I am trying to simulate short rate in Hull White, but have got caught up in this confusion.
I am using a flat forward term structure of 0.05 as input with hull white parameters
a=0.1
sigma=0.05
T= 5 year

Now the problem is that I was expecting the values of short rate to be same as 0.05 along the path, but when I calculate the average it keeps on increasing as time steps

For example, this is my result:
0.050000 0.050512 0.053635 0.058188

Is it necessay for all the short rate values to approach the forward rate values along a path?

It used to work for Libor market model and I was able to match the bond prices using simulated values of forward rate . The realized forward rate values used to match the forward rates implied from the current term structure.

How can I check that my simulated values are right when simulating a shrt rate model?

I think it is ok for r to be increasing. Remember, the expected value of bond price should match with the values of bond prices generated by MC, not that the expected values of rt should match.

E[exp(X)] is not equal to exp[E[X]]

I have updloaded a file to explain the intuition behind Hull White simulation:
MC simulation in Hull White


HTH

I had a similar problem with my implementation - when switching to discrete time, the model wasn't well-behaved. My problem was that the last part of the theta-function was very large (due to a small "a" in a denominator). When I removed that part, the model converged nicely to the term structure as it's supposed to.

I implemented the formula in Matlab with a varying term structure. I would love to hear your comments/suggestions: http://www.quantcode.com/modules/newbb/viewtopic.php?topic_id=471&forum=1