We propose a situation in which a single employee is responsible for processing incoming claims to an insurance company that can be classified as being one of two possible types. More specifically, we consider a priority-based system having separate buffers to store high priority and low priority incoming claims. We construct a mathematical model and perform queueing analysis to evaluate the performance of this priority-based system, which incorporates the possibility of claims being redistributed, lost, or prematurely processed.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

<p>Doktorska disertacija je posvećena re&scaron;avanju nelinearnih diferen&shy; cijalnih jednačina, kao i linearnih diferencijalnih jednačina sa singularite-tim a u okviru prostora Kolomboovih uop&scaron;tenih funkcija. U osnovi, dis&shy;ertacija se može podeliti na dva dela. Prvi deo disertacije je posvećen re&scaron;avanju determinističkih parcijalnih diferencijalnih jednačina primenom teorije polugrupa operatora definisanih na prostorima Kolomboa. Drugi deo disertacije posvećen je re&scaron;avanju stohastičkih običnih i parcijalnih dife&shy;rencijalnih jednačina. Ove jednačine sadrže Kolomboove uop&scaron;tene slučajne procese kao nelinearni deo, ili kao početne uslove.</p> / <p>Doctoral thesis is devoted to nonlinear differential equations, as well as linear differential equations with singularities in the framework of Colombeau generalized function spaces. Basically, the thesis can be devided into two parts. The first part is devoted to solving deterministic partial differential equations applaying semigroup theory where those semigroups are defined on Colombeau spaces. The second part of the thesis is devoted to stochastic ordinary and partial differential equations. Those equations contain Colombeau generalized stochastic processes as nonlinear part, or as initial data.</p>

<p>Doktorska disertacija je posvećena primeni teorije polugrupa operatora na re&scaron;avanje dve klase Cauchy-jevih početnih problema. U prvom delu smo<br />ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji<br />generi&scaron;e <em>C</em>₀&minus;polugrupu i linearnim ograničenim operatorom kombinovanim<br />sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-It&ocirc;-ovom<br />haos ekspanzijom. Dokazali smo postojanje i jedinstvenost re&scaron;enja ove klase<br />SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po<br />vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene<br /><em>C</em>-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim<br />prostorima. Kompleksne stepene operatora smo posmatrali kao integralne<br />generatore uniformno ograničenih analitičkih <em>C</em>-regularizovanih rezolventnih<br />familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema vi&scaron;3eg ili necelog reda.</p> / <p>The doctoral dissertation is devoted to applications of the theory<br />of semigroups of operators on two classes of Cauchy problems. In the first<br />part, we studied parabolic stochastic partial differential equations (SPDEs),<br />driven by two types of operators: one linear closed operator generating a<br /><em>C</em>₀&minus;semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-It&ocirc;<br />chaos expansions. We proved existence and uniqueness of solutions for this<br />class of SPDEs. In particular, we also treated the stationary case when the<br />time-derivative is equal to zero. In the second part, we constructed com-plex powers of <em>C</em>&minus;sectorial operators in the setting of sequentially complete<br />locally convex spaces. We considered these complex powers as the integral<br />generators of equicontinuous analytic <em>C</em>&minus;regularized resolvent families, and<br />incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.</p>