# 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

Even in the multidimensional case, from the Wong-Zakai theorem, the sequence of processes which are solutions of the equation

$X^N_t=\int_0^t \mu(X^N_s) ds+\int_0^t \sigma(X_s^N)dV^N_s$

will converge uniformly in probability on the interval $[0,T]$ to the solution of the Stratonovitch stochastic differential equation

$X_t=\int_0^t \mu(X_s) ds+\int_0^t \sigma(X_s)\circ dW_s$

A survey on Wong-Zakai type theorems is

Wong-Zakai approximations for stochastic differential equations

By using the more recent rough paths theory, we can see that this convergence also holds in the $p$-variation topology for $2<p<3$.

• Unfortunately your link is behind a paywall and I don't have easy access via a university library anymore. I'm quite sure my construction doesn't work for general $V^N$ converging to $W$, and that one needs to impose extra conditions on the sequence. – Simon Lyons Jun 22 '14 at 19:56
• I meant to add - what extra properties of $\{V^N\}$ did they use? – Simon Lyons Jun 22 '14 at 20:06
• Yes, of course, conditions on the approximating sequence V^N are needed. The most general assumptions are given by the rough paths theory: If V^N converges in the p-rough path topology, $2<p<3$ to a Brownian motion then X^N converges to the solution of the Stratonovitch SDE. This is proved in my blog fabricebaudoin.wordpress.com/category/rough-paths-theory The case of the Karhunen-Loeve approximation has been explicitly shown to satisfy the good assumptions in the book by Friz-Victoir page.math.tu-berlin.de/~friz/master4_May6th.pdf See page 437 – Fabrice Baudoin Jun 24 '14 at 22:39
• Oh, very nice. I've been using the approximating process in my thesis (which is in machine learning, where the emphasis is on producing algorithms that 'work' rather than results that are asymptotically correct). A proof wasn't crucial to my thesis, but it is very useful to have in hand. Much appreciated. – Simon Lyons Jun 25 '14 at 12:39