© Hindawi Publishing Corp. MODEL TRACKING FOR RISK PROBLEMS

We assume that we have M candidate insurance models for describing a process. The models considered consist of a risk process driven by right-constant, finite-state spaces, jump processes. Based on observing the history of the risk process, we propose dynamics whose solutions indicate the likelihoods of each candidate model.


Introduction.
Risk theory deals with stochastic models in insurance business, see, for example, Grandell [2].Usually, in such models claims are described by point processes and the amounts claimed by policies holders are sequences of random variables.The profit, or the loss, of the company is the difference between premiums income and the claims.In this paper, we assume that we have M competing models, denoted by {H 1 ,...,H M }, describing the risk process, see Section 2. We are interested in ranking the candidate models based on their likelihood of being most appropriate for describing the risk process and some other processes driving the risk process.This problem as well as others fall within the category of Model Tracking or Detection problems as we are interested in tracking (or detecting) the most appropriate model for describing the proposed risk model, see, for example, Poor [5] and Snyder [6].
In the next section, we present the model of the paper.The main result of the paper is found in Section 3 where the likelihood that our model is best described by a certain candidate model is derived.In Section 4, a filtering problem is discussed.
2. The model.Assume initially that all processes are defined on a probability space (Ω, Ᏺ,P).
Consider an insurance "risk process" R which at time t is the sum of an initial capital R 0 , an integrated premiums process with integrand a nonnegative, bounded, and measurable real-valued function P (•), a new premiums process, a lost premiums process, and a claims process.We also assume that we have M candidate models denoted by {H 1 ,...,H M } representing the dynamics of the risk process.Then, under the hypothesis that model H h is used, h = 1,...,M, we have where Y i H h (•), i = 1, 2, 3, are bounded nonnegative functions and each ν i , i = 1, 2, 3, is an integer-valued random measure which, under probability measure P , has predictable compensator (see Jacod [3]) νi function of Z i t .Here Zi t , i = 1, 2, 3, t ∈ R + , are finite-state spaces processes with right-constant sample paths on the state spaces Si = {s i 1 ,...,s i n i }; s i will denote the (column) vector (s i 1 ,...,s i n i ) .Suppose 1 ≤ ≤ N, and for j ≠ and φ i (x) = π i (x)/π i (s ); then φ i (s j ) = δ j and φ i = (φ Since Z i t is a jump process taking values in the vector space R n i we can write Here (2.4) We assume that each Z i t has almost surely finitely many jumps in any finite interval so that the random measure µ Z i is σ -finite.Let μZ i (dr , e i j ) be the predictable compensator of µ Z i so that (2.5) leads to where (2.6) Now μZ i factors into its Lévy system μZ i dr , e i j = β e i j ,Z i r − ,r dF Z i r − ,r , ( where dF (Z i r − ,r ) represents the conditional probability that the next jump occurs at time r given the previous history of the process.
Assume that the nonnegative measure dF (Z i r − ,r ) is absolutely continuous with respect to Lebesgue measure so that for some nonnegative function f (•).
we have, from (2.8) and (2.9), that (2.9) (2.10) Define the matrix A i (r , ω) = {a i jk (r , ω)}.Then we have the representation (2.11) We assume here that the Z i 's have no common jumps, that is, with denote the complete filtration generated by the risk process and let be the complete filtration generated by the risk process R and the processes Z i , i = 1, 2, 3. Now, given the filtration , and, a set of competing hypotheses {H 1 ,...,H M }, where , we want to determine the dynamics to compute the posterior probabilities (2.15) Consider a simple random variable α, where α ∈{f 1 ,...,f M } and f h = (0,...,1,...,0) ∈ R M .The "1" here is in position h.We suppose α is an indicator function such that α = f h , that is, α, f h = 1 if and only if hypothesis H h holds.Then (2.15) may be rewritten as where the expectation is taken under probability measure P .In Section 3, we propose dynamics to (2.15) whose solution is a solution of some stochastic differential equation.Section 4 is concerned with a filtering problem.

M-ary detection filters.
Suppose P is a reference probability, under which ν i , i = 1, 2, 3, have deterministic compensators H i (dx)dt independent of Z i , i = 1, 2, 3.In order to recover the "real world" probability measure P under which the model dynamics introduced in Section 2 hold, define the Radon-Nikodym derivative Λ such that where (see Jacod and Shiryaev [4]) However, in this section, we will be working under the "reference probability" P .By an abstract version of Bayes' rule (see [1]) The unnormalized probability q h t is given by the equation x −1 q h u− ν i (du, dx)−H(dx)du . (3.5) Here E[ Z i u− ,e i j | u− ] is evaluated under the probability measure P , given that the hypothesis H h holds.
Proof.Using (3.2), we have (3.7) Using (3.4) and [7, Chapter 7, Lemma 3.2] to exchange stochastic integration and conditional expectation under P , we have which, using elementary rules for conditional probabilities and Bayes rule, is Using the notation Z i t = 3 J=1 Z i t ,e i j e i j gives (3.5).
Note that the normalized form of (3.5) is given by (3.12) As an example: suppose that the set of candidate models consists of two models, that is, α ∈ {(1, 0), (0, 1)} and Define the log-likelihood or test statistic process, Large values of l are in favor of model 1 whereas, small values of l are in favor of model 2.

The filtering problem. Equation (3.5) contains E[ Z i u− ,e i j | u− ].
The following result gives the dynamics of the unnormalized version of this filter.Here we assume that the random matrix A is adapted to the filtration .Again we work under the "reference probability" P , under which ν i , i = 1, 2, 3, have deterministic compensators The unnormalized probability process σ t ( ,m,n) satisfies the stochastic integral equation Proof.Note that (2.11) gives Since the processes Z 1 t and Z 2 t share no common jumps, Simplifying the integrand in the stochastic integral in (4.6) gives Conditioning each side of (4.6) on t , under the measure P , and using again [7, Chapter 7, Lemma 3.2] to exchange stochastic integration and conditional expectation establishes the result.
In this paper, a risk model described by M candidate models was discussed.Detection filters were derived using measure change techniques.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: