Applications Of Stochastic Control Theory In Finance And Insurance

Risk theory is the most theoretical part in insurance mathematics which derives from risk management in insurance.It is mainly about several actuarial variables such as ruin probability,the surplus immediately prior to ruin,the deficit at ruin etc.The applications of stochastic process and stochastic analysis,especially the martingale method,in risk theory make risk theory develop rapidly.Many fruitful achievements in characterizing several actuarial variables mentioned above have been made.As the development of finance and insurance markets,these actuarial variables have no longer met the needs of the insurance company.The insurance company is more and more concerned about how to minimize these actuarial variables which represent the risk instead of their expression.In order to minimize them,the insurance company will take the appropriate measures such as reinsurance,investment in finance market.At this time the problem faced by insurance company is how to find the optimal reinsurance or/and investment strategy to minimize the risk.In addition,to the ruin probability,the insurance company also uses the other measures to measure the risk such as the variance of terminal wealth,VaR(Value-at-Risk), etc.At the same time,the insurance company is also concerned about some other variables which represent their utilities and profits such as the utility of the terminal wealth and the expected discounted dividend payments before ruin.All of these belong to stochastic control problems.In the last few decades,in order to solve these problems,stochastic control theory-has been applied into the insurance.For example,[24]and[3]respectively obtained optimal investment strategy and the optimal dividend strategy under diffusion model by the Hamilton-Jacobi-Bellman(HJB) equation approach.Their works are the pioneer of the combination of the insurance and stochastic control theory.Since then,there have been many papers in which the HJB equation was used to solve the optimal control problems in insurance.However,in order to get a perfect result,most of paper usually exclude the possibility of interference factors.For example,there are not any transaction costs when dividends are paid out;there are not any solvency constrains when the optimal dividend is considered;there are not any attention on the interests of reinsurer when the optimal reinsurance is considered,and so on.In effect,these factors can be not ignored since their existence make the optimal strategies presented not to be optimal.Therefore, it is necessary to consider the corresponding optimal strategy again.In most of previous references,only the HJB equation method was applied.Consequently,the original process have to be Markov process;the problems considered have to be like(1.1.4)和(1.1.7);the constructed solution must satisfy the condition given by verification theorem.However, most of the problems which will be considered in my doctoral dissertation do not satisfy these conditions.For example,the optimal problem under Fractional Brownian motion model and the game problem in reinsurance market.Hence,to solve these problems,we must look for the new methods.On the basis of these reasons,my doctoral dissertation will be devoted to doing the following three aspects:Firstly,I will make the model and problem considered more practical.Secondly,the methods will not be limited on HJB equation,and I will try to find the corresponding new optimal control methods according to the current model and problem.Finally,I will try my best to give the very explicit expressions of optimal solutions.In the following,I will introduce the content of every Chapter and Section in detail.A simply introduction of stochastic control theory which will be used in the following chapters are given in the first chapter.Most of the contents are borrowed from[38]and[89].In Chapter 2,a series of optimal dividend problems are considered.Dividend is payments that the company give to the shareholder or the person who provides the initial surplus.Hence,in a sense,the total dividend payments represent company's benefit.Thus,how to pay dividends to maximize the total dividend payments is always one of the most hot topics in finance and insurance.This classical dividend problem has been solved very well and the explicit optimal solution has been given.These results show that the optimal dividend strategies under Brownian motion with drift and compound Poisson model are barrier strategy and band strategy,respectively.These results are obtained under the assumption that there are not transaction costs when the dividends are paid.But,in practice,to prohibit the continuous trade,some transaction costs are required when the dividend are paid.Even though it may be very few,it will affect the optimal strategy.It is obvious that barrier strategy and band strategy are no longer optimal in that case.Therefore,we need to look for the optimal strategy again. Due to the transaction costs,we have to consider not only the optimal dividend payments but also the optimal dividend time.This makes the optimal dividend problem more difficult than the classical optimal dividend problem.Only[28]gives the optimal proportional reinsurance and dividend strategy under the risk model of Brownian motion with drift. Now,there are still many unsolved optimal dividend problems with transaction costs.For example,the optimal dividend problem with the excess-of-loss reinsurance,the optimal dividend problem under the compound Poisson model,and the optimal dividend problem with solvency constrains.We will consider these problems in Section 2.1,Section 2.2 and Section 2.3.In the final section,we relax the restrictive assumptions made in the previous literature,and consider the optimal dividend problem under an extended family of diffusion processes.A completely different result from the previous results is obtained.In section 2.1,we consider the optimal excess-of-loss reinsurance and dividend strategy subject to transaction cost and taxes.Firstly,it is proved that the excess-of-loss reinsurance is better than the proportional reinsurance,that is,the corresponding total discounted expected dividend payments is larger.Then,by solving the corresponding quasi-variational inequality,we obtain the analytical solutions to the optimal return function and the optimal strategy.In[4],they introduced some associated functions to solve the HJB equation. Comparing with[4],the quasi-variational inequality for our problem is more difficult to solve.Moreover a different and more simple method is employed In Section 2.2,we study optimal dividend problem in the classical risk model when payments are subject to both transaction cost and taxes.The optimal dividend problem in the classical risk model is always one of the most difficult topics.The main reason is that the optimal equation has neither the boundary condition nor continuously differentiable solution.In this section, we construct the solution of quasi-variational inequality when the claims are exponentially distributed.We also find a formula for the expected time between dividends.The results show that,as the dividend tax rate decreases,it is optimal for the shareholders to receive smaller but more frequent dividend payments In Section 2.3,we consider a company where surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments.With each dividend payment there are transaction costs and taxes and it is shown in[93]that under some reasonable assumptions,optimality is achieved by using a lump sum dividend barrier strategy,i.e.there is a upper barrier(?)^* and a lower barrier u^* so that whenever surplus reaches(?)^*,it is reduced to u^* through a dividend payment.However,these optimal barriers may be unacceptably low from a solvency point of view.It is argued that in that case one should still look for a barrier strategy,but with barriers that satisfy the given constraints.We propose a solvency constraint similar to that in[92];whenever dividends are paid out the probability of ruin within a fixed time T and with the same strategy in the future,should not exceed a predetermined levelε.It is shown how optimality can be achieved under this constraint, and numerical examples are given.We firstly give lump barrier constrains.This constrains means that there are two barriers: upper barrier and low barrier.The company is allowed to pay dividend only when the reserve exceed the upper barrier and the reserve after dividend is not smaller than the low barrier.We consider the optimal dividend with lump barrier constrains in two steps. We firstly consider the optimal dividend problem with the low barrier constrains.Then, on the basis of the results obtained in the first step,the optimal dividend problem with the upper barrier constrains is considered.The corresponding quasi-variational inequalities are used in both two steps.But,in the second step,the corresponding quasi-variational inequality has no longer continuously differentiable solution.Hence,Ito Formula can not be used,and then the verification theorem does not hold.To tackle this difficulty,we skillfully use the theory of local time to prove that the constructed solution is equal to the optimal value function,and give the optimal strategy.This is the first time to give the verification theorem in non-continuously differentiable case.In the following,we need to look for a lower barrier u_ε＞0 and an upper barrier u_εthat maximize expected discounted Dividends.Meanwhile,the solvency constraints are satisfied as follows:whenever capital is at u_ε,ruin within a fixed time T by following the lump barrier strategy(u_ε,u_ε) should not exceed a small,predetermined numberε.This problem is more difficult than that in[92]since we must look for a pair(u_ε,u_ε),not just a number u_ε.One issue is to find a fast method to calculate the ruin probability for a given lower and upper barrier,and we will show how we can adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicolson algorithm to solve the relevant partial differential equations.The section ends with numerical examples.In Section 2.4,we consider a company where surplus follows an extended family of diffusion processes and whose objective is to maximize expected discounted dividend payments. An extended family of diffusion processes means that we relax restrictive assumptions. In this case,the property and form of the solution of quasi-variational inequality are different from that in previous literature.We provide the new idea to classify the solutions according to their properties,and then prove that there are three possibilities for the optimal dividend strategy:(1) the optimal dividend strategy is a lump sum dividend barrier strategy;(2) the optimal dividend strategy is a lump sum dividend band strategy; (3) the optimal dividend strategy does not exist.Among of them,the most difficult one is the case(2).Since,in this case,all of previous methods to construct the optimal value function and the optimal strategy are no longer valid.To tackle this difficulty,we consider two independent solutions instead of only one solution,and construct the optimal value function respectively in four intervals.Moreover,it is the first time to show clearly that the optimal dividend strategy is lump sum dividend band strategy and give the explicit expression of the strategy.In Chapter 3,the problems of minimizing ruin probability and maximizing the exponential utility are studied.In recent years,the optimization problems of maximizing the exponential utility and minimizing the probability of ruin have been studied by many authors.There exist relationships between the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin.For the case of an ordinary investor(no external risk process) in the discrete time risk model,[37]found the optimal investment strategy that maximizes the exponential utility with utility function of the form u(x)=-e^-mx.He conjectured that such a strategy also minimizes the probability of ruin for some value of m under the assumption that the investor was allowed to borrow an unlimited amount of money,and there was no risk-free interest rate.For the investors which face a random risk process correlated with the risky asset process.[24]show that,with zero interest rate,the investment strategy of maximizing the exponential utility also minimizes the probability of ruin for a specific value of m,which validates Ferguson'conjecture in a strong sense, [24]also show that the conjecture did not hold when the interest rate is positive.Reinsurance is also an important technique for insurance companies seeking to control their risk.To make the utility and ruin analysis even more realistic,we consider incorporating the concept of proportional reinsurance into Browne's model.That is,under these two controls,we consider three optimization problems,namely the problem of maximizing the exponential utility of terminal wealth,the problem of minimizing the probability of ruin,and the problem of minimizing the expected discounted penalty of ruin.By the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that the optimal strategies for maximizing the exponential utility and minimizing the probability of ruin are equivalent.This validates Ferguson's conjecture under current model.For the optimal investment problem,only the single risky asset(stock) is considered in previous literature.Actually,the insurance company,to decrease(increase) their risk (profit),will invest its wealth into multiple risk assets.Hence,we in section 3.2 consider that the insurance company invests its wealth in a financial market consisting of a risk-free asset and n risky assets.The risk model considered is the same as Section 3.1,The insurer is allowed to purchase proportional reinsurance as well as proportional reinsurance.Under the constraint of no-shorting,we consider two optimization problems:the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin.By solving the corresponding Hamilton-Jacobi-Bellman equations,explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risk-free interest rate,the results indicate that Ferguson's conjecture still hold under the current model.In Section 3.3,following the same framework as section 3.2 we consider the excess-of-loss reinsurance instead of proportional reinsurance.First we show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions.Then,by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained.In particular,when there is no risky-free interest rate,the results indicate that Ferguson's conjecture still hold in excess-of-loss reinsurance case.In Chapter 4.Mean-Variance problem in insurance is discussed.The portfolio selection is to allocate the wealth into various assets to disperse risk and ensure the profit.In 1952,Markowitz uses the variance to measure the risk of stock,and provides mean-variance analysis methods for the portfolio selection.This open a prelude to modern finance.Mean-variance problem is that the investors want to find the optimal investment strategy to maximize the mean and minimize the variance.Mean-variance is not only the pioneering work of modern portfolio selection theory but also one of footstone of modern finance.In recent years,finance and insurance market has started to link.In order to increase its profit or decrease its risk,the insurance company will invest the part of its reserve into financial market.There has been many literatures which considered the optimal investment.But the optimal criteria are still several classical actuarial variables such as dividend payments,the probability of ruin and so on.Mean-variance is rarely considered in insurance field,while it is one of the most important and popular conceptions in finance filed.As far as we know,only[120]consider the optimal investment problem under meanvariance criteria and[27]consider the optimal reinsurance/new business under meanvariance criteria.We will do the further research about mean-variance problem in this chapter.In Section 4.1,we study optimal reinsurance/new business and investment(no-shorting) strategy for the mean-variance problem in two risk models:a classical risk model and a diffusion model.The problem is firstly reduced to a stochastic linear-quadratic(LQ) control problem with constraints.Then,the efficient frontiers and efficient strategies are derived explicitly by a verification theorem with the viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations,which is different from that given in[129].Furthermore,by comparisons,we find that the optimal solutions are identical under the two risk models Hitherto,people always model the claim processes by stochastic processes with Markovian. However,in most cases,the insurance claims often present long-range dependence: the behavior of the surplus of an insurer after a given time t not only depends on the information at t but also on the whole history up to time t.This phenomenon is not negligible and likely to have an impact on various issues such as solvency,pricing and optimal retention reinsured,etc.Therefore,fractional Brownian motion has been used recently to model the claims that an insurance company may face.Most of literature consider the ruin problem.Since the property of fractional Brownlan motion is difficult to describe, there are always not very explicit results.A fewer authors consider the optimal control under fractional Brownian motion.It is not only because the fractional Brownian motion is difficult to research,but also it is not Markov process,and then the HJB equation method can be not used.As far as we know,only[63]consider linear-quadratic(LQ) problem. However,most of results are obtained under the Markovian controlled system.Therefore, it is valuable in theory and practice to study the optimal control problems under the more extensive environment.In Section 4.2,we consider the optimal investment problem under mean-variance criteria for an insurer.The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H∈(1/2,1).Due to non-Markovian of fractional Brownian motion, the well-known HJB equation is not applied.However,stochastic calculus for fractional Brownian Motion introduced in[36]make fractional Brownian Motion have many similar properties to Brownian motion such as Zero mean,Girsanov Theorem.By these properties ,we find the method to solve the optimal problem under a classical risk process that is perturbed by a standard fractional Brownian motion.By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained.Moreover,when H→1/2+the results converge to the corresponding(known) results for standard Brownian motion.Section 4.3 considers that the risk process is a compound Poisson process and the insurer can invest in a risk-free asset and multiple risky assets.We obtain the optimal investment policy under mean-variance criteria using the stochastic linear quadratic(LQ) control theory with no-shorting constraint.Then the efficient strategy and efficient frontier are derived explicitly by a verification theorem provided in Section 4.1.In Chapter 5,we consider optimal control problems when risk processes modeled by (fractional) Brownian motion with drift.In Section 5.1,we model the surplus process as a compound poisson process perturbed by Brownian motion,and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem reduces to a version of linear quadratic regulator under jump diffusion process.It is treated by three methods:dynamic programming,completion of square and stochastic maximum principle.The analytic solutions to the optimal control and the corresponding optimal value function are obtained.In this section,we introduce the methods of completion of square and stochastic maximum principle into the insurance filed.These two methods have many differences from the HJB equation method.Especially,the original process is no longer required to be Markov process.Hence,these give some inspiration to solve the optimal problem under Non-Markov process.The following Section 5.2 is a good example.We in this section consider a classical risk model perturbed by a standard fBm with Hurst parameter H∈(1/2,1).The criteria is the same as section 5.1.By using the completion of squares method,the expression of the optimal value function and the corresponding optimal control are derived.In addition,the method in Section 4.2 is also inspired by section 5.1.In Chapter 6,some other optimal control problems are investigated.There has been many literatures which consider the optimal reinsurance,such as[24], [58],[4].However,all of them only consider the insurance company.In practice,if we only consider the insurance company,the optimal reinsurance strategy obtained is often unacceptable for the reinsurance company.This alarms us that there are two parties in the reinsurance contract,their interest collide with each other.Therefore,we need to make a reciprocal reinsurance treaty.This is a game problem.[21]firstly use the game theory to solve the optimal reinsurance problem in insurance filed.The optimal criteria is maximizing the expected exponential utility.Since then,[13],[117]make some research for risk exchanges and reinsurance markets.However,in these papers,only the single period model is considered.It will obviously be of interest to extend the single model to multi-period,or a continuous time model.In this sectioin,we will consider a continuous time model.In this case,the interest of the insurance company and the reinsurance company always collide with in the whole time interval,which is different from the case of the single period.Hence,to solve the optimal problem,we need to look for the new and different methods.In this section,we solve the problem in two steps.Firstly we treat the Pareto optimal problem about the terminal wealth.Then we find the reinsurance strategy to copy the Pareto optimal terminal wealth.By this method,we show that the optimal Pareto reinsurance in continuous time is proportional reinsurance.Moreover,it is proved that the core of the cooperative reinsurance game is non-empty.This section intend to provide one method to solve the optimal problem in continuous time reinsurance market. Many other problems are still unsolved such as the optimal dividend,ruin probability, mean-variance.Section 6.2 deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modeled as an Ornstein-Uhlenbeck process.Here,only the stock price and interest rate can be observable for an investor.It is reduced to a partially observed stochastic control problem.Combining the filtering theory with dynamic programming approach,explicit representations of the optimal value functions and corresponding optimal strategies are derived.Moreover the closed-form solutions are provided in two cases of exponential utility and logarithmic utility.In particular,logarithmic utility is considered under the restriction of short-selling and borrowing.[76]uses martingale method to obtain the optimal strategy.Here,the investor is not allowed to short-sell and borrow,and a simpler method is employed to solve the problem.Moreover,the maximal logarithmic utility is given in the case of a single stock.In section 6.3,the basic claim process is assumed to follow a Brownian motion with drift.In addition,we allow the insurer to invest in a risky asset and to purchase cheap proportional reinsurance.Under these two controls,we consider the problem of minimizing expected time to reach a given capital before ruin.By using the Hamilton-Jacobi-Bellman equation,explicit expressions for optimal value function and optimal strategy are obtained.By the HJB equation,Quasi-variational inequalities,partial differential equation,completion of squares,Pareto optimal,fractional Brownian motion theory,my doctoral dissertation will be devoted to solving optimal dividend with transaction costs and solvency constrains,optimal multi-asset investment,Pareto optimal reinsurance strategy and optimal control problem under fractional Brownian motion insurance model.In the following, we will give the innovation and novelty of every section.1.The innovation in methods(1) In section 2.3,there are two innovations in methods:(a) the verification theorem about non-continuously differentiable solution;(b) to find a fast method to calculate the ruin probability for a given lower and upper barrier,and show how to adapt the Thomas algorithm for solving tridiagonal systems together with the Crank-Nicoison algorithm to solve the relevant partial differential equations.(2) There are two innovations in section 2.4:(a) to give a new idea to classify the solutions of quasi-variational inequality;(b) to give a new method to construct the optimal value function and the optimal strategy.(3) Section 4.1 gives the verification theorem about a class of viscosity solution.(4) Section 6.1 give the method to solve the Pareto optimal reinsurance in continuous time.(5) In section 6.2,it is shown that how to use the HJB equation method to solve the optimal problem with partial information.2.The innovation in skill to solve optimal equation and construct the solution.This part mainly extends some classical and excellent work,but it is not the simple extension on calculation.(1) Section 2.1 extend the work of[28](published in Mathematical finance),and consider the optimal excess-of-loss reinsurance and dividend with transaction costs and taxes. The corresponding quasi-variational inequality is more complicated than that under proportional reinsurance.In section 2.1,we give the different method to construct the solution.(2) In section 2.2,the optimal dividend with transaction costs and taxes is considered under the classical risk process.The construction of the solution is different from Section 2.1 and[28].(3)[24]is known as the pioneering work on the combination of insurance theory and control theory.In third chapter,we extend[24]'s work by incorporating the reinsurance into the model in[24].In this case,the more technical method to construct the solution is needed. 3.To introduce new control method into insurance filed.Section 5.1 use three methods of the HJB equation,stochastic maximum principle, completion of square to solve the optimal problem.4.To consider the optimal control problem under non-markovian processThe fractional Brownian motion is a long-range dependence,non-markovian process. It can model many phenomenon closer to practice.However,since its property is difficult to describe,there are only a few literatures which consider fractional Brownian motion risk model.We in section 4.2 and section 5.2 consider the optimal problem under a classical risk model perturbed by a standard fBm.We find the method to solve the problem and give the explicit solution.Another innovation is that we give the explicit solution for all of problem.