We would usually use a recursion tree to generate possible guesses for the runtime, and then use the substitution method to prove them. More primitive recursion a special case of primitive recursion is for some constant number k. Our main result is that there is an increasing sequence e 0 panjer s recursion formula is used for the evaluation of compound distributions. Choose a web site to get translated content where available and see local events and offers. Package actuar the comprehensive r archive network. Portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance. Transfinite induction is an extension of mathematical induction to wellordered sets, for example to sets of ordinal numbers or cardinal numbers let be a property defined for all ordinals. More formally, by induction on the number of steps used to derive the primitive recursive function. Then aif fn onlog b a for some constant 0, then tn onlog b a.
Recursion a programming strategy for solving large problems think divide and conquer solve large problem by splitting into smaller problems of same kind induction a mathematical strategyfor proving statements about large sets of things first we learn induction. Handling of the general case when a has a nonminimal value, investigate how the results of one or more recursive calls can be combined with the argument. The following recursion formula on compound poisson densities is valid. The simple algebraic proof of this paper yields all the previous results as special case.
Combining panjers recursion with convolution sciencedirect. The panjer recursion scheme 2010 mathematics subject classi. Our proof will show that the above functional has the same characteristic function as that derived in kuelbs 1973. Pdf panjers recursion formula is used for the evaluation of compound. Panjer recursion formula to compute the total amount of claims probability function of a portfolio. Based on your location, we recommend that you select. The first chapter of this thesis introduces panjers recursion that is one of the well known. Approximation and aggregation of risks by variants of panjer. Let i 1 i t ri with multiplicity mi be a solution of the equation. Panjer recursion and fft, discussed in sections 5 and 6, are very e cient numerical methods to calculate these convolutions. A generalization of panjers recursion and numerically stable risk aggregation stefan gerhold, uwe schmock, and richard warnung abstract. Pdf approximation and aggregation of risks by variants.
The use of this algorithm has become a widespread standard technique for life and general insur. The panjer recursion is very easy to implement in practice and numerically cheaper than bruteforce convolution, which asymptotically requires on3 operations to obtain g 0g n compared to the on2 complexity in the panjer case. Pdf panjers recursion for evaluation of compound negative. Panjer derives a recursive method for deriving the aggregate. The panjer method may be viewed as an alternative to more traditional monte. When the severity distribution is continuous with density function, numerical calculation for the compound distribution by applying panjer recursion will involve an approximation of the integration. A vector representing the probability density function of the total amount of claims.
Recursion is used in a variety of disciplines ranging from linguistics to logic. An elementary proof of the adelsonpanjer recursion. Recursive method for functionals of poisson processes 299 recursion method used in proving theorem 1. Kleenes recursion theorem, also called the fixed point theorem, in computability theory. Miller 1 abstract in this paper we use the recursion theorem to show the existence of various in. However, it contains like most recursion theory proofs selfreferences and therefore sometimes hard to visualize for a beginner. It is pretty much recursion schemes for dummies by construction. Panjer recursion versus fft for compound distributions.
Applying the classical panjer recursion in the collective risk model can lead to numerical instabilities, for instance if the claim number distribution is extended negative binomial or extended logarithmic. Then, panjer s recursion can be applied using gerhold et al. Recursive functions stanford encyclopedia of philosophy. Below we will outline the proof presented to our class lecture by professor johanna franklin. Suppose that whenever is true for all kleenes amazing second recursion theorem 419 a1 thm for each n. We prove this by induction on the formation of a primitive recursive function. Then, panjers recursion can be applied using gerhold et al. Panjer s recursion formula is used for the evaluation of compound distributions.
N0 theorem 1 q be a nondegenerate claim number distribution. We will cite some recent work by slaman, hjorth, and harrington in which recursion theoretic thinking was applied to problems in classical descriptive set theory. The recursion theorem final 3 corollary 3 is proved by showing that for all n, there is a function that agrees with fon nnand cannot have a. Refiiiences adelson, r 1966 compound polsson dlstributlons, operatzons research quarlerly, 17. Example of compound geometric and panjer recursion for ruin probabilities typical project topics. In more general cases the distribution of s is a compound distribution. Applying the classical panjer recursion in the collective risk model can lead to.
Recursion theorem proof similar to self construction, with additional ability to compute with its own description we construct recursive machine r abt tis given in the theorem a p bt b on input m, where m is a portion of tm. In this entry, we provide an account of the class of. Panjer derives a recursive method for deriving the aggregate distribution of claims in his article the aggregate claims distribution and stoploss reinsurance published in the transactions of the society of actuaries, volume xxxii, 1980, pages 523 545. An overview of the panjer method for deriving the aggregate. In its most general numerical form the process of recursion consists in defining the value of a function by using other values of the same function. The master theorem analysis of algorithms, about the complexity of divideandconquer algorithms. In actuarial science, panjer recursion 1981 is used in insurance to compute the loss distribution of the compound risk models. Panjer s recursion with convolution now assume only some of the mass points of xsay those with an index in q. Panjers paper motwated the development of an extenswe theory on recurswe methods. Abstract portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance.
Carlo simulation methods as well as an alternative to risk theory methods involving. In 3, 4, 6, and 22 please follow the instructions exactly or you might not get credit. Dickson skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. To really understand 1, one need only understand l. A generalization of panjers recursion and numerically. Panjer class united one formula for the poisson, binomial. Aug 23, 2017 training on collective risk model for ct 6 by vamisdhar ambatipudi. Panjer recursion versus fft for compound distributions paul embrechts marco frei the date of receipt and acceptance should be inserted later abstract numerical evaluation of compound distributions is an important task in insurance mathematics and quantitative risk management qrm. Numerical algorithms for panjer recursion by applying. A recursive method for functionals of poisson processes. A generalization of panjers recursion and numerically stable. Note that we now have a common representation for the pgf. Very helpful for anyone taking risk theory, advanced probabilities panjer 1981.
Oliver 1967 developed a relative stability theory and studied. The recursion theorem the recursion theorem states that some turing machines can reproduce their own descriptions it is implied that we can turn any tm into an equivalent one that has this property. In the present paper we generahse pan lers 1981 recursion for compound. The generalization of multivariate panjer recursion 273 table 1 stoploss reinsurance premium x gx gx ps 0 0. In the present paper, we also obtain an extension of panjer s recursion to the panjer. Refiiiences adelson, r 1966 compound polsson dlstributlons, operatzons research quarlerly, 17, 7375. The question starts by asking a proof of the panjer recursion.
Approximation and aggregation of risks by variants of panjer s recursion. Introducing conditional expectations like in the proof of theorem 1, we. The property of the panjer recursion that we will exploit in developing our simulation algorithms is that 3 can be recognized as a volterra equation of th e second kind panjer, 2006. As recently shown by hipp 2003, recursion 4 can be simpli. The theorem says that for an arbitrary computable function t, there is a turing machine r that computes t on hriand some input. This presentation gives the opportunity to introduce the various morphisms in a simple way, namely as patterns of recursion that are useful in functional programming, instead of the usual approach via category theory, which tends to be needlessly intimidating for the average. They obtain results for a more general recursion than i and obtain results for the case of possibly negative claims. Mathematicians implicitly use this fact whenever they define a function by recursion. Theorem second recursion theorem letf beturingcomputable. For gamma and mixture of gamma distribution the theory has been. Assume the characteristic equation has t k distinct solutions. There exists a turing machine r that computes a function r. An elementary proof of the adelsonpanjer recursion formula. Handling of the base cases for all the minimal values of a, directly without recursion express the result in terms of x 4.
The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. Components for the proof of kleenes recursion theorem. Approximation and aggregation of risks by variants of. A generalization of panjer s recursion and numerically stable risk aggregation stefan gerhold, uwe schmock, and richard warnung abstract. Then you can sum up the numbers in each node to get the cost of the entire algorithm. Implementation of negative binomial claims in panjer recursion. Aug 20, 2009 portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance. It then tells to formulate using panjer recursion, a recursion for esk. It is easy to calculate further quantities in the panu representation, say higher moments. Mar 23, 20 in actuarial science, panjer recursion 1981 is used in insurance to compute the loss distribution of the compound risk models. The generalization of multivariate panjer recursion 269 for a claim number distribution q p n n. A function is primitive recursive if it can be built up using the base functions and the operations of composition and primitive recursion. As recently shown by hipp 2003, recursion 4 can be simpli ed if the severity.
A quick guide on how to compute probabilities for compound poisson using the panjer recursion. The recursive functions, which form a class of computable functions, take their name from the process of recurrence or recursion. We give a new elementary proof of the following wellknown result. The panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable. The use of this algorithm has become a widespread standard technique for life and general insurance problems.
We present a generalization of panjer s recursion that leads to numerically stable algorithms. As recently shown by hipp 2003, recursion 4 can be simpli ed if. The generalization of multivariate panjer recursion. A1 is true, since if maxa, b 1, then both a and b are at most 1.