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Добірка наукової літератури з теми "Ntegral equation for the non - ruin probability"

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Опубліковано: 30 травня 2022
Оновлено: 31 травня 2022

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Статті в журналах з теми "Ntegral equation for the non - ruin probability"

1

Guilbault, Jean-Luc, and Mario Lefebvre. "On a non-homogeneous difference equation from probability theory." Tatra Mountains Mathematical Publications 43, no. 1 (December 1, 2009): 81–90. http://dx.doi.org/10.2478/v10127-009-0027-4.

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Анотація:
Abstract The so-called gambler’s ruin problem in probability theory is considered for a Markov chain having transition probabilities depending on the current state. This problem leads to a non-homogeneous difference equation with non-constant coefficients for the expected duration of the game. This mathematical expectation is computed explicitly.
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2

Quang, Phung Duy. "Ruin Probability in a Generalised Risk Process under Rates of Interest with Homogenous Markov Chains." East Asian Journal on Applied Mathematics 4, no. 3 (August 2014): 283–300. http://dx.doi.org/10.4208/eajam.051013.230614a.

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Анотація:
AbstractThis article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.
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3

Møller, Christian Max. "Stochastic differential equations for ruin probabilities." Journal of Applied Probability 32, no. 01 (March 1995): 74–89. http://dx.doi.org/10.1017/s002190020010258x.

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Анотація:
The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.
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4

Møller, Christian Max. "Stochastic differential equations for ruin probabilities." Journal of Applied Probability 32, no. 1 (March 1995): 74–89. http://dx.doi.org/10.2307/3214922.

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Анотація:
The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.
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5

Asmussen, Søren, and Hanne Mandrup Nielsen. "Ruin probabilities via local adjustment coefficients." Journal of Applied Probability 32, no. 3 (September 1995): 736–55. http://dx.doi.org/10.2307/3215126.

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Анотація:
Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.
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6

Asmussen, Søren, and Hanne Mandrup Nielsen. "Ruin probabilities via local adjustment coefficients." Journal of Applied Probability 32, no. 03 (September 1995): 736–55. http://dx.doi.org/10.1017/s0021900200103171.

Повний текст джерела
Анотація:
Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.
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7

Bondarev, B. V., and V. O. Boldyreva. "Deriving the Equation for the Non-Ruin Probability of the Insurance Company in (B, S)-market. Stochastic Claims and Stochastic Premiums." Cybernetics and Systems Analysis 50, no. 5 (September 2014): 750–58. http://dx.doi.org/10.1007/s10559-014-9665-x.

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8

Assaf, David, Yuliy Baryshnikov, and Wolfgang Stadje. "Optimal strategies in a risk selection investment model." Advances in Applied Probability 32, no. 02 (June 2000): 518–39. http://dx.doi.org/10.1017/s0001867800010065.

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Анотація:
We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.
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9

Assaf, David, Yuliy Baryshnikov, and Wolfgang Stadje. "Optimal strategies in a risk selection investment model." Advances in Applied Probability 32, no. 2 (June 2000): 518–39. http://dx.doi.org/10.1239/aap/1013540177.

Повний текст джерела
Анотація:
We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.
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Дисертації з теми "Ntegral equation for the non - ruin probability"

1

Федчишина, Ірина Юріївна. "Уточнення апроксимації де Вілдера для оцінки ймовірності банкрутства у страховій моделі Крамера-Лундберга". Master's thesis, Київ, 2018. https://ela.kpi.ua/handle/123456789/23449.

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Анотація:
В магістерській дисертації запропонований новий підхід до наближеного знаходження ймовірності банкрутства страхової компанії на нескінченному часовому горизонті. Необхідність такого наближеного знаходження зумовлюється тим, що точне значення ймовірності банкрутства, будучи розв’язком складного інтегрального рівняння, часто не може бути виражене в явній аналітичній формі. Ідея розробленого методу полягає в заміні процесу страхового ризику на інший процес ризику зі страховими виплатами, розподіленими за законом, що є сумішшю двох експоненціальних розподілів. Для такого процесу ризику ймовірність банкрутства відома в аналітичній формі. Заміна реалізується шляхом прирівнювання перших п’яти кумулянтів початкового та нового процесів ризику.
In the master's thesis a new approach to the approximate finding of the ruin probability of an insurance company on an infinite time horizon is proposed. The need for such an approximate finding is due to the fact that the exact value of the ruin probability, being a solution to a complex integral equation, can often not be expressed in explicit analytical form. The idea of the developed method is to replace the process of risk with another risk process with insurance payments distributed according to the law, which is a mixture of two exponential distributions. For such a risk process, the ruin probability is known in analytical form. Replacement is realized by equating the first five cumulants of the initial and new risk processes.
В магистерской диссертации предложен новый поход к приближенному нахождению вероятности банкротства страховой компании на бесконечном временном горизонте. Необходимость такого приближенного нахождения обусловлено тем, что точное значение вероятности банкротства, будучи решением сложного интегрального уравнения, часто не может быть выражено в явной аналитической форме. Идея разработанного метода заключается в замене процесса страхового риска на другой процесс риска со страховыми выплатами, распределенными по закону, который является смесью двух экспоненциальных распределений. Для такого процесса риска вероятность банкротства известна в аналитической форме. Замена реализуется путем приравнивания первых пяти кумулянтов начального и нового процессов риска.
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2

Ni, Ying. "Perturbed Renewal Equations with Non-Polynomial Perturbations." Licentiate thesis, Mälardalen University, School of Education, Culture and Communication, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-9354.

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Анотація:

This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.

The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.

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