First order ODE with input delay

I am new to Julia / Turing.jl and wondering if it is possible to estimate the affine parameters a,b of the following model.
The model is a stochastic differential equation (SDE) of first order with variable input delay T_d in the external control input u_1(t) (control engineering nomenclature):

T\dot{x} =x+ u_1(t-Td) + \epsilon
T_d(t) = a + bu_2(t)

with \epsilon being white, Gaussian noise.
I have time-discrete measurements of x(t). The external inputs u_1(t),u_2(t) are known as well.

I have seen SDE in DifferentialEquations.jl but I am not sure, if time-dependent delays are possible.


That’s a stochastic delay differential equation.