The recent surge of the insurance products such as Universal Variable Life poses a challenging problem of finding a "fair" price in an incomplete financial market. This Thesis applies the "Principle of Equivalent Utility" to price a general life insurance. The benefit of the insurance can depend on the investment market as well as the policy status which is modelled by a continuous time Markov chain. The so-called "indifference price" can be determined by solving an equation involving two value functions, resulting from the stochastic control problems with and without insurance liabilities. These value functions are expected to be the viscosity solutions to the corresponding Hamilton-Jacobi-Bellman equations. Using a dynamic programming argument this Thesis shows that the value functions involved may depend on the policy status, and the resulting HJB equation is in the form of a system of fully nonlinear second order partial differential-difference equations (PDDE). Further, this Thesis gives a detailed discussion on the notion of viscosity solutions for the systems of PDDEs and proves that the value function of the stochastic control problem with insurance risk is indeed a unique viscosity solution to the HJB equation. Finally, this Thesis provides a numerical scheme to approximate the viscosity solutions via finite difference methods. Some numerical results and convergence analysis are also given.
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