Simultaneous ruin probability for two-dimensional brownian risk model

2020 ◽  
Vol 57 (2) ◽  
pp. 597-612 ◽  
Author(s):  
Krzysztof Dȩbicki ◽  
Enkelejd Hashorva ◽  
Zbigniew Michna
Keyword(s):  
Ruin Probability ◽  
Time Horizon ◽  
Asymptotic Theory ◽  
Risk Model ◽  
Two Dimensional ◽  
Infinite Time ◽  
Ruin Time ◽  
Initial Capital ◽  
Infinite Time Horizon

AbstractThe ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.

scholarly journals Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model

Risks ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 83 ◽  
Author(s):  
Krzysztof Dȩbicki ◽  
Lanpeng Ji ◽  
Tomasz Rolski
Keyword(s):  
Time Horizon ◽  
Optimization Problem ◽  
Adjustment Coefficient ◽  
Two Dimensional ◽  
Infinite Time ◽  
Initial Capital ◽  
Brownian Motion With Drift ◽  
Surplus Process ◽  
Logarithmic Asymptotics ◽  
Brownian Model

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P ( u ) for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where u is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of - ln P ( u ) / u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.

scholarly journals The Ruin Problem with a Finite Time Horizon

1984 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Marc-Henri Amsler
Keyword(s):  
Upper Bound ◽  
Finite Time ◽  
Ruin Probability ◽  
Time Horizon ◽  
Infinite Time ◽  
Ruin Theory ◽  
Middle Term ◽  
Infinite Time Horizon ◽  
Finite Time Horizon ◽  
Ruin Problem

AbstractThe paper presents an extension of the classical Cramér–Lundberg ruin theory: the famous upper bound for the ruin probability with an infinite time horizon can be extended in a certain sense to the short and middle term case. Furthermore, a relation between the average values of lifetime and ruin amount is given.

scholarly journals Pandemic-type failures in multivariate Brownian risk models

Extremes ◽  
2021 ◽  
Author(s):  
Krzysztof Dȩbicki ◽  
Enkelejd Hashorva ◽  
Nikolai Kriukov
Keyword(s):  
Time Horizon ◽  
Risk Model ◽  
Point Of View ◽  
Asymptotic Approximations ◽  
Applied Probability ◽  
Infinite Time ◽  
Risk Models ◽  
Practical Point ◽  
Multiple Failures ◽  
Infinite Time Horizon

AbstractModelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018).

Unique Open Loop Nash Equilibria on the Infinite Time Horizon

2001 ◽  
Vol 34 (20) ◽  
pp. 29-34
Author(s):  
Gerhard Jank ◽  
Dirk Kremer ◽  
Gábor Kun
Keyword(s):  
Time Horizon ◽  
Nash Equilibria ◽  
Open Loop ◽  
Infinite Time ◽  
Infinite Time Horizon

scholarly journals Dynamic Proportional Reinsurance and Approximations for Ruin Probabilities in the Two-Dimensional Compound Poisson Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Yan Li ◽  
Guoxin Liu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Optimal Strategies ◽  
Proportional Reinsurance ◽  
Two Dimensional ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman ◽  
Poisson Risk Model

We consider the dynamic proportional reinsurance in a two-dimensional compound Poisson risk model. The optimization in the sense of minimizing the ruin probability which is defined by the sum of subportfolio is being ruined. Via the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal value function and prove the verification theorem. In addition, we obtain the Lundberg bounds and the Cramér-Lundberg approximation for the ruin probability and show that as the capital tends to infinity, the optimal strategies converge to the asymptotically optimal constant strategies. The asymptotic value can be found by maximizing the adjustment coefficient.

scholarly journals An inventory model for deteriorating items under time varying demand condition

2013 ◽  
Vol 8 (1) ◽  
pp. 1273-1278
Author(s):  
Srichandan Mishra ◽  
S.P. Mishra ◽  
N. Mishra ◽  
J. Panda
Keyword(s):  
Time Horizon ◽  
Cost Minimization ◽  
Inventory Model ◽  
Deteriorating Items ◽  
Time Varying ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Demand Rate ◽  
Demand Condition ◽  
Varying Demand

In this paper we discuss the development of an inventory model for deteriorating items which investigates an instantaneous replenishment model for the items under cost minimization. A time varying type of demand rate with infinite time horizon, exponential deterioration and with shortage in considered. The result is illustrated with numerical example.

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