Hello Ruben
The problem with Gamma mixture resides in the parameter definition. For Gamma process, both parameters have to be positive.
This is ensured when sigma (in the context of other parameters defined in this example) is at least 80 bp
In[79]:= pm1 =
ParameterMixtureDistribution[GammaDistribution[a, b],
b \[Distributed] ExponentialDistribution[\[Lambda]]];
In[80]:= sl5 =
Solve[{Mean[pm1] == x0 + \[Mu]*t,
StandardDeviation[pm1] == \[Sigma] Sqrt[t]}, {a, \[Lambda]}];
In[81]:= sl5vals = {a /. sl5[[1]], \[Lambda] /. sl5[[1]]};
In[82]:= evpm1 =
Subscript[A, 0]
Expectation[If[x > k, x - k, 0], x \[Distributed] pm1,
Assumptions -> k > 0 && t > 0] // Simplify
In[69]:= evpmcal =
evpm1 /. {a -> sl5vals[[1]], \[Lambda] -> sl5vals[[2]]};
In[78]:= evpmcal /. {t -> 5, x0 -> 0.014,
k -> 0.014, \[Sigma] -> 0.008, \[Mu] -> 0, Subscript[A, 0] -> rann}
Out[78]= 0.0269472
Hope this answers your question. If not, please let me know.
Kind regards
Igor