I suspect it may be confused by the

D[Subscript[m, d][v],v]

at the end of your integral.

While you might mathematically understand what the integral of ln(x) dsin(x^3) means, I think if you look at all the documentation and examples I think you will see they all show integration with respect to a simple variable, not a function and not a subscripted variable.

Can you express your integral in such a form and try evaluating the integration again?

Sorry, I was just trying to write a syntactically correct input. I still don't know if it is what you wanted, and how to get a useful output.

Dear Professor:

Thanks for such elegant coding. i am wondering is there any way to track how Mathematica get the parameters of hpergeometric function, i.e show the calculation process. I tried "Trace", however, it does not give me anything.

Just a first attempt at fixing the syntax problems:

$Assumptions = \[CapitalTheta] > 0 && t > 0 && \[Delta] > 0 && 
   s < \[CapitalTheta];
Subscript[m, d][t_] = 
  Subscript[\[Lambda], 2] t + (
    Subscript[\[Lambda], 1] - Subscript[\[Lambda], 2])/
    Subscript[\[Lambda], 1] (1 - E^(-Subscript[\[Lambda], 1] t));
Subscript[M, X][s_] = 
 Integrate[
  E^(s x) \[CapitalTheta]*E^(-\[CapitalTheta] x), {x, 0, Infinity}]    
Subscript[M, Subscript[Z, 0][t_]][s_] = E^(
 Subscript[\[Lambda], 2]
   Integrate[
   Subscript[M, X][s E^(-\[Delta] v)], {v, 0, 
    t}])      (*Equation (3.6)*)

Subscript[M, Subscript[Z, d][t_]][s_] = 
 1 + Integrate[(Subscript[M, X][s E^(-\[Delta] v)] - 1) Subscript[M, 
     Subscript[Z, 0] (t - v)][sE^(-\[Delta] v)] D[Subscript[m, d][v], 
     v], {v, 0, t}]