Give the basic definition of insurance payable at the moment of death.

Use the death benefit function on [0,ω-x)

- b is piecewise constant

- b(t) death benefit paid at time t is benefit occurred than.

Insurance payable on the moment of death with a death benefit function b(t) on the interval [0,ω−x), where b(t) is piecewise constant, refers to a type of life insurance policy where the insurer agrees to pay out a predetermined sum of money to the designated beneficiary upon the insured individual's death.

This type of insurance provides financial protection to the beneficiary or beneficiaries, typically family members or dependents, in the event of the insured's death.

In this context:

• ω represents the maximum possible lifespan of the insured individual.
• x represents any waiting or exclusion period before the death benefit becomes payable.
• b(t) denotes the death benefit paid at time t, which is the benefit that occurs at that moment. The function b(t) is piecewise constant, meaning it remains constant over specific time intervals.

This type of insurance ensures that upon the insured's death, the designated beneficiaries receive the specified death benefit according to the terms of the policy, provided the death occurs within the coverage period [0,ω−x).