Stochastic Control For Constrained Lévy Processes In Insurance And Finance

In practical risk markets, there always exist regulations such as minimum cash requirement, investment constraints and so on. In the context of stochastic control in finance and insurance, this thesis investigates Levy processes under regulations and its applications. By use of renewal argument, PDMP method, martingale theory, QVI and numerical approximations, this thesis focuses on value functions, including the long run average profit function per unit time of an insurance firm and the related solvency studies (e.g. the expected discounted dividends, the moment generating function of total dividends, the Gerber-Shiu expected discounted penalty function) and so on. The thesis is organized as follows.In chapters 2-4, the risk model of an insurance firm is investigated under regulation imposed by the regulatory authority. That is, the regulator exercises a minimum cash requirement level and penalties for violating it to regulate the insurance firm. These three chapters extend and deepen the studies in Tapiero et al. (1983), where a joint insurance corporation-regulatory authority problem was investigated in a classical risk model. In the fourth chapter, the dividend optimization problem is investigated for a diffusion model under linear investment constraints. At last, the Gerber-Shiu expected discounted penalty function is obtained with stochastic income and a barrier dividend strategy.First, we consider a classical risk model with heavy tailed claims, included in a regulation mechanism of minimum cash requirement. The problem of the insurance firm is to establish an investment and risk exposure policy as well as a barrier dividend strategy, which maximizes the long run average profit per unit time. The strategy of the insurance firm is a function of the strategy used by the regulator. For regularly varying tailed claim size distributions, we find the asymptotics of the stationary distribution of the risk model and derive fundamental asymptotic results of the insurance firm's problem. In the special case of Pareto claim size distributions with special parameters setting, the asymptotic optimal control policy is found in closed form, as well as numerical results.Then, chapter 3 investigates a Levy risk model with the same regulation as in Chap-ter 2. Under the given regulation, the insurance corporation maximizes its long run av-erage profit per unit time, by choosing its investment/(non-cheap) reinsurance/dividend policy. In addition, it is assumed that proportional transaction cost occurs, when short term investments is converted into cash. Explicit expressions of the long run average profit per unit time, of the regulatory authority's cost function are derived. For the case of non transaction cost, a joint insurance corporation-regulatory authority problem is also investigated, which is in the concept of Stackelberg strategies. Finally, by variable transformations in the numerical solution of Volterra integral equations for the station-ary distributions, the resulting values of the optimal control policy without traction cost are approximated numerically.The PDMP method and martingales are used to solvency studies in Chapter 4 for the classical risk model under the same regulation as in Tapiero et al. (1983). Chap-ter 4 focuses on the discounted value functions, which are different from the long run average profit function in Chapter 1 and Chapter 2. The risk model includes three features, namely debit interest, short-term and long-term invested interest, barrier div-idend strategy. We derive integro-differential equations under absolute ruin for the ex-pected discounted dividends and its moment generating function, and the Gerber-Shiu expected discounted penalty function. In the case of exponential claim amounts, explicit expressions of the corresponding value functions are obtained, as well as their numerical illustrations.Chapter 5 investigates the dividend optimization problem of a linear diffusion model with linear investment constraints and dividend transaction costs. Moreover a corpora-tion as a small investor can invest its reserve in a classical Black-Scholes market without paying transaction fees. The main feature of this chapter is that there exists general linear constraints on investments including the special case of short-sale and borrowing constraints. This results in a regular-impulse stochastic control problem. By character-izing the value function (the expected discounted dividends), then it is a once continuous viscosity solution of the corresponding quasi-variational inequalities (QVI). The nontriv-ial case is that the investment can't meet the loss of wealth due to discounting with positive market risk price. In this case, delicate analysis is carried out on QVI w.r.t three possible situations, leading to an explicit construction of the value functions to-gether with the optimal investment/dividend policies. We also give a brief conclusion of other trivial cases and apply the derived results into explicit examples numerically.At last, the ruin problem is investigated with stochastic income and barrier dividend strategy, which extends the results of Albrecher and Kainhofer (2002) and Bao (2006). Firstly, this chapter considers the expected discounted penalty with common distributed claim amounts and non-linear dividend barrier, and obtains the limit solution without barrier dividends; then the results of stationary renewal process and PH renewal process are derived for fixed constant dividend barrier. Finally, the conclusions are applied in ruin probabilities, probability distribution of surplus prior to ruin and the probability of surplus arriving at dividend barrier before ruin, as well as numerical examples.