\mathbb{R}}_{+}$
is a continuous function for every $i_{1}\in A$, $m\notin A\setminus{i_{1}}$.
The result follows from (11) and the formula for the geometric series. $\Box$

The strictly convex function ${\Upsilon}$ is less or equal to unit on
$[0,\beta_{*}]$, finite on $[0,r_{*}[$ and tends to infinity at $r_{*}$.
Hence, there is unique $\gamma\in]\beta_{*},r_{*}[$ such that
${\Upsilon}(\gamma)=1$. Moreover, ${\Upsilon}(\gamma+\epsilon)<\infty$ for
some ${\varepsilon}>0$.

##  4 Integrability of $Q_{1}$

The following identity is obvious:

$Y_{{\upsilon}_{n}}=-\sum_{k=1}^{n}\prod_{j=1}^{k-1}e^{-(V_{{\upsilon}_{j}}-V_{{\upsilon}_{j-1}})}\int_{]{\upsilon}_{k-1},{\upsilon}_{k}]}e^{-(V_{s}-V_{{\upsilon}_{k-1}})}dP_{s}.$

Using the abbreviations

$Q_{k}:=-\int_{]{\upsilon}_{k-1},{\upsilon}_{k}]}e^{-(V_{s}-V_{{\upsilon}_{k-1}})}dP_{s},\qquad
M_{j}:=e^{-(V_{{\upsilon}_{j}}-V_{{\upsilon}_{j-1}})}$

we rewrite it in a more transparent form as

$Y_{{\upsilon}_{n}}=Q_{1}+M_{1}Q_{2}+M_{1}M_{2}Q_{3}+...+M_{1}...M_{n-1}Q_{n}.$
(12)

Note that the random variables ${\upsilon}_{k}-{\upsilon}_{k-1}$, that is, the
lengths of intervals between successive returns to the initial state, form and
i.i.d. sequence. The random variables $(Q_{k},M_{k})$ have the same law and
are independent on the $\sigma$-algebra
$\sigma\\{(M_{1},Q_{1}),...,(M_{k-1},Q_{k-1})\\}$.

First, we study integrability properties of $Y$. For this we need a general
lemma involving parameters $\beta,\sigma>0$ and an independent on $W$
exponential random variable $\tau$ with parameter $\lambda>0$.

###### Lemma 2

Let $0<q<r(\lambda,\beta,\sigma)$ where $r(\lambda,\beta,\sigma)$ is given by
(9). Then

$\displaystyle C(q,\lambda,\beta,\sigma):={\bf
E}\left[\left(\int_{0}^{\tau}e^{-(\sigma W_{s}+(1/2)\sigma^{2}\beta
s)}ds\right)^{q}\right]<\infty.$

Proof. Put $W^{(\sigma\beta/2)}_{s}:=W_{s}+(1/2)\sigma\beta s$. Take
$\rho,\rho^{\prime}>1$ such that $1/\rho+1/\rho^{\prime}=1$ and $\rho
q<r(\lambda,\beta,\sigma)$. Dominating the integrant by its supremum and using
the Hölder inequality we get that

$C(q,\lambda,\beta,\sigma)\leq{\bf E}\left[\tau^{q}\sup_{s\leq\tau}e^{-\sigma
qW^{(\sigma\beta/2)}_{s}}\right]\leq\left({\bf
E}\left[\tau^{q\rho^{\prime}}\right]\right)^{1/\rho^{\prime}}\left({\bf
E}\left[\sup_{s\leq\tau}e^{-\sigma\rho
qW^{(\sigma\beta/2)}_{s}}\right]\right)^{1/\rho}.$

Since an exponential random variable has moments of any order, the first
multiplier in the right-hand side is finite. According to the formula (1.2.1)
in Ch. 2 of the reference book BS ,

${\bf E}\left[\sup_{s\leq\tau}e^{-\sigma\rho
qW^{(\sigma\beta/2)}_{s}}\right]={\bf E}\left[e^{-\sigma\rho
q\inf_{s\leq\tau}W^{(\sigma\beta/2)}_{s}}\right]=\frac{r(\lambda,\beta,\sigma)}{r(\lambda,\beta,\sigma)-\rho
q}<\infty.$

The lemma is proven. $\Box$

Note that the condition ${\Upsilon}(q)<\infty$ holds only
$q<r_{*}:=\min_{j}r(\lambda_{j},\beta_{j},\sigma_{j})$ and the above lemma
implies the following useful

###### Corollary 1

If ${\Upsilon}(q)<\infty$, then
$C^{*}(q):=\max_{j}C(q,\lambda_{j},\beta_{j},\sigma_{j})<\infty$.

###### Lemma 3

Let $q>0$ be such that ${\Upsilon}(q)<\infty$. Then

${\bf E}\left[\int_{0}^{{\upsilon}_{1}}e^{-qV_{s}}ds\right]<\infty,\qquad{\bf
E}\left[\Big{(}\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\Big{)}^{q}\right]<\infty.$
(13)

Proof. Let $\tau$ be a random variable exponentially distributed with
parameter $\lambda>0$ and let $W$ be a Wiener process independent on $\tau$.
Then

$\displaystyle{\bf E}\left[\int_{0}^{\tau}e^{-q(\sigma
W_{s}+(1/2)\sigma^{2}\beta s)}ds\right]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\bf P}[s\leq\tau]{\bf E}\left[e^{-q(\sigma
W_{s}+(1/2)\sigma^{2}\beta s)}ds\right]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{-(\lambda+(1/2)\sigma^{2}q(\beta-q))s}ds=\frac{1}{\lambda+(1/2)\sigma^{2}q(\beta-q)},$

if the denominator in the right-hand side is strictly greater than zero, and
infinity otherwise.

Using the conditioning with respect to ${\cal
F}_{k-1}=\sigma\\{\vartheta_{1},...,\vartheta_{k-1}\\}$ and the Markov
property we get from (10) with the abbreviation
$\bar{f}_{k}(q):=f_{0}(q)f_{\vartheta_{1}}(q)...f_{\vartheta_{k-1}}(q)$,
$k\geq 1$, that

$\displaystyle{\Upsilon}(q)$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{\infty}{\bf
E}\left[\bar{f}_{k}(q)I_{\\{\varpi\geq{k},\vartheta_{k}=0\\}}\right]=\sum_{k=2}^{\infty}{\bf
E}\left[\bar{f}_{k}(q)I_{\\{\varpi\geq{k}\\}}{\bf
E}[I_{\\{\vartheta_{k}=0\\}}|{\cal F}_{k-1}]\right]$ $\displaystyle=$
$\displaystyle{\bf
E}\left[\sum_{k=2}^{\infty}\bar{f}_{k}(q)I_{\\{\varpi\geq{k}\\}}{\bf
E}[I_{\\{\vartheta_{k}=0\\}}|\vartheta_{k-1}]\right]\geq p_{*}{\bf
E}\left[\sum_{k=2}^{\infty}\bar{f}_{k}(q)I_{\\{\varpi\geq{k}\\}}\right]$

where $p_{*}>0$ is the minimal of values $P_{j,0}$ different from zero.

Note that

$\sum_{j=2}^{\varpi}\bar{f}_{j}(q)=\sum_{j=2}^{\infty}I_{\\{\varpi=j\\}}\sum_{k=2}^{j}\bar{f}_{k}(q)=\sum_{k=2}^{\infty}\bar{f}_{k}(q)\sum_{j=k}^{\infty}I_{\\{\varpi=j\\}}=\sum_{k=2}^{\infty}\bar{f}_{k}(q)I_{\\{\varpi\geq{k}\\}}.$

It follows that

${\bf E}\left[\sum_{j=2}^{\varpi}\bar{f}_{j}(q)\right]={\bf
E}\left[\sum_{k=2}^{\infty}\bar{f}_{k}(q)I_{\\{\varpi\geq{k}\\}}\right]\leq\frac{1}{p_{*}}{\Upsilon}(q).$

Since

$\displaystyle{\bf E}\left[\int_{0}^{{\upsilon}_{1}}e^{-qV_{s}}ds\right]$
$\displaystyle=$ $\displaystyle{\bf
E}\left[\sum_{k=2}^{\infty}I_{\\{\varpi=k\\}}\sum_{j=0}^{k-1}e^{-qV_{\tau_{j}}}\int_{\tau_{j}}^{\tau_{j+1}}e^{-q(V_{s}-V_{\tau_{j}})}ds\right]$
$\displaystyle=$ $\displaystyle{\bf
E}\left[\sum_{k=2}^{\infty}I_{\\{\varpi=k\\}}\sum_{j=1}^{k}\bar{f}_{j}(q)\frac{1}{\lambda_{\vartheta_{j-1}}}\right]\leq\frac{1}{\lambda_{*}}\left(f_{0}(q)+{\bf
E}\left[\sum_{j=2}^{\varpi}\bar{f}_{j}(q)\right]\right),$

where $\lambda_{*}:=\min_{i}\lambda_{i}$, we obtain from the above
inequalities that

${\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{-qV_{s}}ds\right]\leq\frac{f_{0}(q)}{\lambda_{*}}+\frac{1}{\lambda_{*}p_{*}}{\Upsilon}(q)<\infty.$
(14)

The first integrability property of (13) is proven.

To prove the second property of (13) we start with the case $q\leq 1$. Using
the elementary inequality $(\sum|x_{i}|)^{q}\leq\sum|x_{i}|^{q}$ and Corollary
1 we get that

$\displaystyle{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\right)^{q}\right]$
$\displaystyle\leq$ $\displaystyle{\bf
E}\left[\sum_{k=2}^{\infty}I_{\\{\varpi=k\\}}\sum_{j=0}^{k-1}e^{-qV_{\tau_{j}}}\left(\int_{\tau_{j}}^{\tau_{j+1}}e^{-(V_{s}-V_{\tau_{j}})}ds\right)^{q}\right]$
$\displaystyle=$ $\displaystyle{\bf
E}\left[\sum_{k=2}^{\infty}I_{\\{\varpi=k\\}}\sum_{j=0}^{k-1}\bar{f}_{k}(q){\bf
E}\left[\left(\int_{\tau_{j}}^{\tau_{j+1}}e^{-(V_{s}-V_{\tau_{j}})}ds\right)^{q}\right]\right]$
$\displaystyle\leq$ $\displaystyle C^{*}(q){\bf
E}\left[\sum_{k=2}^{\infty}I_{\\{\varpi=k\\}}\sum_{j=0}^{k-1}\bar{f}_{k}(q)\right]$
$\displaystyle\leq$ $\displaystyle C^{*}(q)\left(1+f_{0}(q)+{\bf
E}\left[\sum_{k=2}^{\varpi}\bar{f}_{k}(q)\right]\right)$ $\displaystyle\leq$
$\displaystyle
C^{*}(q)\left(1+f_{0}(q)+\frac{1}{p_{*}}{\Upsilon}(q)\right)<\infty.$

Let $q>1$. Due to the continuity of ${\Upsilon}$ there exists $q^{\prime}>q$
such that ${\Upsilon}(q^{\prime})<\infty$. Applying, first, the Hölder
inequality with the conjugate exponents $q$ and $p:=q/(q-1)$ and then the
Young inequality for products with the conjugate exponents $q^{\prime}/q$ and
$q^{\prime}/(q^{\prime}-q)$, we get that

$\displaystyle{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\right)^{q}\right]$
$\displaystyle\leq$ $\displaystyle{\bf
E}\left[\int_{0}^{{\upsilon}_{1}}{\upsilon}_{1}^{q-1}e^{-qV_{s}}ds\right]$
$\displaystyle\leq$ $\displaystyle(1-q/q^{\prime}){\bf
E}\left[{\upsilon}_{1}^{q_{1}}\right]+(q/q^{\prime}){\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{-q^{\prime}V_{s}}ds\right],$

where $q_{1}:=(q-1)q^{\prime}/(q^{\prime}-q)$. The expectation of the integral
in the right-hand side is finite in virtue of the first inequality in (13),
applied with $q^{\prime}$. It remains to recall that the first return time of
the finite state Markov process $\theta$ has moments of any order.

For the reader convenience we give the proof of the above fact. Take an
arbitrary $m>1$ and let denote by $\varrho_{j}:=\tau_{j}-\tau_{j-1}$ the
interjump times of the Markov process $\theta$. Recall that the conditional
distribution of the vector $(\varrho_{1},\dots,\varrho_{k})$ given
$\vartheta_{1}=i_{1},\dots,\vartheta_{k-1}=i_{k-1}$ is the same as the
distribution of the vector $\tilde{\varrho}_{1},\dots,\tilde{\varrho}_{k})$
with independent components having, respectively, exponential distributions
with the parameters $\lambda_{0}$, $\lambda_{i_{1}}$ …, $\lambda_{i_{k-1}}$.
Using the Hölder inequality (now for the sum) and this fact we get that

$\displaystyle{\bf E}\left[{\upsilon}_{1}^{m}\right]$ $\displaystyle=$
$\displaystyle{\bf
E}\left[\left(\sum_{j=1}^{\varpi}\varrho_{j}\right)^{m}\right]\leq{\bf
E}\left[\varpi^{m-1}\sum_{j=1}^{\varpi}\varrho_{j}^{m}\right]$
$\displaystyle=$ $\displaystyle{\bf E}\left[\sum_{k\geq 2}\sum_{i_{1}\neq
0,i_{2}\neq
0,...,i_{k}=0}I_{\\{\vartheta_{1}=i_{1},\dots,\vartheta_{k-1}=i_{k-1},\vartheta_{k}=0\\}}k^{m-1}\sum_{j=1}^{k}\varrho_{j}^{m}\right]$
$\displaystyle=$ $\displaystyle\Gamma(m+1){\bf
E}\left[\varpi^{m-1}\left(\sum_{j=1}^{\varpi}\frac{1}{\lambda_{\vartheta_{j-1}}}\right)^{m}\right]\leq\Gamma(m+1)\lambda_{*}^{-m}{\bf
E}\left[\varpi^{m}\right]$

where $\Gamma$ is the Gamma function and $\lambda_{*}:=\min_{j}\lambda_{j}$.
It remains to make a reference to the fact that the first return time $\varpi$
for the Markov chain $\vartheta$ has moments of any order, see, e.g. F , Ch.
XV, exer. 18-20. $\Box$

The following lemma provides the required integrability property of $Q_{1}$.

###### Lemma 4

Suppose that $\Pi(|x|^{\gamma}):=\int|x|^{\gamma}\Pi(dx)<\infty$. Then ${\bf
E}\left[|Q_{1}|^{\gamma}\right]<\infty$.

Proof. Case where $\gamma\leq 1$. The inequality
$(|x|+|y|)^{\gamma}\leq|x|^{\gamma}+|y|^{\gamma}$ allows us to check
separately finiteness of moments of the integral of $e^{-V}$ with respect to
the Lebesgue mesure (this is already done, see the second property in (13))
and the integral with respect to the jump component of the process $P$. The
latter integral is just a sum. Since the jump measure $\pi(dt,dx)$ has the
compensator $\tilde{\pi}(dt,dx)=\Pi(dx)dt$, we have that

$\displaystyle{\bf E}\left[|e^{-V}x*\pi_{{\upsilon}_{1}}|^{\gamma}\right]$
$\displaystyle\leq$ $\displaystyle{\bf E}\left[e^{-\gamma
V}|x|^{\gamma}*\pi_{{\upsilon}_{1}}\right]={\bf E}\left[e^{-\gamma
V}|x|^{\gamma}*\tilde{\pi}_{{\upsilon}_{1}}\right]$ $\displaystyle\leq$
$\displaystyle\Pi(|x|^{\gamma}){\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{-\gamma V_{s}}ds\right]<\infty$

in virtue of the first property in (13).

Case where $\gamma>1$. Now we shall split integrals using the elementary
inequality

$(|x|+|y|)^{\gamma}\leq 2^{\gamma-1}(|x|^{\gamma}+|y|^{\gamma}).$

Because of the second property in (13), we need to consider only the integral
with respect to the jump component of $P$. Note that
$e^{-V}|x|*\tilde{\pi}_{{\upsilon}_{1}}<\infty$. Then

${\bf E}\left[\left(e^{-V}|x|*\pi_{{\upsilon}_{1}}\right)^{\gamma}\right]\leq
2^{\gamma-1}\left({\bf
E}\left[\big{|}e^{-V}|x|*(\pi-\tilde{\pi})_{{\upsilon}_{1}}\big{|}^{\gamma}\right]+{\bf
E}\left[\left(e^{-V}|x|*\tilde{\pi}_{{\upsilon}_{1}}\right)^{\gamma}\right]\right).$

Due to the first property in (13)

${\bf
E}\left[\left(e^{-V}|x|*\tilde{\pi}_{{\upsilon}_{1}}\right)^{\gamma}\right]\leq\big{(}\Pi(|x|)\big{)}^{\gamma}{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\right)^{\gamma}\right]<\infty.$

Let $I_{s}:=e^{-V}|x|*(\pi-\tilde{\pi})_{s}$. According to the Novikov
inequalities111 See Novikov  and a discussion in KP2020 . with $\alpha=1$ the
moment of the order $\gamma>1$ of the random variable $I^{*}_{t}:=\sup_{s\leq
t}|I_{s}|$ admits the bound

$\displaystyle{\bf E}\left[I^{*\gamma}_{{\upsilon}_{1}}\right]$
$\displaystyle\leq$ $\displaystyle C_{\gamma,1}\left({\bf
E}\left[(e^{-V}|x|*\tilde{\pi}_{{\upsilon}_{1}})^{\gamma}\right]+{\bf
E}\left[e^{-\gamma V}|x|^{\gamma}*\tilde{\pi}_{{\upsilon}_{1}}\right]\right)$
$\displaystyle\leq$ $\displaystyle C^{\prime}_{\gamma,1}{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\right)^{\gamma}\right]+C^{\prime}_{\gamma,1}{\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{-\gamma V_{s}}ds\right]$

where $C^{\prime}_{\gamma,1}:=C_{\gamma,1}(\Pi(|x|))^{\gamma}<\infty$,
$C^{\prime\prime}_{\gamma,1}:=C_{\gamma,1}\Pi(|x|^{\gamma})<\infty$ due to our
assumption. The both integrals in the right-hand side, as we proved, are
finite. $\Box$

##  5 Study of the process $Y$

###### Lemma 5

The process $Y$ has the following properties:

$(i)$ $Y_{t}$ converges almost surely as $t\to\infty$ to a finite random
variable $Y_{\infty}$.

$(ii)$ $Y_{\infty}=Q_{1}+M_{1}Y_{1,\infty}$ where $Y_{1,\infty}$ is a random
variable independent on $(Q_{1},M_{1})$ and having the same law as
$Y_{\infty}$.

$(iii)$ $Y_{\infty}$ is unbounded from above.

Proof. $(i)$ Take $p\in]0,\gamma\wedge 1[$. Then $r:={\bf
E}\left[M_{1}^{p}\right]<1$ and, Lemma 4, ${\bf
E}\left[|Q_{1}|^{p}\right]<\infty$. It follows that ${\bf
E}\left[|Y_{{\upsilon}_{n+1}}-Y_{{\upsilon}_{n}}|^{p}\right]={\bf
E}\left[M_{1}^{p}\dots M_{n}^{p}Q_{n+1}^{p}\right]=r^{n}{\bf
E}\left[|Q_{1}|^{p}\right]$ and, therefore,

${\bf E}\left[\left(\sum_{n\geq
0}|Y_{{\upsilon}_{n+1}}-Y_{{\upsilon}_{n}}|\right)^{p}\right]\leq\sum_{n\geq
0}{\bf E}\left[|Y_{{\upsilon}_{n+1}}-Y_{{\upsilon}_{n}}|^{p}\right]<\infty.$

Thus, $\sum_{n}|Y_{{\upsilon}_{n+1}}-Y_{{\upsilon}_{n}}|<\infty$ a.s. implying
that $Y_{{\upsilon}_{n}}$ converges a.s. to some finite random variable we
shall denote $Y_{\infty}$.

Let $Y_{t}^{*}:=\sup_{s\leq t}|Y_{s}|$. Then

${\bf E}\left[Y^{*p}_{{\upsilon}_{1}}\right]\leq c^{p}{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\right)^{p}\right]+(\Pi(|x|))^{p}{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{-pV_{s}}ds\right)^{p}\right]<\infty.$

Put

$\Delta_{n}=\sup_{v\in[{\upsilon}_{2n},{\upsilon}_{n+1}]}\Big{|}\int_{{\upsilon}_{n}}^{v}e^{-V_{s}}dP_{s}\Big{|}.$

Then

${\bf E}\left[\Delta_{n}^{p}\right]={\bf E}\left[M_{1}^{p}\dots
M_{n}^{p}\sup_{v\in[{\upsilon}_{n},{\upsilon}_{n+1}]}\left|\int_{{\upsilon}_{n}}^{v}e^{-(V_{s}-V_{{\upsilon}_{n}})}dP_{s}\right|^{p}\right]\leq
r^{n}{\bf E}\left[Y^{*p}_{{\upsilon}_{1}}\right]$

and, therefore, for any ${\varepsilon}>0$

$\sum_{n\geq 0}{\bf P}[\Delta_{n}\geq{\varepsilon}]\leq e^{-p}\sum_{n\geq
0}{\bf E}\left[\Delta_{n}^{p}\right]<\infty.$

By the Borel–Cantelli lemma for all $\omega$ except a null-set
$\Delta_{n}(\omega)\leq{\varepsilon}$ for all $n\geq n(\omega)$. This implies
that $Y_{t}$ converges a.s. to the same limit as the sequence
$Y_{{\upsilon}_{n}}$.

$(ii)$ Rewriting (12) in the form

$Y_{{\upsilon}_{n}}=Q_{1}+M_{1}(Q_{2}+M_{2}Q_{3}+...+M_{2}...M_{n-1}Q_{n})$

and observing that the sequence of random variables in the parentheses
converges almost surely to a random variable with same law as $Y_{\infty}$ and
independent on $(Q_{1},M_{1})$ we get the needed assertion.

$(iii)$ In virtue of $(ii)$ it is sufficient to check that the set
$\\{Q_{1}\geq N,\ M_{1}\leq 1/N\\}$ is non-null whatever is $N\geq 1$.

Recall that

$Q_{1}=-e^{-V}x*\pi_{{\upsilon}_{1}}-c\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds,$

where
$dV_{s}=\sigma_{\theta_{s}}dW_{s}+(1/2)\sigma^{2}_{\theta_{s}}\beta_{\theta_{s}}ds$.

We consider several cases.

$1)$ $c<0$. Using conditioning with respect to $\theta$ we may argue as
$\theta$ would be deterministic, i.e. assuming that $V$ is a process with a
deterministic switching of parameters and ${\upsilon}_{1}$ is just a number,
say, $t>0$. On the set $\\{T_{1}>{\upsilon}_{1}\\}$ we have
$Q_{1}=-c\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds$. Since $T_{1}$ is independent
on $W$ and the set $\\{T_{1}>{\upsilon}_{1}\\}$ we need to check only that the
set

$B_{N}(t):=\left\\{-c\int_{0}^{t}e^{-V_{s}}ds\geq N,\ e^{-V_{t}}\leq
1/N\right\\}$

is non-null. In the case where $\theta$ has no jumps on $[0,t]$ the process
$V_{s}=\sigma_{0}W_{t}+(1/2)\sigma^{2}_{0}\beta_{0}t$ we get the latter
property using conditioning with respect to $W_{t}=x$. Indeed, the conditional
distribution of $(W_{s})_{s\leq t}$ given $W_{t}=x$ is the same as the
(unconditional) distribution of the Brownian bridge $B^{x}=(B^{x})_{s\leq t}$
ending at $t$ at the value $x$. The latter is a continuous Gaussian process.
This implies that the conditional distribution of the integral involved in
$B_{N}(t)$ is unbound from above. Integrating over a suitable set with respect
to the distribution of $W_{t}$ shows that $B_{N}(t)$ is non-null. In the case
of several jumps at the moments $t_{1},\dots t_{k}$ we can show that the
integral over the interval $[0,t_{1}]$ has unbounded conditional distribution
given
$(W_{t_{1}},W_{t_{2}}-W_{t_{1}},\dots,W_{t}-W_{t_{k}})=(x_{t_{1}},x_{t_{2}},\dots
x_{t_{k}+1})$ and conclude by integrating with respect to the distribution of
the increments of $W$ over a set $[x,\infty[^{k+1}$ for sufficiently large
$x\in{\mathbb{R}}_{+}$.

$2)$ $c\geq 0$. Put $\sigma^{*}:=\max_{j}\sigma_{j}$,
$\kappa^{*}:=\max_{j}(1/2)\sigma_{j}^{2}\beta_{j}$,
$\kappa_{*}:=\min_{j}(1/2)\sigma_{j}^{2}\beta_{j}$. Let $\delta>0$ and let
$r_{N}:=(\sigma^{*}\delta+\ln N)/\kappa_{*}$. The set

$A_{N}:=\\{|W_{s}|<\delta,\ \forall\,s\leq
r_{N}+1\\}\cap\\{r_{N}\leq{\upsilon}_{1}\leq r_{N}+1\\}$

is non-null. On this set for all $s\in[0,{\upsilon}_{1}]$ we have the bounds

$-\sigma^{*}\delta+\kappa_{*}r_{N}\leq
V_{s}\leq\sigma^{*}\delta+\kappa^{*}(r_{N}+1)$

implying that

$M_{1}=e^{-V_{{\upsilon}_{1}}}\leq e^{\sigma^{*}\delta-\kappa_{*}r_{N}}=1/N$

and

$c\int_{0}^{{\upsilon}_{1}}e^{-V_{s}}ds\leq(c/N)(r_{N}+1)=:C_{N}.$

Since $P$ is not an increasing process, $\Pi(]-\infty,0[)>0$. Hence, the set

$\Big{\\{}e^{-\sigma^{*}\delta-\kappa^{*}(r_{N}+1)}|x|I_{\\{x<0\\}}*\pi_{r_{N}}\geq
C_{N}+N,\ xI_{\\{x>0\\}}*\pi_{r_{N}}=0\Big{\\}}$

is non-null and its intersection with $A_{N}$ is also non-null. But this
intersection is a subset of the set $\\{Q_{1}\geq N,\ M_{1}\leq 1/N\\}$.
$\Box$

##  6 Bounds for the ruin probability

###### Lemma 6

For every $u>0$

$\bar{G}_{i}(u)\leq\,\Psi_{i}(u)=\frac{\bar{G}_{i}(u)}{{\bf
E}\big{[}\bar{G}_{\theta_{\tau^{u,i}}}(0)|\tau^{u,i}<\infty\big{]}}\leq\frac{\bar{G}_{i}(u)}{\min_{j}\bar{G}_{j}(0)},$
(15)

where $\bar{G}_{i}(u):={\bf P}[Y_{\infty}^{i}>u]$.

Proof. Let $\tau$ be an arbitrary stopping time with respect to the filtration
$({\cal F}^{P,R,\theta}_{t})$. As the finite limit $Y^{i}_{\infty}$ exists,
the random variable

$Y_{\tau,\infty}^{i}:=\begin{cases}-\lim_{N\to\infty}\int_{]\tau,\tau+N]}\,e^{-(V_{s}-V_{\tau})}dP_{s},&\tau<\infty,\\\
0,&\tau=\infty,\end{cases}$

is well defined. On the set $\\{\tau<\infty\\}$

$Y_{\tau,\infty}^{i}=e^{V_{\tau}}(Y^{i}_{\infty}-Y^{i}_{\tau})=X_{\tau}^{u}+e^{V_{\tau}}(Y^{i}_{\infty}-u).$
(16)

Let $\xi$ be a ${\cal F}_{\tau}^{P,R,\theta}$-measurable random variable. Note
that the conditional distribution of $Y_{\tau,\infty}^{i}$ given
$(\tau,\xi,\theta_{\tau})=(t,x,j)\in{\mathbb{R}}_{+}\times{\mathbb{R}}\times\\{0,1,K-1\\}$
is the same as the distribution of $Y_{\infty}^{j}$. It follows that

${\bf P}\big{[}Y_{\tau,\infty}^{i}>\xi,\ \tau<\infty,\
\theta_{\tau}=j\big{]}={\bf
E}\left[\bar{G}_{j}(\xi)\,I_{\\{\tau<\infty,\;\theta_{\tau}=j\\}}\right].$

Thus, if ${\bf P}[\tau<\infty]>0$, then

${\bf P}\big{[}Y_{\tau,\infty}^{i}>\xi,\ \tau<\infty\big{]}={\bf
E}\big{[}\bar{G}_{\theta_{\tau}}(\xi)\,|\,\tau<\infty\big{]}\,{\bf
P}\big{[}\tau<\infty\big{]}\,.$

Noting that $\Psi_{i}(u):={\bf P}\big{[}\tau^{u,i}<\infty\big{]}\geq{\bf
P}\big{[}Y^{i}_{\infty}>u\big{]}=\bar{G}_{i}(u)>0$, we deduce from here using
(16) that

$\displaystyle\bar{G}_{i}(u)$ $\displaystyle={\bf P}\big{[}Y^{i}_{\infty}>u,\
\tau^{u,i}<\infty\big{]}={\bf
P}\big{[}Y_{\tau^{u,i},\infty}^{i}>X_{\tau^{u,i}}^{u,i},\
\tau^{u,i}<\infty\big{]}$ $\displaystyle={\bf
E}\big{[}\bar{G}_{\theta_{\tau^{u,i}}}(X_{\tau^{u,i}}^{u,i})\,|\,\tau^{u,i}<\infty\big{]}\,{\bf
P}\big{[}\tau^{u,i}<\infty\big{]}$

implying the equality in (15). Also,

${\bf
E}\big{[}\bar{G}_{\theta_{\tau^{u,i}}}(0)|\tau^{u,i}<\infty\big{]}=\sum_{j=0}^{K-1}\bar{G}_{j}(0){\bf
P}\big{[}\theta_{\tau^{u,i}}=j|\tau^{u,i}<\infty\big{]}\leq{\min_{j}\bar{G}_{j}(0)}$

implying the result. $\Box$

##  7 Ruin with probability one

Assuming that $\beta^{*}:=\max_{j}\beta_{j}$ is strictly negative, we give a
sufficient condition under which the ruin is imminent.

###### Theorem 7.1

Suppose that $\beta^{*}<0$, $\Pi(]-\infty,-{\varepsilon}])>0$ for all
${\varepsilon}>0$, and there exists $\delta\in]0,|\beta^{*}|\wedge 1[$ for
which $\Pi(|x|^{\delta})<\infty$. Then $\Psi_{i}(u)=1$ for any $u>0$ and $i$.

Proof. Put $\widetilde{X}_{n}=\widetilde{X}^{i}_{n}:=X^{i}_{\upsilon_{n}}$.
Note that (3) implies that the sequence $\widetilde{X}_{n}$ satisfies the
difference equation

${\tilde{X}}_{n}={A}_{n}{\tilde{X}}_{n-1}+{B}_{n},\qquad n\geq 1,\
{\tilde{X}}_{0}=u,$ (17)

where ${A}_{n}:=M^{-1}_{n}:=e^{V_{{\upsilon}_{n}}-V_{{\upsilon}_{n-1}}}$ and

${B}_{n}:=-M^{-1}_{n}\,Q_{n}=\int_{]{\upsilon}_{n-1},{\upsilon}_{n}]}e^{V_{{\upsilon}_{n}}-V_{s}}dP_{s}.$

According to Corollary 6.2 in KP2020 , $\inf_{n}\widetilde{X}_{n}<0$ a.s. if
the ratio $B_{1}/A_{1}$ is unbounded from below and there is $\delta\in]0,1[$
such that ${\bf E}[A^{\delta}_{1}]<1$ and ${\bf E}[|B_{1}|^{\delta}]<\infty$.
By our assumption the event that on a fixed finite interval the process $P$
has arbitrary many downward jumps of the size larger than ${\varepsilon}$ and
has no jumps upward is of strictly positive probability. Due to independence
of $P$ and $(W,\theta)$ this implies that $-Q_{1}=B_{1}/A_{1}$ is unbounded
from below.

Noting that

$\tilde{f}_{j}(\delta):=\frac{\lambda_{j}}{\lambda_{j}+(1/2)\sigma^{2}_{j}\delta(|\beta_{j}|-\delta)}<1,$
(18)

we get that

${\bf E}[A^{\delta}_{1}]={\bf E}\big{[}e^{\delta
V_{{\upsilon}_{1}}}\big{]}={\bf E}\left[e^{\delta
V_{\tau_{1}}}\prod_{i=2}^{\varpi}e^{\delta(V_{\tau_{i}}-V_{\tau_{i-1}})}\right]={\bf
E}\left[\sum_{i=2}^{\varpi}\tilde{f}_{0}(\delta)\tilde{f}_{\theta_{1}}(\delta)\dots
f_{\theta_{i-1}}(\delta)\right]<1.$

Finally, the property ${\bf E}[|B_{1}|^{\delta}<\infty$ can be proved by the
same arguments as in the proof of Lemma 4 with $\gamma$ and $V$ replaced by
$\delta$ and $V_{{\upsilon}_{1}}-V$ and the reference to (13) replaced by the
reference to (19) in Lemma 7 below. $\Box$

###### Lemma 7

Suppose that $\beta^{*}<0$. Then for any $\delta\in]0,|\beta^{*}|[$

${\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{\delta(V_{\upsilon_{1}}-V_{s}})ds\right]<\infty,\quad\quad{\bf
E}\left[\left(\int_{0}^{{\upsilon}_{1}}e^{(V_{\upsilon_{1}}-V_{s})}ds\right)^{\delta}\right]<\infty.$
(19)

Proof. The arguments are very similar to that of Lemma 3 and we only sketch
them. The only new feature is that we need to consider processes of the form
$(V_{T}-V_{s})_{s\in[0,T]}$ rather than $(V_{s})_{s\in[0,T]}$. The crucial
observation is that the process $(W_{T}-W_{s})_{s\in[0,T]}$ in the reversed
time $s^{\prime}:=T-s$ is a Wiener process.

First, observe that

${\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{\delta(V_{\upsilon_{1}}-V_{s})}ds\right]={\bf
E}\left[\sum^{\varpi}_{k=1}e^{\delta(V_{\upsilon_{1}}-V_{\tau_{k}})}\int_{\tau_{k-1}}^{\tau_{k}}e^{\delta(V_{\tau_{k}}-V_{s})}ds\right]$

Given a trajectory of $\theta$, the exponential and the integral in each
summand are conditionally independent and their conditional expectations admit
explicit expressions. For the integral it is
$1/\lambda_{\vartheta_{k-1}}\tilde{f}_{\vartheta_{k-1}}(\delta)$ where
$\tilde{f}_{j}$ in (18). Note that for $\delta\in]0,|\beta^{*}|[$ the
conditional expectation of the integral is dominated by $1/\lambda_{*}$
implying that

$\displaystyle{\bf
E}\left[\int_{0}^{{\upsilon}_{1}}e^{\delta(V_{\upsilon_{1}}-V_{s})}ds\right]$
$\displaystyle\leq$ $\displaystyle\frac{1}{\lambda_{*}}{\bf
E}\left[\sum^{\varpi}_{k=1}e^{\delta(V_{\tau_{\varpi}}-V_{\tau_{k}})}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{\lambda_{*}}{\bf
E}\left[\sum^{\varpi}_{k=1}\prod_{n=k}^{\varpi-1}e^{\delta(V_{\tau_{n+1}}-V_{\tau_{n}})}\right]=\frac{1}{\lambda_{*}}{\bf
E}\left[\sum^{\varpi}_{k=1}\prod_{n=k}^{\varpi-1}\tilde{f}_{\theta_{n}}(\delta)\right].$

Due to the choice of $\beta$, we have that
$\tilde{f}^{*}:=\max_{j}\tilde{f}_{j}(\delta)<1$ and, therefore,

${\bf
E}\left[\sum^{\varpi}_{k=1}\prod_{n=k}^{\varpi-1}\tilde{f}_{\theta_{n}}(\delta)\right]\leq{\bf
E}\left[\sum^{\varpi}_{k=1}(\tilde{f}^{*})^{\varpi-k}\right]\leq\sum_{k=1}^{\infty}(\tilde{f}^{*})^{k}<\infty.$

The first property in (19) is proven.

Let $\tau$ be an exponential random variable with parameter $\lambda>0$. For
any $\delta\in]0,\widetilde{r}[$, where

$\widetilde{r}:=\sqrt{{2\lambda}/{\sigma^{2}}+{\beta^{2}}/{4}}+|\beta|/{2},$

we have, according to (1.1.1) in Ch. 2 of the reference book BS , that

$\tilde{C}(\delta,\lambda,\beta,\sigma):={\bf
E}\left[e^{\delta\sigma\sup_{s\leq\tau}W^{(\sigma\beta/2)}_{s}}\right]=\frac{\widetilde{r}}{\widetilde{r}-\delta}<\infty.$

We get (as in Corollary 1) that for all $k\geq 1$

${\bf
E}\left[\left(\int_{\tau_{k-1}}^{\tau_{k}}e^{V_{\tau_{k}}-V_{s}}ds\right)^{\delta}\right]\leq\tilde{C}^{*}(\delta),$

with some constant $\tilde{C}^{*}(\delta)<\infty$, and we complete the proof
of the second property in (19) as in Lemma 3. $\Box$

Acknowledgement. This work was supported by the Russian Science Foundation
associated grants 20-68-47030 and 20-61-47043.

## References

  * (1) Albrecher, H., Gerber, H., Yang, H.: A direct approach to the discounted penalty function. North American Actuarial Journal, 14, 4, 420–434 (2010)
  * (2) Anderson, W.J.: Continuous-time Markov chains. An Applications-Oriented Approach. Springer, Berlin (1991)
  * (3) Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific, Singapore (2010)
  * (4) Borodin, A.N., Salminen, P.: Handbook of Brownian Motion – Facts and Formulae. 2nd Edition, Basel–Boston–Berlin, Springer Basel AG, Birkhaüser (2002)
  * (5) Buraczewski, D., Damek, E.: A simple proof of heavy tail estimates for affine type Lipschitz recursions. Stochastic processes and their applications, 127, 657 – 668 (2017)
  * (6) Di Masi, G.B., Kabanov, Yu.M., Runggaldier, W.J.: Mean-square hedging of options on a stock with Markov volatilities. Theory Probab. Appl., 39, 1, 172–182 (1994)
  * (7) Ellanskaya, A., Kabanov, Yu.: On ruin probabilities with risky investments in a stock with stochastic volatility. Extremes, DOI 10.1007/s10687-021-00420-8
  * (8) Feller, W.: An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition, Wiley (1991)
  * (9) Frolova, A., Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous. Finance and Stochastics, 6, 2, 227 – 235 (2002)
  * (10) Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1, 1, 126-166 (1991)
  * (11) Grandell, I.: Aspects of Risk Theory. Springer, Berlin (1990)
  * (12) Guivarc’h, Y., Le Page, E.: On the homogeneity at infinity of the stationary probability for affine random walk. In: Bhattacharya S., Das T., Ghosh A., Shah R. (eds). “Recent trends in ergodic theory and dynamical systems”. Contemporary Mathematics AMS, 119 – 130 (2015)
  * (13) Kabanov, Yu., Pergamenshchikov, S.: In the insurance business risky investments are dangerous: the case of negative risk sums. Finance and Stochastics, 20, 2, 355 – 379 (2016)
  * (14) Kabanov, Yu., Pergamenshchikov, S.: Ruin probabilities for a Lévy-driven generalized Ornstein–Uhlenbeck process. Finance and Stochastics, 24, 1, 39 – 69 (2020)
  * (15) Kabanov, Yu., Pukhlyakov, N.: Ruin probabilities with investments: smoothness, IDE and ODE, asymptotic behavior. Preprint (2020). https://arxiv.org/abs/2011.07828
  * (16) Novikov, A.A.: On discontinuous martingales, Theory of Probability and its Applications, 20, 11 – 26 (1975)
  * (17) Paulsen, J.: Risk theory in stochastic economic environment. Stochastic processes and their applications, 46, 327 – 361 (1993)
  * (18) Paulsen, J.: Sharp conditions for certain ruin in a risk process with stochastic return on investments, Stoch. Process. Appl., 75, 135 – 148 (1998)
  * (19) Paulsen, J.: On Cramér-like asymptotics for risk processes with stochastic return on investments. Annals of Applied Probability, 12, 1247 – 1260 (2002)
  * (20) Paulsen, J., Gjessing, H. K.: Ruin theory with stochastic return on investments. Advances in Applied Probability, 29, 965 – 985 (1997)
  * (21) Pergamenshchikov, S., Zeitouni, O.: Ruin probability in the presence of risky investments. Stoch. Process. Appl., 116, 267–278 (2006). Erratum to: “Ruin probability in the presence of risky investments”. Stoch. Process. Appl., 119, 305 – 306 (2009)