Stochastic process example with Girsanov likelihood

example.jl

struct StoProMeasure
    θ::Float64
end

struct WienerMeasure
end

function logdensity(P::StoProMeasure, tr, ::WienerMeasure)
    som = 0.0
    t, x = tr
    for i in 1:length(x)-1
       b = - x[i] + P.θ*atan(x[i])
       som += b*(x[i+1] - x[i] - 0.5 * b * (t[i+1]-t[i]))
    end
    som
end

function sample(P::StoProMeasure, T, dt)
    x = 0.0
    t = 0.0
    ts = [t]
    xs = [x]
    for i in 1:round(Int,T/dt)
        x += - x*dt + P.θ*atan(x)*dt + sqrt(dt)*randn()
        t += dt
        push!(ts, t)
        push!(xs, x)
    end
    tr = ts, xs
end

tr = sample(StoProMeasure(2.0), 1000, 0.1) # true parameter 2.0

loglikelihood = [x => logdensity(StoProMeasure(x),tr, WienerMeasure()) for x in -1.5:0.1:2.5]

_, i = findmax(last.(loglikelihood))
θml = loglikelihood[i][1]

println("Maximum likelihood estimate of θ: ", θml)