  This paper considers an optimal life insurance for a householder subject to
mortality risk. The household receives a wage income continuously, which is
terminated by unexpected (premature) loss of earning power or (planned and
intended) retirement, whichever happens first. In order to hedge the risk of
losing income stream by householder's unpredictable event, the household enters
a life insurance contract by paying a premium to an insurance company. The
household may also invest their wealth into a financial market. The problem is
to determine an optimal insurance/investment/consumption strategy in order to
maximize the expected total, discounted utility from consumption and terminal
wealth. To reflect a real-life situation better, we consider an incomplete
market where the householder cannot trade insurance contracts continuously. To
our best knowledge, such a model is new in the insurance and finance
literature. The case of exponential utilities is considered in detail to derive
an explicit solution. We also provide numerical experiments for that particular
case to illustrate our results.
