a known function set by the modeller, whereas $g(\cdot)$ is often unknown (e.g. when $\{X_t\}$ is a diffusion process, i.e. the solution of a stochastic differential equation) except for very simple toy models. Goal of our work is to estimate the parameters $\theta=(\eta,\phi)$ using observations $y=(y_0,y_1,...,y_n)$ from $\{Y_t\}_{t\geq t_0}$ collected at discrete times $\{t_0,t_1,...,t_n\}$. As an example, $\{Y_t\}_{t\geq t_0}$ may be defined as $$Y_{t_i}=X_{t_i} + \epsilon_{t_i}, \quad \epsilon_{t_i}\sim p_{\epsilon}(\phi), \qquad i=0,1,...,n \label{eq:measurement-model}$$ with $\{\epsilon_t\}$ representing unobservable noise sources (e.g. measurement errors) having distribution with probability density function (pdf) $p_\epsilon(\cdot)$.

In Bayesian inference the goal is to analytically derive the posterior distribution $\pi(\theta|y)$ or, most frequently, implement an algorithm for sampling draws from the posterior distribution. Sampling procedures are often carried out using Markov chain Monte Carlo (MCMC) or Sequential Monte Carlo (SMC) methods embedded in MCMC procedures [@andrieu2010particle]. However the last ten years have seen an explosion of methodological advances for the so-called approximate Bayesian computational methods. We propose to use an ABC-MCMC sampler as a “workhorse” to obtain an approximate MLE for $\theta$. Notice that (approximate) Bayesian algorithms leading to an approximate MLE have also been considered in [@rubio2013simple]. See [@grazian-liseo(2015)] for ABC strategies for “integrated likelihoods”, where nuisance 