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  "body": "\n## Definition\n\nA *Brownian bridge* is a real stochastic process $(X_t)_{0 \\le t \\le 1}$ such that:\n\n1. $(X_t)$ is a [[Gaussian process|22WEM6RTW0]] with $E[X_t] = 0$ for all $t \\in [0, 1]$.\n2. Its [[covariance|0KBFAWYRTE]] is $E[X_s X_t] = \\min(s, t) - st$ for all $s, t \\in [0, 1]$.\n3. Almost surely, the path $t \\mapsto X_t$ is continuous.\n\n## Remarks\n\nIf $W$ is a [[Wiener process|552AWX3QXN]], then $X_t = W_t - t W_1$ is a Brownian bridge.\nCheck: $X$ is a linear image of the Gaussian process $W$, hence Gaussian and centred, with continuous paths, and $E[X_s X_t] = \\min(s, t) - st - st + st = \\min(s, t) - st$.\nThe Wiener process $W$ restricted to $[0, 1]$ satisfies clauses 1 and 3 but fails clause 2, since $E[W_s W_t] = \\min(s, t)$; for instance $\\operatorname{Var}(W_1) = 1$, while clause 2 forces $X_1 = 0$.\nClause 2 gives $X_0 = X_1 = 0$ almost surely and $\\operatorname{Var}(X_t) = t(1 - t)$.\nThe law of a Brownian bridge is the conditional law of $(W_t)_{0 \\le t \\le 1}$ given $W_1 = 0$, in a sense made precise by a limit.\n",
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