Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$.
We wish to explain the returns through a set of $J$ common tradeable factors, $f = (f_1, \dots , f_J )'$ which are observed at the same time points.
We assume the following conditional factor model explains the returns of stock $k$, $(k = 1, \dots, M)$ at times $t_i$, (i = 1, ..., n): \begin{align} R_{k,i} = \alpha_k(t_i) + \beta_k(t_i)'f_i + ω_{kk}(t_i)z_{k,i}. \end{align} Where $R_{k,i}$ and $f_i$ are the observed returns and factors at time $t_i$.
This can be rewritten in matrix notation as: \begin{align} R_{i} = \alpha(t_i) + \beta(t_i)'f_i + \Omega^{1/2}(t_i)z_{i}. \end{align}
In the paper on Testing conditional factor models they propose a continuous-time stochastic differential equation version of the above discrete time factor model to do theoretical analysis \begin{align} \mathrm{d}s(t) = \alpha(t)\mathrm{d}t + \beta(t)'\mathrm{d}F(t) + \Sigma^{1/2}(t)\mathrm{d}B(t). \end{align}
Where $s(t) = \log S(t)$ are the $M$ log prices, $F(t)$ are the $J$ factors and $B(t)$ is an $M$-dimensional Brownian motion.
We have observed the $s(t)$ and $F(t)$ at the $t_i$'s
My question is whether there are any efficient ways to do inference and possibly simulation of the diffusion process.
My main interest lies in sample paths of $\alpha(t)$ and $\beta(t)$
