I'm trying to derive the risk neutral process for a stock with both continuous and discrete dividends. In particular, suppose the forward level process at time, $t$ is given by $F(S_t, t, T) = e^{(r-y)(T-t)}(S_t-d(t,T))$, where $r$ is the risk free interest rate, $y$ is a continuous dividend yield and $d(t,T)$ is the sum of all dividends paid in $(t, T]$. Under the R.N. measure, the stock process is given by $\dfrac{dS_t}{S_t}=(r-\mu-y)dt+\sigma dW_t$. Then in distribution, we have that:
$$S_T=S_te^{(-0.5\sigma^2+r-\mu-y)(T-t) + \sigma\sqrt{T-t}Z}=(F(S_t,t,T)-d(t,T))e^{(-0.5\sigma^2)(T-t)+ \sigma\sqrt{T-t}Z} $$
If we put $X_T:=\dfrac{S_T}{F(S_t,t,T)-d(t,T)}$, then we have that:
$$X_T=e^{-0.5\sigma^2(T-t)+ \sigma\sqrt{T-t}Z}$$
Something doesn't seem correct. I'm trying to incorporate all the discrete divs into the forward process and skip the jumps in the simulation. Is this wrong? Please help or correct me. Thanks.
