# Karhunen–Loève approximation of Brownian motion and diffusions

The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:

$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$ where $Z_n \sim \mathcal{N}(0,1)$ \re i.i.d random variables.

Suppose we truncate this sum at some finite $N$, and define the process

$V_t = \sum_{n=1}^N Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi}.$

Let $V^{(k)}$ be a vector of $k$ i.i.d copies of $V$.

We can define a new process

$X_t = \int_0^t \mu(X_t) dt + \int_0^t \sigma(X_t)dV^{(k)}_t,$

where the second integral is defined pathwise as a Lebesgue-Stieltjes integral. As $N \rightarrow \infty$, will this process converge weakly to a diffusion in either the Ito or Stratonovich sense?

Karatzas and Shreve's book discusses the the one-dimensional case - one can show strong convergence, so clearly weak convergence follows too. They give a caveat about strong convergence of multidimensional processes, which is partly what prompted this question.

I'm interested in weak convergence for multidimensional processes, since this might give an interesting alternative to the Euler-Maruyama numerical approximation scheme. This is a fairly straightforward idea, so I'm sure there are results in the literature. I'd appreciate any references you could provide.

Many thanks.

-
@Simon: I will take a look on the book you referred - so maybe you can put an extended reference (which chapter/page?). Before looking into this: is there significant difference which does not allow to repeat the proof of strong convergence by Karatzas and Shreve for the multidimensional case? –  Ilya Jul 18 '11 at 20:03
You can find the material in the second edition of Karatzas and Shreve, section 5.2, subsection D. Remark 2.25 in particular. It is stated explicitly that the one-dimensional proof does not carry over. They include a reference to Ikeda and Watanabe's book, but I don't have easy access to a copy. –  Simon Lyons Jul 19 '11 at 12:54