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Suppose I have an $n$-dimensional Itô SDE $$dX_t = \sigma(X_t) dW_t + \lambda(X_t)dt$$ and I'm interested in diffusion bridges from $X_0=a\in\mathbb R^n$ to $X_T=b\in\mathbb R^n$.

Now let $Y_t$ be a diffusion bridge of the SDE $$dY_t=\sigma(Y_t)dW_t - \lambda(Y_t) dt$$ from $Y_0=b$ to $Y_T=a$.

Based on a (very) non-rigorous approximation argument I would conjecture that the laws of $X_t$ and $Y_{T-t}$ agree.

To make my proof rigorous would involve $\varepsilon$'s and $\delta$'s that I think would quite long. Is this a known result? Is there a proof using a theorem about diffusions. Or is it wrong?

Pointers etc are very much appreciated.

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    $\begingroup$ Note: as you've written it, they are not independent processes (same Brownian motion $W_t$). Is it not sufficient to just point out that every path of $Y_t$ corresponds (bijectively) to a path of $X_t$ by reversing time? (After time reversion it is the same SDE.) $\endgroup$
    – Kirill
    Commented Jul 18, 2013 at 0:31
  • $\begingroup$ @Kirill Thanks for the comment. You're right, I think that is sufficient. I knew it would be something obvious like that that I'd missed. $\endgroup$
    – Tim
    Commented Jul 18, 2013 at 7:49
  • $\begingroup$ @Kirill I'm unsure how the detail of this works. Would you mind posting a proof? Thanks! $\endgroup$
    – Ben Derrett
    Commented Jul 20, 2013 at 12:09

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The laws of the processes do not agree. Time-reversal for stochastic processes is more complicated than just to swich signs in the drift compnent. Think of an Ornstein-Uhlenbeck process started from an equilibrium for example. Just google for time-reversed diffusions for more help.

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