Stochastic Differential equation: CAPM

Let $$R=(R_1, \dots ,R_M)$$′ denote a vector of excess returns of M assets observed at $$n$$ time points, $$0, within a time span $$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 $$k$$, $$(k=1,…,M)$$ 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 + \Sigma^{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 $$\beta(t)$$.