An Analysis and Implementation of the Hidden Markov Model to Technology Stock Prediction*Risks* **2017**, *5*(4), 62; doi:10.3390/risks5040062 - 24 November 2017**Abstract **

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Future stock prices depend on many internal and external factors that are not easy to evaluate. In this paper, we use the Hidden Markov Model, (HMM), to predict a daily stock price of three active trading stocks: Apple, Google, and Facebook, based on

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Future stock prices depend on many internal and external factors that are not easy to evaluate. In this paper, we use the Hidden Markov Model, (HMM), to predict a daily stock price of three active trading stocks: Apple, Google, and Facebook, based on their historical data. We first use the Akaike information criterion (AIC) and Bayesian information criterion (BIC) to choose the numbers of states from HMM. We then use the models to predict close prices of these three stocks using both single observation data and multiple observation data. Finally, we use the predictions as signals for trading these stocks. The criteria tests’ results showed that HMM with two states worked the best among two, three and four states for the three stocks. Our results also demonstrate that the HMM outperformed the naïve method in forecasting stock prices. The results also showed that active traders using HMM got a higher return than using the naïve forecast for Facebook and Google stocks. The stock price prediction method has a significant impact on stock trading and derivative hedging.
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Bounded Brownian Motion*Risks* **2017**, *5*(4), 61; doi:10.3390/risks5040061 - 17 November 2017**Abstract **

Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed to describe ratios of asset prices under a martingale measure. We provide a simple example of a tractable driftless diffusion which also has a bounded state space.
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Optimal Claiming Strategies in Bonus Malus Systems and Implied Markov Chains*Risks* **2017**, *5*(4), 58; doi:10.3390/risks5040058 - 8 November 2017**Abstract **

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In this paper, we investigate the impact of the accident reporting strategy of drivers, within a Bonus-Malus system. We exhibit the induced modification of the corresponding class level transition matrix and derive the optimal reporting strategy for rational drivers. The hunger for bonuses

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In this paper, we investigate the impact of the accident reporting strategy of drivers, within a Bonus-Malus system. We exhibit the induced modification of the corresponding class level transition matrix and derive the optimal reporting strategy for rational drivers. The hunger for bonuses induces optimal thresholds under which, drivers do not claim their losses. Mathematical properties of the induced level class process are studied. A convergent numerical algorithm is provided for computing such thresholds and realistic numerical applications are discussed.
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A Review and Some Complements on Quantile Risk Measures and Their Domain*Risks* **2017**, *5*(4), 59; doi:10.3390/risks5040059 - 7 November 2017**Abstract **

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure *Q* on the open unit interval and a wide class ${\mathcal{L}}_{Q}$ of random variables, we define the quantile risk measure ${\varrho}_{}$

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In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure *Q* on the open unit interval and a wide class ${\mathcal{L}}_{Q}$ of random variables, we define the quantile risk measure ${\varrho}_{Q}$ as the map that integrates the quantile function of a random variable in ${\mathcal{L}}_{Q}$ with respect to *Q*. The definition of ${\mathcal{L}}_{Q}$ ensures that ${\varrho}_{Q}$ cannot attain the value $+\infty $ and cannot be extended beyond ${\mathcal{L}}_{Q}$ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall.
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An EM Algorithm for Double-Pareto-Lognormal Generalized Linear Model Applied to Heavy-Tailed Insurance Claims*Risks* **2017**, *5*(4), 60; doi:10.3390/risks5040060 - 7 November 2017**Abstract **

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Generalized linear models might not be appropriate when the probability of extreme events is higher than that implied by the normal distribution. Extending the method for estimating the parameters of a double Pareto lognormal distribution (DPLN) in Reed and Jorgensen (2004), we develop

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Generalized linear models might not be appropriate when the probability of extreme events is higher than that implied by the normal distribution. Extending the method for estimating the parameters of a double Pareto lognormal distribution (DPLN) in Reed and Jorgensen (2004), we develop an EM algorithm for the heavy-tailed Double-Pareto-lognormal generalized linear model. The DPLN distribution is obtained as a mixture of a lognormal distribution with a double Pareto distribution. In this paper the associated generalized linear model has the location parameter equal to a linear predictor which is used to model insurance claim amounts for various data sets. The performance is compared with those of the generalized beta (of the second kind) and lognorma distributions.
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Non-Parametric Integral Estimation Using Data Clustering in Stochastic dynamic Programming: An Introduction Using Lifetime Financial Modelling*Risks* **2017**, *5*(4), 57; doi:10.3390/risks5040057 - 31 October 2017**Abstract **

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This paper considers an alternative way of structuring stochastic variables in a dynamic programming framework where the model structure dictates that numerical methods of solution are necessary. Rather than estimating integrals within a Bellman equation using quadrature nodes, we use nodes directly from

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This paper considers an alternative way of structuring stochastic variables in a dynamic programming framework where the model structure dictates that numerical methods of solution are necessary. Rather than estimating integrals within a Bellman equation using quadrature nodes, we use nodes directly from the underlying data. An example of the application of this approach is presented using individual lifetime financial modelling. The results show that data-driven methods lead to the least losses in result accuracy compared to quadrature and Quasi-Monte Carlo approaches, using historical data as a base. These results hold for both a single stochastic variable and multiple stochastic variables. The results are significant for improving the computational accuracy of lifetime financial models and other models that employ stochastic dynamic programming.
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Optional Defaultable Markets*Risks* **2017**, *5*(4), 56; doi:10.3390/risks5040056 - 23 October 2017**Abstract **

The paper deals with defaultable markets, one of the main research areas of mathematical finance. It proposes a new approach to the theory of such markets using techniques from the calculus of optional stochastic processes on *un*usual probability spaces, which was not

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The paper deals with defaultable markets, one of the main research areas of mathematical finance. It proposes a new approach to the theory of such markets using techniques from the calculus of optional stochastic processes on *un*usual probability spaces, which was not presented before. The paper is a foundation paper and contains a number of fundamental results on modeling of defaultable markets, pricing and hedging of defaultable claims and results on the probability of default under such conditions. Moreover, several important examples are presented: a new pricing formula for a defaultable bond and a new pricing formula for credit default swap. Furthermore, some results on the absence of arbitrage for markets on *un*usual probability spaces and markets with default are also provided.
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Optimal Form of Retention for Securitized Loans under Moral Hazard*Risks* **2017**, *5*(4), 55; doi:10.3390/risks5040055 - 21 October 2017**Abstract **

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We address the moral hazard problem of securitization using a principal-agent model where the investor is the principal and the lender is the agent. Our model considers structured asset-backed securitization with a credit enhancement (tranching) procedure. We assume that the originator can affect

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We address the moral hazard problem of securitization using a principal-agent model where the investor is the principal and the lender is the agent. Our model considers structured asset-backed securitization with a credit enhancement (tranching) procedure. We assume that the originator can affect the default probability and the conditional loss distribution. We show that the optimal form of retention must be proportional to the pool default loss even in the absence of systemic risk when the originator can affect the conditional loss given default rate, yet the current regulations propose a constant retention rate.
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Exposure as Duration and Distance in Telematics Motor Insurance Using Generalized Additive Models*Risks* **2017**, *5*(4), 54; doi:10.3390/risks5040054 - 25 September 2017**Abstract **

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In Pay-As-You-Drive (PAYD) automobile insurance, the premium is fixed based on the distance traveled, while in usage-based insurance (UBI) the driving patterns of the policyholder are also considered. In those schemes, drivers who drive more pay a higher premium compared to those with

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In Pay-As-You-Drive (PAYD) automobile insurance, the premium is fixed based on the distance traveled, while in usage-based insurance (UBI) the driving patterns of the policyholder are also considered. In those schemes, drivers who drive more pay a higher premium compared to those with the same characteristics who drive only occasionally, because the former are more exposed to the risk of accident. In this paper, we analyze the simultaneous effect of the distance traveled and exposure time on the risk of accident by using Generalized Additive Models (GAM). We carry out an empirical application and show that the expected number of claims (1) stabilizes once a certain number of accumulated distance-driven is reached and (2) it is not proportional to the duration of the contract, which is in contradiction to insurance practice. Finally, we propose to use a rating system that takes into account simultaneously exposure time and distance traveled in the premium calculation. We think that this is the trend the automobile insurance market is going to follow with the eruption of telematics data.
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Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks*Risks* **2017**, *5*(4), 53; doi:10.3390/risks5040053 - 22 September 2017**Abstract **

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The main objective of this work is to develop a detailed step-by-step guide to the development and application of a new class of efficient Monte Carlo methods to solve practically important problems faced by insurers under the new solvency regulations. In particular, a

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The main objective of this work is to develop a detailed step-by-step guide to the development and application of a new class of efficient Monte Carlo methods to solve practically important problems faced by insurers under the new solvency regulations. In particular, a novel Monte Carlo method to calculate capital allocations for a general insurance company is developed, with a focus on coherent capital allocation that is compliant with the Swiss Solvency Test. The data used is based on the balance sheet of a representative stylized company. For each line of business in that company, allocations are calculated for the one-year risk with dependencies based on correlations given by the Swiss Solvency Test. Two different approaches for dealing with parameter uncertainty are discussed and simulation algorithms based on (pseudo-marginal) Sequential Monte Carlo algorithms are described and their efficiency is analysed.
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The Impact of Risk Management in Credit Rating Agencies*Risks* **2017**, *5*(4), 52; doi:10.3390/risks5040052 - 21 September 2017**Abstract **

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An empirical study was conducted to determine the impact of different types of risk on the performance management of credit rating agencies (CRAs). The different types of risks were classified as operational, market, business, financial, and credit. All these five variables were analysed

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An empirical study was conducted to determine the impact of different types of risk on the performance management of credit rating agencies (CRAs). The different types of risks were classified as operational, market, business, financial, and credit. All these five variables were analysed to ascertain their impact on the performance of CRAs. In addition, apart from identifying the significant variables, the study focused on setting out a structured framework for future research. The five independent variables were tested statistically using structural equation modelling (SEM). The results indicated that market risk, financial risk, and credit risk have a significant impact on the performance of CRAs, whereas operational risk and business risk, though important, do not have a significant influence. This finding has a significant implication for the examination and inter-firm evaluation of CRAs.
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An Integrated Approach to Pricing Catastrophe Reinsurance*Risks* **2017**, *5*(3), 51; doi:10.3390/risks5030051 - 19 September 2017**Abstract **

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We propose an integrated approach straddling the actuarial science and the mathematical finance approaches to pricing a default-risky catastrophe reinsurance contract. We first apply an incomplete-market version of the no-arbitrage martingale pricing paradigm to price the reinsurance contract as a martingale by a

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We propose an integrated approach straddling the actuarial science and the mathematical finance approaches to pricing a default-risky catastrophe reinsurance contract. We first apply an incomplete-market version of the no-arbitrage martingale pricing paradigm to price the reinsurance contract as a martingale by a measure change, then we apply risk loading to price in—as in the traditional actuarial practice—market imperfections, the underwriting cycle, and other idiosyncratic factors identified in the practice and empirical literatures. This integrated approach is theoretically appealing for its merit of factoring risk premiums into the probability measure, and yet practical for being applicable to price a contract not traded on financial markets. We numerically study the catastrophe pricing effects and find that the reinsurance contract is more valuable when the catastrophe is more severe and the reinsurer’s default risk is lower because of a stronger balance sheet. We also find that the price is more sensitive to the severity of catastrophes than to the arrival frequency; implying (re)insurers should focus more on hedging the severity than the arrival frequency in their risk management programs.
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Interest Rates Term Structure under Ambiguity*Risks* **2017**, *5*(3), 50; doi:10.3390/risks5030050 - 14 September 2017**Abstract **

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After financial crisis, the role of uncertainty in decision making processes has largely been recognized as the new variable that contributes to shaping interest rates and bond prices. Our aim is to discuss the impact of ambiguity on bonds interest rates (yields). Starting

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After financial crisis, the role of uncertainty in decision making processes has largely been recognized as the new variable that contributes to shaping interest rates and bond prices. Our aim is to discuss the impact of ambiguity on bonds interest rates (yields). Starting from the realistic assumption that investors ask for an ambiguity premium depending on the efficacy of government interventions (if any), we lead to an exponential multi-factor affine model which includes ambiguity as well as an ambiguous version of the Heath-Jarrow-Morton (HJM)model. As an example, we propose the realistic economic framework given by ^{Ulrich} (^{2008}, ^{2011}), and we recover the corresponding ambiguous HJM framework, thus offering a large set of interest rate models enriched with ambiguity. We also give a concrete view of how different simulated scenarios of ambiguity can influence the economic cycle (through rates and bond prices).
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Model Uncertainty in Operational Risk Modeling Due to Data Truncation: A Single Risk Case*Risks* **2017**, *5*(3), 49; doi:10.3390/risks5030049 - 13 September 2017**Abstract **

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Over the last decade, researchers, practitioners, and regulators have had intense debates about how to treat the data collection threshold in operational risk modeling. Several approaches have been employed to fit the loss severity distribution: the empirical approach, the “naive” approach, the shifted

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Over the last decade, researchers, practitioners, and regulators have had intense debates about how to treat the data collection threshold in operational risk modeling. Several approaches have been employed to fit the loss severity distribution: the empirical approach, the “naive” approach, the shifted approach, and the truncated approach. Since each approach is based on a different set of assumptions, different probability models emerge. Thus, model uncertainty arises. The main objective of this paper is to understand the impact of model uncertainty on the value-at-risk (VaR) estimators. To accomplish that, we take the bank’s perspective and study a single risk. Under this simplified scenario, we can solve the problem analytically (when the underlying distribution is exponential) and show that it uncovers similar patterns among VaR estimates to those based on the simulation approach (when data follow a Lomax distribution). We demonstrate that for a fixed probability distribution, the choice of the truncated approach yields the lowest VaR estimates, which may be viewed as beneficial to the bank, whilst the “naive” and shifted approaches lead to higher estimates of VaR. The advantages and disadvantages of each approach and the probability distributions under study are further investigated using a real data set for legal losses in a business unit (Cruz 2002).
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A Cointegrated Regime-Switching Model Approach with Jumps Applied to Natural Gas Futures Prices*Risks* **2017**, *5*(3), 48; doi:10.3390/risks5030048 - 12 September 2017**Abstract **

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Energy commodities and their futures naturally show cointegrated price movements. However, there is empirical evidence that the prices of futures with different maturities might have, e.g., different jump behaviours in different market situations. Observing commodity futures over time, there is also evidence for

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Energy commodities and their futures naturally show cointegrated price movements. However, there is empirical evidence that the prices of futures with different maturities might have, e.g., different jump behaviours in different market situations. Observing commodity futures over time, there is also evidence for different states of the underlying volatility of the futures. In this paper, we therefore allow for cointegration of the term structure within a multi-factor model, which includes seasonality, as well as joint and individual jumps in the price processes of futures with different maturities. The seasonality in this model is realized via a deterministic function, and the jumps are represented with thinned-out compound Poisson processes. The model also includes a regime-switching approach that is modelled through a Markov chain and extends the class of geometric models. We show how the model can be calibrated to empirical data and give some practical applications.
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Assessment of Policy Changes to Means-Tested Age Pension Using the Expected Utility Model: Implication for Decisions in Retirement*Risks* **2017**, *5*(3), 47; doi:10.3390/risks5030047 - 9 September 2017**Abstract **

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Means-tested pension policies are typical for many countries, and the assessment of policy changes is critical for policy makers. In this paper, we consider the Australian means-tested Age Pension. In 2015, two important changes were made to the popular Allocated Pension accounts: the

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Means-tested pension policies are typical for many countries, and the assessment of policy changes is critical for policy makers. In this paper, we consider the Australian means-tested Age Pension. In 2015, two important changes were made to the popular Allocated Pension accounts: the income means-test is now based on deemed income rather than account withdrawals, and the income-test deduction no longer applies. We examine the implications of the new changes in regard to optimal decisions for consumption, investment and housing. We account for regulatory minimum withdrawal rules that are imposed by regulations on Allocated Pension accounts, as well as the 2017 asset-test rebalancing. The policy changes are considered under a utility-maximising life cycle model solved as an optimal stochastic control problem. We find that the new rules decrease the advantages of planning the consumption in relation to the means-test, while risky asset allocation becomes more sensitive to the asset-test. The difference in optimal drawdown between the old and new policy is only noticeable early in retirement until regulatory minimum withdrawal rates are enforced. However, the amount of extra Age Pension received by many households is now significantly higher due to the new deeming income rules, which benefit wealthier households who previously would not have received Age Pension due to the income-test and minimum withdrawals.
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Optimal Insurance Policies in the Presence of Costs*Risks* **2017**, *5*(3), 46; doi:10.3390/risks5030046 - 6 September 2017**Abstract **

We reconsider costs in insurance, and suggest a new type of cost function, which we argue is a natural choice when there are relatively small, but frequent, claims. If a fixed cost is incurred each time a claim is made, we obtain a

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We reconsider costs in insurance, and suggest a new type of cost function, which we argue is a natural choice when there are relatively small, but frequent, claims. If a fixed cost is incurred each time a claim is made, we obtain a Pareto optimal deductible even if the cost function does not vary with the indemnity. The classical result says that deductibles appear if and only if costs are variable. This implies that when the claims are relatively small, it is not optimal for the insured to be compensated, since the costs outweigh the benefits and a deductible will naturally occur. When we constrain the contract to contain a cap, a non-trivial deductible is Pareto optimal regardless of the assumptions about the cost structure, which is what is known as an XL-contract.
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Existence and Uniqueness for the Multivariate Discrete Terminal Wealth Relative*Risks* **2017**, *5*(3), 44; doi:10.3390/risks5030044 - 28 August 2017**Abstract **

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In this paper, the multivariate fractional trading ansatz of money management from Vince (Vince 1990) is discussed. In particular, we prove existence and uniqueness of an “optimal *f*” of the respective optimization problem under reasonable assumptions on the trade return matrix. This

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In this paper, the multivariate fractional trading ansatz of money management from Vince (Vince 1990) is discussed. In particular, we prove existence and uniqueness of an “optimal *f*” of the respective optimization problem under reasonable assumptions on the trade return matrix. This result generalizes a similar result for the univariate fractional trading ansatz. Furthermore, our result guarantees that the multivariate optimal *f* solutions can always be found numerically by steepest ascent methods.
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A Low Price Correction for Improved Volatility Estimation and Forecasting*Risks* **2017**, *5*(3), 45; doi:10.3390/risks5030045 - 28 August 2017**Abstract **

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In this work, we focus on volatility estimation which plays a crucial role in risk analysis and management. In order to improve value at risk (VaR) forecasts, we discuss the concept of *low price effect* and introduce the *low price correction* which does

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In this work, we focus on volatility estimation which plays a crucial role in risk analysis and management. In order to improve value at risk (VaR) forecasts, we discuss the concept of *low price effect* and introduce the *low price correction* which does not require any additional parameters and instead of returns it takes into account the prices of the asset. Judgement on the forecasting quality of the proposed methodology is based on both the relative number of violations and VaR volatility. For illustrative purposes, a real example from the Athens Stock Exchange is fully explored.
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On the First Crossing of Two Boundaries by an Order Statistics Risk Process*Risks* **2017**, *5*(3), 43; doi:10.3390/risks5030043 - 18 August 2017**Abstract **

We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields

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We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields new expressions for the ruin probability in the insurance and the dual risk models with dependence between the claim severities or capital gains respectively.
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