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| In [[actuarial science]] and [[applied probability]] '''ruin theory''' (sometimes '''collective risk theory''') uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
| | It involves expertise and knowledge of various tools and technologies used for creating websites. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Creating a website from scratch can be such a pain. When you write a new post, you'll see a small bar that goes across the text input area. Which is perfect for building a mobile site for business use. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. all the necessary planning and steps of conversion is carried out in this phase, such as splitting, slicing, CSS code, adding images, header footer etc. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Customization of web layout is easy due to the availability of huge selection of templates. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. When you sign up with Wordpress, you gain access to several different templates and plug-in that allow you to customize your blog so that it fits in with your business website design seamlessly. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. If you cherished this post and also you want to acquire guidance regarding [http://www.playpromedia.com/members/delilobryan/activity/183534/ wordpress backup] kindly stop by our own web-page. Change the entire appearance of you blog using themes with one click. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
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| ==Classical model==
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| [[Image:Samplepathcompoundpoisson.JPG|thumb|right|350px|A sample path of compound Poisson risk process]]
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| The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process<ref>{{cite doi|10.1016/0167-6687(87)90019-9}}</ref> or Poisson risk process) was introduced in 1903 by the Swedish actuary [[Filip Lundberg]].<ref>Lundberg, F. (1903) Approximerad Framställning av Sannolikehetsfunktionen, Återförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala.</ref> Lundberg's work was republished in the 1930s by [[Harald Cramér]].<ref>{{cite doi|10.1214/aos/1176350596}}</ref>
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| The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate ''c'' > 0 from customers and claims arrive according to a [[Poisson process]] with intensity ''λ'' and are [[independent and identically distributed non-negative random variables]] with distribution ''F'' and mean ''μ'' (they form a [[compound Poisson process]]). So for an insurer who starts with initial surplus ''x'',<ref>{{cite doi|10.1007/978-3-540-31343-4_1}}</ref>
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| :<math>X_t = x + ct - \sum_{i=1}^{N_t} \xi_i \quad \text{ for t} \geq 0.</math>
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| The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as | |
| :<math>\psi(x)=\mathbb{P}^x\{\tau<\infty\}</math>
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| where the time of ruin is <math>\scriptstyle \tau=\inf\{t>0 \,:\, X(t)<0\}</math> with the convention that <math>\scriptstyle \inf\varnothing=\infty</math>. This can be computed exactly using the [[Pollaczek–Khinchine formula]] as<ref>{{cite jstor|4141346}}</ref> (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an [[M/G/1 queue]]<ref name="rolski">{{cite doi|10.1002/9780470317044.ch5}}</ref>)
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| :<math>\psi(x)=\left(1-\frac{\lambda \mu}{c}\right) \sum_{n=0}^\infty \left(\frac{\lambda \mu}{c}\right)^n (1-F^{\ast n}_l(x))</math>
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| where <math>\scriptstyle F^{\ast n}_l(x)</math> is the transform of the tail distribution of ''F'',
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| :<math>F_l(x) = \frac{1}{\mu} \int_0^x \left(1-F(u)\right) \text{d}u.</math>
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| In the case where the claim sizes are exponentially distributed, this simplifies to<ref name="rolski" />
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| :<math>\psi(x) = \frac{\lambda \mu}{c}e^{-\left( \frac{1}{\mu}-\frac{\lambda}{\beta}\right)x}.</math>
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| ==Sparre Andersen model==
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| E. Sparre Andersen extended the classical model in 1957<ref>Andersen, E. Sparre. "On the collective theory of risk in case of contagion between claims." ''Transactions of the XVth International Congress of Actuaries''. Vol. 2. No. 6. 1957.</ref> by allowing claim inter-arrival times to have arbitrary distribution functions.<ref>Thorin, Olof. "[http://casact.net/library/astin/vol8no1/104.pdf Some comments on the Sparre Andersen model in the risk theory]" ''The ASTIN bulletin: international journal for actuarial studies in non-life insurance and risk theory'' (1974): 104.</ref>
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| ::<math>X_t = x + ct - \sum_{i=1}^{N_t} \xi_i \quad \text{ for t} \geq 0.</math>
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| where ''N''(''t'') is a [[renewal process]] and ''X''<sub>''i''</sub> independent and identically distributed non-negative random variables with no probability mass at zero.
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| ==Expected discounted penalty function==
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| [[Michael R. Powers]]<ref name="powers">{{cite doi|10.1016/0167-6687(95)00006-E}}</ref> and Gerber and Shiu<ref name="gerber-shiu">{{cite doi|10.1080/10920277.1998.10595671}}</ref> analyzed the behavior of the insurer's surplus through the '''expected discounted penalty function''', which is commonly referred to as Gerber-Shiu function in the ruin literature. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.<ref name="powers" />
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| In [[Michael R. Powers|Powers]]’ notation, this is defined as
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| :<math>m(x)=\mathbb{E}^x[e^{-\delta\tau}K_{\tau}]</math>,
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| where <math>\delta</math> is the discounting force of interest, <math>K_{\tau}</math> is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation <math>\mathbb{E}^x</math> corresponds to the probability measure <math>\mathbb{P}^x</math>. The function is called expected discounted cost of insolvency in Powers.<ref name="powers" />
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| In Gerber and Shiu’s notation, it is given as
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| :<math>m(x)=\mathbb{E}^x[e^{-\delta\tau}w(X_{\tau-},X_{\tau})\mathbb{I}(\tau<\infty)]</math>,
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| where <math>\delta</math> is the discounting force of interest and <math>w(X_{\tau-},X_{\tau})</math> is a penalty function capturing the economic costs to the insurer at the time of ruin (assumed to depend on the surplus prior to ruin <math>X_{\tau-}</math> and the deficit at ruin <math>X_{\tau}</math>), and the expectation <math>\mathbb{E}^x</math> corresponds to the probability measure <math>\mathbb{P}^x</math>. Here the indicator function <math>\mathbb{I}(\tau<\infty)</math> emphasizes that the penalty is exercised only when ruin occurs.
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| It is quite intuitive to interpret the expected discounted penalty function. Since the function measures the actuarial present value of the penalty that occurs at <math>\tau</math>, the penalty function is multiplied by the discounting factor <math>e^{-\delta\tau}</math>, and then averaged over the probability distribution of the waiting time to <math>\tau</math>. While Gerber and Shiu<ref name="gerber-shiu" /> applied this function to the classical compound-Poisson model, Powers<ref name="powers" /> argued that an insurer’s surplus is better modeled by a family of diffusion processes.
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| There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.
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| {| class="wikitable"
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| |-
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| ! Special case
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| ! Mathematical representation
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| ! Choice of penalty function
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| |-
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| | Probability of ultimate ruin
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| | <math>\mathbb{P}^x\{\tau<\infty\}</math>
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| | <math>\delta=0,w(x_1,x_2)=1</math>
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| |-
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| | Joint (defective) distribution of surplus and deficit
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| | <math>\mathbb{P}^x\{X_{\tau-}<x,X_\tau<y\}</math>
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| | <math>\delta=0,w(x_1,x_2)=\mathbb{I}(x_1<x,x_2<y)</math>
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| | Defective distribution of claim causing ruin
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| | <math>\mathbb{P}^x\{X_{\tau-}-X_\tau<z\}</math>
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| | <math>\delta=0,w(x_1,x_2)=\mathbb{I}(x_1+x_2<z)</math>
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| |-
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| | Trivariate Laplace transform of time, surplus and deficit
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| | <math>\mathbb{E}^x[e^{-\delta \tau -s X_{\tau-}-z X_\tau}]</math>
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| | <math>w(x_1,x_2)=e^{-s x_1-z x_2}</math>
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| |-
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| | Joint moments of surplus and deficit
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| | <math>\mathbb{E}^x[X_{\tau-}^jX_\tau^k]</math>
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| | <math>\delta=0,w(x_1,x_2)=x_1^jx_2^k</math>
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| |}
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| Other finance-related quantities belonging to the class of the expected discounted penalty function include the perpetual American put option,<ref>{{cite journal | last1=Gerber | first1= H.U. |last2= Shiu |first2= E.S.W. | year=1997 | title= From ruin theory to option pricing | journal = AFIR Colloquium, Cairns, Australia 1997| url = http://www.actuaries.org/AFIR/Colloquia/Cairns/Gerber_Shiu.pdf}}</ref> the contingent claim at optimal exercise time, and more.
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| ==Recent developments==
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| *Compound-Poisson risk model with constant interest
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| *Compound-Poisson risk model with stochastic interest
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| *Brownian-motion risk model
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| *General diffusion-process model
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| *Markov-modulated risk model
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| *Accident Probability Factor(APF) Calculator - Risk analysis model (@SBH)
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| ==See also==
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| *[[Poisson process]]
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| *[[Continuous-time Markov process]]
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| *[[Stochastic processes]]
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| *[[Queueing theory]]
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| ==Bibliography==
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| *{{cite book
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| | author = Gerber, H.U.
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| | year = 1979
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| | title = An Introduction to Mathematical Risk Theory
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| | publisher = S.S. Heubner Foundation Monograph Series 8
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| | location = Philadelphia}}
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| *{{cite book
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| | author = Asmussen S.
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| | year = 2000
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| | title = Ruin Probabilities
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| | publisher = World Scientific Publishing Co.
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| | location = Singapore}}
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| ==References==
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| {{reflist}}
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| [[Category:Actuarial science]]
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| [[Category:Probability theory]]
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| [[Category:Stochastic processes]]
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| [[Category:Mathematical finance]]
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| [[Category:Risk]]
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It involves expertise and knowledge of various tools and technologies used for creating websites. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.
Creating a website from scratch can be such a pain. When you write a new post, you'll see a small bar that goes across the text input area. Which is perfect for building a mobile site for business use. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available.
The least difficult and very best way to do this is by acquiring a Word - Press site. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. all the necessary planning and steps of conversion is carried out in this phase, such as splitting, slicing, CSS code, adding images, header footer etc. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Customization of web layout is easy due to the availability of huge selection of templates.
If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. This includes enriching the content with proper key words, tactfully defining the tags and URL.
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