Optimal consumption, investment and life insurance with surrender option guarantee

Optimal investment-consumption and life insurance with capital constraints

“…The feasible strategy (3.4) means that it is allowed to draw an infinite utility from the strategy (π, c, p) ∈ A ′ , but only if the expectation over the negative parts of the utility function is finite. It is clear that for a positive utility function, the sets A and A ′ are equal ( see e.g., Kronborg and Steffensen [15]). In order to solve the unrestricted control problem (3.3), one can use the Hamilton-Jacobi-Bellman (HJB) equation (e.g.…”

“…Mnif [20]) or the combination of HJB equation with backward stochastic differential equation (BSDE) with jumps (Guambe and Kufakunesu [7]). In this paper, we use the martingale approach applied in (Karatzas et al [12], Castaneda-Leyva and Hernández-Hernández [1], Kronborg and Steffensen [15]). This is due to the restricted problem in the next section, where its terms are derived from the martingale method in the unrestricted problem.…”

“…Other references include (Huang et al [10], Pliska and Ye [21], Liang and Guo [17]). Recently, Kronborg and Steffensen [15] extended this problem to include capital constraints, previously introduced by Teplá [27] and El Karoui et. al.…”

“…where Z(t) is given by (2.4) and τ ∧ T := min{τ, T }. The expression µ(t)(p(t) − X(t))dt from the wealth process (2.7), corresponds to the risk premium rate to pay for the life insurance p at time t. Notice that choosing p > X corresponds to buying a life insurance and p < X corresponds to selling a life insurance, that is, buying an annuity (Kronborg and Steffensen [15]).…”

Application of Generalized Geometric It&#244;-L&#233;vy Process to Investment-Consumption-Insurance Optimization Problem under Inflation Risk

“…Their jump-diffusion model which considered Poisson jumps, proved the existence of the optimal strategy using martingale approach. [10] Paper extended the work of Kronborg & Steffensen [11] which considered an investor with uncertain lifetime, endowed with deterministic labor income who has a possibility to continuously invest in a Black-Scholes market and to buy life insurance or annuities. The paper [10] studies the optimal consumption, investment and life insurance when the investor is restricted to fulfil an American capital guarantee.…”