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It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that $\operatorname{Cov}(W_t,W_s)=\max(t,s)$?

Let me ask it in another way: it is known that $k(x,y)=\min(x,y)$ is the reproducing kernel of the Cameron Martin RKHS. What is the RKHS (if any) of the kernel $k(x,y)=\max(x,y)$?

Thanks for your help!

EDIT: Please recall that $$\operatorname{Cov}(B_s,B_t)=\operatorname{Cov}(sB_{1/s},tB_{1/t})$$

but I didn't manage to go further.

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2 Answers 2

So that this has a (non-deleted) answer:

No such process can exist. If $\operatorname{Var}(W_0) = \max(0,0) = 0$ then $W_0$ is a constant. In particular, $W_0$ and $W_1$ are independent, so we have $\operatorname{Cov}(W_0, W_1)= 0 \ne 1= \max(0,1)$.

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If $0<s<t$ then $$ \operatorname{cov}(B_t,B_s) = \operatorname{cov}(B_s + (B_t-B_s),B_s) = \operatorname{cov}(B_s,B_s) + \operatorname{cov}(B_t-B_s,B_s) = \operatorname{var}(B_s) + 0 = s. $$

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thanks! but how would you represent this backward-brownian-motion in terms of Ito process or Ito integral? –  Troy McClure Dec 28 '12 at 1:09
    
@MichaelHardy The covariance is never max(s,t). –  Did Dec 28 '12 at 22:19
    
@did : Let $C_t=B_t-B_1$. Then for $0<s<t<1$, you've got $\operatorname{cov}(C_s,C_t) = t$, i.e. it's the maximum. –  Michael Hardy Dec 28 '12 at 23:37
    
@MichaelHardy Not at all. For the reason why this example fails and why any other would fail as well, see my now deleted answer. –  Did Dec 28 '12 at 23:39
    
@did : OK, I see: it's the distance from the point where the process is fixed (in the example, $1$) and the nearest of the two indices to that point. –  Michael Hardy Dec 28 '12 at 23:47

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