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However, the formulas IV. Given this can be done, Ab u , O u, T can then be calculated numerically by some method for transform inversion, say the fast Fourier transform FFT as implemented in Grubel [] for infinite horizon ruin probabilities for the renewal model. Differential- and integral equations The idea is here to express 'O u or ' u, T as the solution to a differential- or integral equation, and carry out the solution by some standard numerical method.

One example where this is feasible is the renewal equation for tl' u Corollary III. However, most often it is more difficult to come up with reasonably simple equations than one may believe at a first sight, and in particular the naive idea of conditioning upon process behaviour in [0, dt] most often leads to equations involving both differential and integral terms. An example where this idea can be carried through by means of a suitable choice of supplementary variables is the case of state-dependent premium p x and phase-type claims, see VIII.

It has generalizations to the models with renewal arrivals, a Markovian environment or periodically varying parameters. However, in such cases the evaluation of C is more cumbersome. In fact, when the claim size distribution is of phase-type, the exact solution is as easy to compute as the Cramer-Lundberg approximation at least in the first two of these three models. Diffusion approximations Here the idea is simply to approximate the risk process by a Brownian motion or a more general diffusion by fitting the first and second moment, and use the fact that first passage probabilities are more readily calculated for diffusions than for the risk process itself.

Diffusion approximations are easy to calculate, but typically not very precise in their first naive implementation. However, incorporating correction terms may change the picture dramatically. In particular, corrected diffusion approximations see IV. Large claims approximations In order for the Cramer-Lundberg approximation to be valid, the claim size distribution should have an exponentially decreasing tail B x. In the case of heavy-tailed distributions, other approaches are thus required. Approximations for O u as well as for 1 u, T for large u are available in most of the models we discuss.

See Chapter IX. This list of approximations does by no means exhaust the topic; some further possibilities are surveyed in We return to various extensions and sharpenings of Lundberg's inequality finite horizon versions, lower bounds etc. When comparing different risk models, it is a general principle that adding random variation to a model increases the risk.

For example, one expects a model with a deterministic claim size distribution B, say degenerate at m, to have smaller ruin probabilities than when B is non-degenerate with the same mean m. This is proved for the compound Poisson model in However, empirical evidence shows that the general principle holds in a broad variety of settings, though not too many precise mathematical results have been obtained.

3. Probability Theory

In practice, they have however to be estimated from data, obtained say by observing the risk process in [0, T]. This procedure in itself is fairly straightforward; e. However, the difficulty comes in when drawing inference about the ruin probabilities. How do we produce a confidence interval? And, more importantly, can we trust the confidence intervals for the large values of u which are of interest? In the present author's opinion, this is extrapolation from data due to the extreme sensitivity of the ruin probabilities to the tail of the claim size distribution in particular in contrast, fitting a parametric model to U1,.

For example, one may question whether it is possible to distinguish between claim size distributions which are heavy-tailed or have an exponentially decaying tail. Simulation may be used just to get some vague insight in the process under study: simulate one or several sample paths, and look at them to see whether they exhibit the expected behaviour or some surprises come up. However, the more typical situation is to perform a Monte Carlo experiment to estimate probabilities or expectations or distributions which are not analytically available. For example, this is a straightforward way to estimate finite horizon ruin probabilities.

The infinite horizon case presents a difficulty, because it appears to require an infinitely long simulation. Truncation to a finite horizon has been used, but is not very satisfying. Still, good methods exist in a number of models and are based upon representing the ruin probability zb u as expected value of a r. We look at a variety of such methods in Chapter X, and also discuss how to develop methods which are efficient in terms of producing a small variance for a fixed simulation budget.

A main problem is that ruin is typically a rare event i. The chapter number is specified only when it is not the current one. Thus Proposition 4. References like Proposition A. Abbreviations c. E expectation. R s the real part of a complex number s. D [0, oo the space of R-valued functions which are right-contionuous and have left limits.

Unless otherwise stated, all stochastic processes considered in this book are assumed to have sample paths in this space. Usually, the processes we consider are piecewise continuous, i. Then the assumption of D-paths just means that we use the convention that the value at each jump epoch is the right limit rather than the left limit. In the French-inspired literature, often the term 'cadlag' continues a droite avec limites a gauche is used for the D-property. N it, a2 the normal distribution with mean p and variance oa2. Matrices and vectors are denoted by bold letters.

Usually, matrices have uppercase Roman or Greek letters like T, A, row vectors have lowercase Greek letters like a, 7r, and column vectors have lowercase Roman letters like t, a. In particular: I is the identity matrix e is the column vector with all entries equal to 1 ei is the ith unit column vector, i. Thus, the ith unit row vector is e'i. F o r a given set x1, Notation like f3i and 3 t in Chapter VI has a similar , though slightly more complicated, intensity interpretation.

FL, EL the probability measure and its corresponding expectation corresponding to the exponential change of measure given by Lundberg conjugation, cf. Chapter II Some general tools and results The present chapter collects and surveys some topics which repeatedly show up in the study of ruin probabilities. Due to the generality of the theory, the level of the exposition is, however, somewhat more advanced than in the rest of the book. The reader should therefore observe that it is possible to skip most of the chapter, in particular at a first reading of the book.

More precisely, the relevance for the mainstream of exposition is the following: The martingale approach in Section 1 is essentially only used here. All results are proved elsewhere , in most cases via likelihood ratio arguments. The likelihood ratio approach in Section 2 is basic for most of the models under study. When encountered for the first time in connection with the compound Poisson model in Chapter III, a parallel self-contained treatment is given of the facts needed there. The general theory is, however, used in Chapter VI on risk processes in a Markovian or periodic environment.

The duality results in Section 3 and, in part, Sections 4, 5 are, strictly speaking, not crucial for the rest of the book. The topic is, however, fundamental at least in the author' s opinion and the probability involved is rather simple and intuitive.

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Sections 4, 5 on random walks and Markov additive processes can be skipped until reading Chapter VI on the Markovian environment model. The more general Theorem 6. A simple calculation see Proposition III. From this it is readily seen see III. Corollary 1. However, as shown by the following example this set-up is too restrictive: typically', the parameters of the two processes can be reconstructed from a single infinite path, and F, P are then singular concentrated on two disjoint measurable sets. The interesting concept is therefore to look for absolute continuity only on finite time intervals possibly random, cf.

Theorem 2. Tt is absolutely continuous w. Pt The following result gives the connection to martingales. Proposition 2. F such that 2. This proves i. The precise meaning of this is the following: being. T9-measurable Y8. The converse follows since the class of r. F8-measurable r. A more elementary version along the lines of Theorem 2. The formulation has applications to virtually all the risk models studied in this book.

The result is a sample path relation, and thus for the moment no parametric assumptions on say the structure of the arrival process are needed. That is, instead of 3. The sample path relation between these two processes is illustrated in Fig. Theorem 3. The results can be viewed as special cases of Siegmund duality, see Siegmund []. Historically, the connection between risk theory and other applied probability areas appears first to have been noted by Prabhu [] in a queueing context.

Nevertheless, one may feel that the interaction between the different areas has been surprisingly limited even up to today. For discrete time random walks , there is an analogue of Theorem 3. For a given i. R -valued sequence Z1, Z2 , Theorem 4. Proof By 4. Proof Since YN, Thus the assertion of Theorem 4. Remark 4. Similarly, there is a more general version of Corollary 4. One then assumes Yn to be a stationary sequence, w.

Next consider change of measure via likelihood ratios. For a random walk, a Markovian change of measure as in Theorem 2. Then the change of measure in Theorem 2. In particular, 4. Breiman [78] p. We get: Corollary 4. In risk theory, they arise as models for the reserve or claim surplus at a discrete sequence of instants, say the beginning of each month or year , or imbedded into continuous time processes , say by recording the reserve or claim surplus just before or just after claims see Chapter V for some fundamental examples.

However, the tradition in the area is to use continuous time models. The appropriate generalization of random walks to continuous time is processes with stationary independent increments Levy processes. Note that the structure of such a process admits a complete description. Now consider reflected versions of processes with stationary independent increments.

Corollary 4. Proposition 4. Then the Markov process given by Theorem 2. Example 4. Recalling that U1, 0. As for processes with stationary independent increments , the structure of MAP's is completely understood when E is finite: 2and only there ; one reason is that in parts of the applied probability literature, MAP stands for the Markovian arrival process discussed below. That a process with this description is a MAP is obvious; the converse requires a proof, which we omit and refer to Neveu [] or cinlar [87].

If E is infinite a MAP may be much more complicated. As a generalization of the m. Proposition 5. By Perron-Frobenius theory see A. The corresponding left and right eigenvectors v " , h " may be chosen with strictly positive components. The function ic a plays in many respects the same role as the cumulant g. In particular, its derivatives are 'asymptotic cumulants', cf. Corollary 5. Proof By Perron-Frobenius theory see A. We also get an analogue of the Wald martingale for random walks: Proposition 5. Furthermore, Jeast- tK a h a J jj it L o is a martingale.

The argument is slightly heuristic e. For the second , we differentiate 5. Squaring in Corollary 5. Remark 5. From 5. In view of this discussion , we take the martingale property as our basic condition below though this is automatic in the finite case. An example beyond the finite case occurs for periodic risk processes in VI. Usually, some extra conditions are imposed, in particular that f is bounded;,for the present purposes, this is, however, inconvenient due to the unboundedness of ea8 so we shall not aim for complete rigour but interpret C in a broader sense.

In the infinite case , one can directly verify that 5. We omit the details. Then the MAP in Proposition 5. Bi [0] Remark 5. First note that the ijth element of Ft[a] is h. This shows that F, is absolutely continuous w. F:j with a density proportional to eei. Similarly, in continuous time 5.

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  5. Notes and references The earliest paper on treatment of MAP's in the present spirit we know of is Nagaev []. Though the literature on MAP's is extensive, there is, however, hardly a single comprehensive treatment; an extensive bibliography on aspects of the theory can be found in Asmussen [16]. Conditions for analogues of Corollary 5. For the Wald identity in Corollary 5.

    The closest reference on exponential families of random walks on a Markov chain we know of within the more statistical oriented literature is Hoglund [], which, however, is slightly less general than the present setting. On Fig. In simple cases like the compound Poisson model, the ladder heights are i. In other cases like the Markovian environment model, they have a semi-Markov structure but in complete generality, the dependence structure seems too complicated to be useful. In any case, at present we concentrate on the first ladder height. The main result of this section is Theorem 6.

    To illustrate the ideas, we shall first consider the compound Poisson model in the notation of Example 1. Theorem 6. But since St -4 -oo a. Figure 6. As above , the r. It follows that we can describe the Palm version M as follows. Note in particular that the Palm distribution of the mark size i. The result is notable by giving an explicit expression for a ruin in great generality and by only depending on the parameters of the model through the arrival rate 0 and the average in the Palm sense claim size EU0. The last property is referred to as insensitivity in the applied probability literature.

    Proof of Theorem 6. A standard argument for stationary processes [78] p. A further relevant reference related to Corollary 6. Some possibilities are numerical Laplace transform inversion via Corollary 3. For finite horizon ruin probabilities , see Chapter IV. It is worth mentioning that much of the analysis of this chapter can be carried over in a straightforward way to more general Levy processes.

    A common view of the literature is to consider such processes as perturbed compound Poisson risk processes , i. We do not spell out in detail such generalizations. See e. Proof It was noted in Chapter I that p - 1 is the expected claim surplus per unit time, and this immediately yields a. Obviously, the Uk - Tk are i. We return to this approach in Chapter V. Here is one immediate application: Proposition 1.

    For the proof, we need the following lemma: Lemma 1. The right hand inequality in 1. Thus using Lemma 1. A similar argument for lim sup proves a , and b , c are immediate consequences of a. Part d follows by a slightly more intricate general random walk result [APQ], p.

    Considering the next downcrossing which occurs w. There is also a central limit version of Proposition 1. The general case now follows either by another easy application of Lemma 1. Remark 1. Notes and references All material of the present section is standard. Summing over n, the formula for the distribution of M follows. Note that the distribution B0 with density bo is familiar from renewal theory as the limiting stationary distribution of the overshoot forwards recurrence time , [APQ] Ch.

    As a vehicle for computing tIi u , 2. The following results generalizes the fact that the conditional distribution of the deficit ST o just after ruin given that ruin occurs i. Theorem A1. Again, there is a general marked point process version, cf. In the risk theory literature, the Pollaczeck-Khinchine formula is often referred to as Beekman 's convolution formula, cf. Beekman [61], [62]. As shown in But claims are exponential , hence without memory, and hence this overshoot has the same distribution as the claims themselves.

    Integrating from u to oo, the result follows. Alternatively, use Laplace transforms. The result can, however , also be seen probabilistically without summing infinite series. For a failure at x, the current ladder step must terminate which occurs at rate S and there must be no further ones which occurs w. Example VIII. A variety of proofs are available. Corollary 3. The case of 3. We omit the details see, e.

    Of course, it is not surprising that such arguments are more cumbersome since the ladder height representation is not used. Also 3. In fact, either of these sets of formulas are what many authors call the Pollaczeck-Khinchine formula. In view of 3. F and c.

    The question then naturally arises whether ie is the c. The answer is yes: inserting in 4. Then FB denotes the probability measure governing the compound Poisson risk process with arrival intensity,0e and claim size distribution Be; the corresponding expectation operator is E9. The following result Proposition 4. Then the Xk are i. The identity 4.

    Ruin Probabilities (Advanced Series on Statistical Science and Applied Probability Series)

    Now consider a general G. The behaviour at zero is given by the first order Taylor expansion c a r. Equation 5. An established terminology is to call -y the adjustment coefficient but there are various alternatives around, e. Example 5. It is then readily seen that the non-zero solution of 5. Thus, Lundberg conjugation corresponds to interchanging the rates of the interarrival times and the claim sizes.

    Using 5. Inserting 5. Therefore, extensions and generalizations are main topics in the area of ruin probabilities, and in particular numerous such results can be found later in this book; in particular, see Sections IV. The mathematical approach we have taken is less standard in risk theory some of the classical ones can be found in the next subsection.

    The techniques are basically standard ones from sequential analysis, see for example Wald [] and Siegmund []. Of further proofs of Lundberg's inequality, we mention in particular the martingale approach, see II.

    Next consider the Cramer-Lundberg approximation. Here the most standard proof is via the renewal equation in Corollary 3. Note that by 5. It is then a matter of routine to verify the conditions of the key renewal theorem Proposition A1. Easy calculus now gives 5. This excludes heavy-tailed distributions like the log-normal or Pareto, but may in many other cases not appear all that restrictive, and the following possibilities then occur: 1.

    Case 3 may be felt to be rather atypical, but some non-pathological examples exist, for example the inverse Gaussian distribution see Example 9. Notes and references Ruin probabilities in case 3 with y non-existent are studied, e. To the present authors mind, this is a somewhat special situation and therefore not treated in this book. Section 7c : Proposition 6. Hence by Taylor expansion, the inequality in 6. However, it needs to be used with caution say in Lundberg's inequality or the Cramer-Lundberg approximation, in particular when u is large.

    Example 6. According to Corollary 3. Notes and references The approximation was introduced by Beekman [60], with the present version suggested by Bowers in the discussion of [60]. In order to make the processes look so much as possible alike, we make the first three cumulants match, which according to Proposition 1. Notes and references The approximation 7. Though of course it is based upon purely empirical grounds, numerical evidence e. Grandell [] pp. Proposition 7. Letting Bo be the stationary excess life distribution, we have according to the Pollaczeck-Khinchine formula in the form 3.

    Notes and references Heavy traffic limit theory for queues goes back to Kingman []. The present situation of Poisson arrivals is somewhat more elementary to deal with than the renewal case see e. We return to heavy traffic from a different point of view diffusion approximations in Chapter IV and give further references there. In the setting of risk theory, the first results of heavy traffic type seem to be due to Hadwiger []. Numerical evidence shows that the fit of 7.

    Mathematically, we shall represent this situation with a limit where 3 10 but B is fixed. Of course, in risk theory heavy traffic is most often argued to be the typical case rather than light traffic. However , light traffic is of some interest as a complement to heavy traffic , as well as it is needed for the interpolation approximation to be studied in the next subsection.

    Indeed, by monotone time T of the first claim , i. Again, the Poisson case is much easier than the renewal case. Sigman []. Light traffic does not appear to have been studied in risk theory. Let OLT u denote the light traffic approximation given by Proposition 7. Thus , even if the safety loading is not very small, one may hope that some correction of the heavy traffic approximation has been obtained.

    Another main queueing paper is Whitt [], where further references can be found. The adaptation to risk theory is new; no empirical study of the fit of 7. Recall that B ' is said to be stochastically smaller than B 2 in symbols, B ' f BM y dy x x for all x; an equivalent characterization is f f dB ' r 2 u probabilities, the proof is complete. Example 8. Let ua denote the a fractile of the ruin function, i. One then obtains the following table: U U0. Note to make the figures comparable, all distributions have mean 1. A standard example from queueing theory is 9. Example 9. Similar conclusions will be found below.

    Proof This is an easy time transformation argument in a similar way as in Proposition 1. Similar notation for partial derivatived are used below, e. Proposition 9. Differentiating w. Of course, we cannot expect in general to find explicit expressions like in Example 9. Barndorff-Nielsen [58].

    Stochastic Systems

    By dominated convergence and 9. Here 9. That it is no restriction to assume one of the ti x to be linear follows since the whole set-up requires exponential moments to be finite thus we can always extend the family if necessary by adding a term Ox. That it is no restriction to assume k 2, we can just fix k - 2 of the parameters. Notes and references The general area of sensitivity analysis gradient estimation is currently receiving considerable interest in queueing theory. However, the models there e. Thus, the main tool is simulation, for which we refer to X. Comparatively less work seems to have been done in risk theory; thus, to our knowledge, the results presented here are new.

    Van Wouve et al. For the proof, we need a lemma. Lemma Proof of Theorem Then r. Combining Theorem Notes and references Theorem Asmussen [23] can then be used to produce an estimate of ry. This approach in fact applies also for many models more general than the compound Poisson one. The notation is essentially as in Chapter III. In particular, the premium rate is 1, the Poisson intensity is 0 and the claim size distribution is B with m. Further let 'Yo be the unique point in 0, 'y where c a attains it minimum value.

    See Fig. By the likelihood identity III. FL and independent of T u. Thus the l. But by the fundamental likelihood ratio identity Theorem Note that it follows from Proposition 1.

    Digital book Ruin probabilities (2nd edition) (Advanced Series on Sta…

    Y u belonging to a convolution semigroup. Corollary Since U1 ,T, U2,T, EN has an Erlang distribution with parameters N, 1 , i. Then see Prabhu [] pp. Notes and references Proposition 1. Related formulas are in Takacs []. Seal [] gives a different numerical integration fomula for 1 - 0 u,T which, however, is numerically unstable for large T. Obviously, this integral is 0 if STv.

    The proof is combined with the proof of Theorem Notes and references For Theorems 2. Lemma 3. Then by Proposition 2. Laplace transform of the ruin time T u : Corollary 3. For the general compound Poisson model, the known results are even less explicit than for the exponential claims case, and take basically the form of approximations and inequalities. Later results then deal with more precise and refined versions of this statement. For the proof, we need the following auxiliary result: Proposition 4.

    Proposition A1. This proves the first assertion of 4. For 4. Theorem 7. Notes and references Theorem 4. Nummelin , Large deviations of uniformly recurrent Markov additive processes. Latouche , M. Remiche and P. Taylor , Transient Markov arrival processes. Lee and C. Un , A study of on-off characteristics of conversation speech. Lehtonen and H. Nyrhinen , On asymptotically efficient simulation of ruin probabilities in a Markovian environment.

    MR 93h MR Zbl Queueing Systems 36 Markopoulou , F. Tobagi and M. Karam , Assessing the quality of voice communications over Internet backbones. Mogulskii , Large deviations for trajectories of multi-dimensional random walks. Nagaev , Large deviations of sums of independent random variables. Hamburg 25 Nummelin , Markov additive processes. Eigenvalue properties and limit theorems. Nummelin , Markov additive processes II.

    Large deviations. Soyer , Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Methods Oper. Pacheco and N.

    Søren Asmussen

    Prabhu , Markov-additive processes of arrivals , Advances in queueing, Probab. Stochastics Ser. Puhalskii , Large deviation analysis of the single server queue. Queueing Systems 21 Puhalskii and W. Whitt , Functional large deviation principles for first-passage-time proc esses. Rieder and N. Journals Seminars Books Theses Authors. Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. Article First Online: 20 September This is a preview of subscription content, log in to check access. Albrecher H, Kortschak D Asymptotic results for the sum of dependent non-identically distributed random variables.

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