# Simulation of Itô integral processes where integrand depends on terminal (Volterra process)

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?)

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.

• Can you provide some examples of the integrand $f(s,t)$? What sort of regularity does $f$ have? The choice of integration method seems to depend on this. – Nawaf Bou-Rabee Apr 28 '19 at 17:39
• @NawafBou-Rabee Done. – Luke Thorburn Apr 28 '19 at 23:02
• The first hit when entering "simulation fractional brownian motion" into Google is a whole thesis devoted to the problem. Have you read this? What do you want to achieve which the methods presented there don't? – Martin Hairer Apr 29 '19 at 0:04
• @MartinHairer Yes, I have read that thesis. I'm not interested in simulating fBm in general, but in approximating well this particular representation of it by a (possibly complicated/algorithmic) function of its standard Bm driver. – Luke Thorburn Apr 29 '19 at 0:14

## 1 Answer

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, Potential Anal. 10, No. 2, 177-214 (1999). ZBL0924.60034.