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.
Best Answer
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 $2<p<3$.