4

What is an example of a continuous semi-martingale that cannot be written as a stochastic integral with respect to Brownian Motion?

random walk
  • 41
  • 2
  • 1
    If you mean, cannot be written in the form $X_t = \int_0^t Y_t,dB_t$, then: any semimartingale that is not a local martingale starting at 0. For instance, $X_t = 1$ or $X_t = t$. Or do you mean to allow a bounded variation term? How would it be related to the Brownian motion? – Nate Eldredge Nov 16 '13 at 00:51
  • Yes, I did mean brownian motion plus some bounded variation term. So it does not seem plausible that every continuous semimartingale can be written that way, so I supposed a counter-example would be fairly well known. – random walk Nov 18 '13 at 19:26
  • So in that case, the stochastic integral would correspond to the local martingale part of your semimartingale. Thus I suppose we could ask either of the following more precisely stated questions: (1) Does there exist a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_t, P)$ (satisfying the usual conditions, let us say), and a continuous local martingale $X_t$ defined on $\Omega$, such that there do not exist a Brownian motion $B_t$ and an adapted process $A_t$ for which $\int_0^t A_s,dB_s = X_t$ almost surely? (It could be that $\Omega$ cannot support a BM at all.) – Nate Eldredge Nov 18 '13 at 19:59
  • Or, (2): Does there exist $\Omega, X_t$ as above, for which there does not exist another filtered probability space $\Omega'$ and a Brownian motion $B_t$ and an adapted process $A_t$ defined on $\Omega'$, such that $X_t = \int_0^t A_s,dB_s$ in law? (I don't know the answer to either question.) – Nate Eldredge Nov 18 '13 at 20:01

1 Answers1

0

Let X(t) = B(g(t)) where g is the Cantor function. This is a time change of Brownian motion and is easily seen to be a martingale. Note that while X' exists and is zero a.e., X is not of bounded variation in fact [X] = g. If X = HB (stochastic integral) for some adapted square integrable process H, then we would have 0 = g' = H^2 a.e. => H = 0 a.e. => X = 0 a.e.

random walk
  • 41
  • 2