Anano's maximum worship for geometric Brownian motion

Question

Anano's maximum worship for geometric Brownian motion

I am trying to make a maximum likelihood assessment (MLE) for the geometric Buryat movement with Theano. I know that Theano is primarily an ML library, but it should work ... (of course, I'm just trying to use Theano, and my ultimate goal is to make MLE for a slightly more complex model ....)

Writing math in SO is a bitch, so look for GBM, for example. here .

Here is the Theano code, which is slightly modified from the logistic regression example in Theano's tutorial:

training_steps = 1000
dt = 1/390
NT = len(Xdata)
rng = np.random
rate = 0.001

# Declare Theano symbolic variables
X = T.vector("X")
dX = T.vector("dX")
mu = theano.shared(rng.randn(), name="mu")
sigma2 = theano.shared(rng.random(), name="sigma2")

print("Initial model:")
print(mu.get_value())
print(sigma2.get_value())

# Construct Theano expression graph

eps = (dX - mu*dt*X)/X
summand1 = eps**2
loghood = summand1.sum()/sigma2/dt + NT*T.log(sigma2)
gmu, gsigma2 =  T.grad(loghood, [mu, sigma2])

# Compile
train = theano.function(
          inputs=[X,dX],
          outputs=[loghood],
          updates=((mu, mu - rate * gmu), (sigma2, sigma2 - rate * gsigma2)))

# Train
costlist = []
mulist = []
siglist = []
for i in range(training_steps):
    cost = train(Xdata, dXdata)
    costlist.append(cost[0])
    mulist.append(mu.get_value())
    siglist.append(sigma2.get_value())

print("Final model:")
print(mu.get_value())
print(sigma2.get_value())

plot(costlist)