Double integration of Brownian Motion transition probability

Questions
1

Hello,

I am trying to integrate the transition density of the Brownian Motion - but Mathematica gives me values above one. If I understand correctly, the values that I get should all be less or equal to one.

What am I doing wrong?

bm[\[Mu]_, \[Sigma]_] := 
 ItoProcess[\[DifferentialD]x[
     t] == \[Mu] \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t], 
  x[t], {x, 0}, t, w \[Distributed] WienerProcess[]]

In[46]:= pdf = PDF[gProc2[\[Mu], \[Sigma]][t], x]

Out[46]= E^(-((x - t \[Mu])^2/(2 t \[Sigma]^2)))/(Sqrt[2 \[Pi]] Sqrt[
 t \[Sigma]^2])

In[47]:= NIntegrate[E^(-((x - t \[Mu])^2/(2 t \[Sigma]^2)))/(
  Sqrt[2 \[Pi]] Sqrt[t \[Sigma]^2]), {t, 0, 5}, {x, 0, 
   5}] /. {\[Mu] -> 0.04, \[Sigma] -> 0.7}

During evaluation of In[47]:= NIntegrate::inumr: The integrand E^(-((x-<<1>>)^2/(2 t \[Sigma]^2)))/(Sqrt[2 \[Pi]] Sqrt[t \[Sigma]^2]) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,5},{0,5}}. >>

Out[47]= 2.66889

Thank you in advance,

Edvin

2

Not familiar with the subject but it’s not clear what your pdf under your integral is. The first two equations you post seem to have nothing to do with your integral. In your integral you have an unknown variable x.

3

Thank you for the reply. The first equation is the diffusion equation. I put it up there to show that I am not making anything up. On the last equations the pdf its integrated on x, and t.

I checked my equation again and it seems right.

Also, I found out that replacing the list inside the integral solves the warning messages. Nevertheless, the values are greater than 1.

4

perhaps there is just a constant factor missing in your pdf under the integral. A pdf integrated over the entire domain should equal 1.