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 $k$, $(k=1,…,M)$ 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 $\beta(t)$.


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