Harold lives a whole year after he dies? What have you done, so far?
Note: "end of the year" is good. It's a nice, discrete problem. We don't really need the continuous functions. We can use them if desired.
Hint: Year 1
p(Julio is living) = 0.2 * (4-1) = 0.2 * 3 = 0.6
p(Harold is living) = 0.3 * (4-1) = 0.3 * 3 = 0.9
p(both alive) = 0.6 * 0.9 = 0.54 Pay Nothing and move on to year 2.
p(Julio Alive and Harold Dead) = 0.6 * (1 - 0.9) = 0.6 * 0.1 = 0.06 Pay 100,000 at the end of the year and terminate the policy.
p(Harold Alive and Julio Dead) = (1 - 0.6) * 0.9 = 0.4 * 0.9 = 0.36 Pay 100,000 at the end of the year and terminate the policy.
p(Both Dead) = (1 - 0.6) * (1 - 0.9) = 0.4 * 0.1 = 0.04 Pay 100,000 at the end of the year and terminate the policy.
Checking: 0.54 + 0.06 + 0.36 + 0.04 = 1.00 -- Looks like we got everything.
Price at Year 0, then: (0.54 * 0 + 0.06 * 100000 + 0.36 * 100000 + 0.04 * 100000)/1.07 = (0.54 * 0 + 0.46 * 100000)/1.07 = ... You do the arithmetic.
Year 2
Careful. We care ONLY about both alive at the end of year 1. We already paid, otherwise.
Let's see what you get.