I need to simulate a process of the form


$$X\_t=\int\_0^t f(s,t)\mathop{dW\_s}$$


where $f$ is deterministic and the integral is an Itô integral. I know I can simply take finite Itô sums of discrete increments of the Brownian motion driver, but I am wondering if there are more sophisticated approaches. Common methods such as the Euler-Maruyama method do not appear to be applicable because the integrand depends on the upper terminal $t$ and so $X\_t$ is not an Itô process.


Are there known approaches for simulating this kind of process?


(Answers to related questions that would help to find relevant literature would also be useful: eg. Does this kind of process have a name? Is there a way of writing it as a SDE? If so, does that class of SDEs have a name?)




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**EDIT**


The particular integral I'm interested in is the Molchan-Golosov representation of fractional Brownian motion. Up to a multiplicative constant, this is 
$$\int\_0^t (t-s)^{H-1/2}F(1/2-H,H-1/2,H+1/2,\frac{s-t}{s})\mathop{dW\_s}$$
where $H\in(0,1)$ and $F$ is the Gauss hypergeometric function.


In further reading I have found that processes of this form are known as *Volterra processes*, but I haven't found any discussion of simulation algorithms.



Simulation of Itô integral processes where integrand depends on terminal (Volterra process)
Here is an approximation scheme that uses a chain of independent Brownian bridges. For $t>0$ fixed, consider the following partition of the time interval $[0,t]$ $$
t\_0 = 0 < t\_1 < t\_2 < \dots < t\_{n} = t \;.
$$ At these discrete values, compute a discretized Brownian motion $W\_i = W(t\_i)$ in the standard way $$
W\_{i} = W\_{ i-1} + \sqrt{t\_{i+1}-t\_i} \xi\_i \;, i=1, \dots, n \;,
$$ where the $\xi\_i$'s are independent standard normal random variables. Let $\mathcal{G}\_n$ denote the $\sigma$-field generated by this discretized Brownian motion.
Then a pathwise accurate approximation to $X\_t$ that converges in the $L^2$ sense is given by $$
\tilde X\_t = \mathbb{E} \left( \int\_0^t f(s,t) dW\_s \mid \mathcal{G}\_n \right)
$$
and since a Brownian motion pinned at the $t\_i$'s are independent Brownian bridges with mean $$
W\_i + \frac{s-t\_i}{t\_{i+1}-t\_i} (W\_{i+1} - W\_i)
$$
we obtain 
 $$
\tilde X\_t = \sum\_{i=0}^{n-1} \frac{W\_{i+1}-W\_i}{t\_{i+1}-t\_i} \int\_{t\_i}^{t\_{i+1}} f(s,t) ds \;.
$$ This approximation is based on Proposition 3.1 of the following paper.


*Decreusefond, Laurent; Üstünel, Ali Süleyman*, [**Stochastic analysis of the fractional Brownian motion**](http://dx.doi.org/10.1023/A:1008634027843), Potential Anal. 10, No. 2, 177-214 (1999). [ZBL0924.60034](https://zbmath.org/?q=an:0924.60034).


