I’d like to model prices as following a kind of stochastic difference equation, where jumps from one time step to the next follow

p_{t} = p_{t - 1} + c \exp(- b t_0) \Delta t + X(t)

X ~ \text{student_t}(3, 0, \sqrt{\Delta t} \times (\sigma_0 + \sigma \exp(- \gamma t))

Here is the model

```
d_price_mean(c::Real, b::Real, t0::Real, dt::Real)::Real = begin
c * exp(- b * t0) * dt
end
perturbation_volatility(
sigma1::Real,
sigma::Real,
gamma::Real,
t0::Real, dt::Real)::Real = begin
sigma1 + sigma * exp(- gamma * t0) * sqrt(dt)
end
@model price_model(
d_price,
selldown_time,
d_selldown_time,
n
) = begin
c ~ LogNormal(2, 1)
b ~ LogNormal(2, 1)
sigma1 ~ LogNormal(-1.3862943611198906, 1)
sigma ~ LogNormal(-0.6931471805599453, 1)
gamma ~ LogNormal(2, 1)
price_diff = d_price .- d_price_mean.(Ref(c), Ref(b), selldown_time, d_selldown_time)
vol = perturbation_volatility.(
Ref(sigma1),
Ref(sigma),
Ref(gamma),
selldown_time,
d_selldown_time)
std_price_diff = price_diff ./ vol
for i = 1:n
std_price_diff[i] ~ TDist(3)
end
end
```

When I try sampling from this model, it never finishes even with a trivial amount of data, and when I try running the MAP to test, each of the individual parameters (c, b, \sigma, \sigma_{1}, \gamma) all converge to exactly their prior modes. This indicates to me that I must be defining the likelihood in a way such that its gradient is exactly the zero vector.

My guess is that for Turing to pick up on a likelihood, it must have an argument to the left of the sampling statement, or else it just defines a new parameter, even if its been defined in the model scope. What would be the right way to write these kinds of reparameterizations?