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As in the title, question is whether a Brownian bridge: $X_{t} = B_{t} - tB_{1}$ is a Markov process. I could sort of prove it by the markov property, but not sure whether it's sufficient. Does anyone have any good ideas? Thanks.

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Yes, the Brownian bridge process is a Markov process with respect to its own filtration, because $X_t$ is the Ito process with SDE $\mathrm{d} X_t = \frac{X_t}{1-t} \mathrm{d} t + \mathrm{d} B_t$ with initial condition $X_0 = 0$ on time domain $0 \le t < 1$. This is exercise 5.11 in Oksendal's book "Stochastic Differential Equations".

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