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The general case variance does not agree with the most common example of a brownian bridge. Bt-tBt in the 0-1 interval. it seems to me that the variance for the general case should be (T-T0)(T1-T)^2/(T2-T1). it is possible to derive the variance of the general process using integration by parts,
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You have the distribution for any one point along the bridge, so you can simulate it recursively by first simulating one point in between the first pair, then you have two adjacent pairs and can simulate another two points for those two bridges, etc.. CHF
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There seems to be an error in obtaining the general brownian bridge W(t) from brownian motion B(t): the text says B(t) = W(t) - tW(T), but this doesn't satisfy B(T) = 0. Should this be B(t) = W(t) - tW(t)/T? (I've fixed this)
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Yes indeed. This was a mistake. If B is a
Brownian bridge, and W(t) = B(t) + t Z is a Brownian motion, then B(t) = W(t) - t W(1) is independent of W(1) = Z. So Z must be independent of B. Sorry to answer so late...
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In the construction W(t) = B(t) + t Z, it doesn't say that Z must be independent of B. Could someone confirm that independence is required (or at least that certain dependencies are not allowed)?
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I would expect the variance for the general brownian bridge W(T) between T1 and T2 to be (T-T1)(T2-T) / ((T2-T1)^2), rahter than / (T2-T1)
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This is really poorly written. I'm not sure what the second paragraph is trying to get across otherwise I'd try to edit it. Rewrite anyone?
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I see someone reverted back and removed the square exponent, and I completely agree. Otherwise, notice that the variance in any
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Is a
Brownian bridge a diffusion process, using E.B. Dynkin's definition? If so, where is the proof?
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It would be nice if there was a comment on how one might simulate a
Brownian bridge.
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Is there an analogue of the
Brownian bridge, but for a
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I incorporated a reference where this is proved. --
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360:{\displaystyle T\in (T_{1},T_{2})}
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