There are problems with your code. You give no value to g. The integral defining V:
NIntegrate[Exp[-B[t]]*
((1 - c*g)*H/K + (1 - c*g)*g*(1 - H/K)),
{t, 0, v}]
has the variable t only in Exp[-B[t]] and it has v in the other factor. Did you really mean that?
As for speeding repeated integrals, you may try the NDSolve way:
B = NDSolveValue[{y'[v] ==
b*(1 - c*g)*H/K + b*g*(1 - c*g)*(1 - H/K) - d*x/K,
y[0] == 0},
y, {v, 0, 1000}]
