# Widow annuity- change of mortality

Take widow annuity for her (x), which pay 1000 per year from the death of him (y). Premium will be paying as continuous annuity until the time of first death with intensity $$P$$. We have: $$\bar{a}_{x}=12,5296 ; \bar{a}_{x:y}=4,0525 ; \bar{I}\bar{a}_{x}=127,667; \bar{I}\bar{a}_{x:y}=12,0561$$;
The mortality changed in both populations and now we have: $$m_{x}^*=m_x-0.002$$ for women $$m_{y}^*=m_y+0.001$$ for men
Calculate new premium $$P^*$$. Their life are independent.
I tried to calculate $$\bar{a}_{x}^*=\int_{0}^{\infty}{e^{0.002t}e^{-t\delta}p_x(t)dt}$$