Brownian Bridge Drift and Failure of Terminal Equivalence

The Bridge Law Before Its Endpoint

A Brownian bridge from \(0\) to \(0\) over \([0,T]\) starts at zero and is constrained to return to zero at the fixed time \(T\). Before \(T\), it has continuous random fluctuations, but its drift increasingly pulls it toward the endpoint. The coefficient of this drift becomes singular as \(t\) approaches \(T\). This is also where a key limitation of Girsanov’s theorem appears: the bridge law and Brownian law are equivalent on every strictly shorter interval, but not on the full interval including \(T\).

Work on the canonical space \(C([0,T],\mathbb{R})\), with coordinate process \(B_t\), its natural filtration \((\mathcal{F}_t)\), and Wiener measure \(\mathbb{P}\). Thus \(B\) is a Brownian motion under \(\mathbb{P}\). To define the bridge measure, one construction is to take a Brownian motion \(\widehat B\) on an auxiliary probability space and form \(X_t=\widehat B_t-(t/T)\widehat B_T\). The law of \(X\) on the canonical space is denoted by \(\mathbb{Q}\). Its paths satisfy \(X_0=X_T=0\), and its covariance is \(\mathbb{E}^{\mathbb{Q}}[B_sB_t]=\min(s,t)-st/T\).

At each fixed \(t<T\), this density is finite and strictly positive. Its explicit expression is

The formula has a conditional-probability interpretation. Given the Brownian path up to \(t\), its conditional density for ending at zero at time \(T\) is \(p_{T-t}(-B_t)\). Dividing by the unconditional endpoint density \(p_T(0)\) gives the relative likelihood of the bridge law up to time \(t\). The endpoint event itself has probability zero under Brownian motion, so this ratio is a density on \(\mathcal{F}_t\), not a conditioning on a positive-probability event.

Why the Finite-Time Density Is a True Martingale

Proof. Fix \(0\leq s<t<T\). Brownian increments give the conditional density of \(B_t\) given \(\mathcal{F}_s\), and the convolution identity for Gaussian densities gives \(\int_{\mathbb{R}}p_{t-s}(y-B_s)p_{T-t}(-y)\,dy=p_{T-s}(-B_s)\). Therefore \(\mathbb{E}[Z_t\mid\mathcal{F}_s]=Z_s\). Taking expectations, or using the same identity at \(s=0\), gives \(\mathbb{E}[Z_t]=Z_0=1\). The displayed formula for \(Z_t\) is strictly positive and finite for \(t<T\). Since \(d\mathbb{Q}|_{\mathcal{F}_u}=Z_u\,d\mathbb{P}|_{\mathcal{F}_u}\), strict positivity implies that the two restricted measures have the same null sets: if \(\mathbb{P}(A)=0\), then \(\mathbb{Q}(A)=\mathbb{E}[\mathbf{1}_A Z_u]=0\); if \(\mathbb{Q}(A)=0\), positivity of \(Z_u\) implies \(\mathbb{P}(A)=0\). \(\square\)

The density’s stochastic differential makes its drift interpretation explicit. On any fixed interval \([0,u]\), \(u<T\), the process \(H_s=-B_s/(T-s)\) is progressively measurable and satisfies \(\mathbb{E}\int_0^u H_s^2\,ds=\int_0^u s/(T-s)^2\,ds<\infty\). Thus it belongs to the Ito-integrand class \(\mathcal{L}^2\) on that interval. Applying Ito’s formula to \(f(t,x)=\sqrt{T/(T-t)}\exp(-x^2/[2(T-t)])\), its derivatives are

In particular, \(f_t+\tfrac12 f_{xx}=0\). The Ito formula, including its second-derivative correction, is \(dZ_t=(f_t+\tfrac12 f_{xx})(t,B_t)\,dt+f_x(t,B_t)\,dB_t\), hence

This calculation identifies the density as the stochastic exponential associated with \(H\) on \([0,u]\). Its true-martingale property has already been verified directly by conditional expectation, so no exponential-integrability test is needed here. In particular, one should not infer that Novikov’s condition holds: that sufficient condition requires \(\mathbb{E}^{\mathbb{P}}\!\left[\exp(\tfrac12\int_0^u H_s^2\,ds)\right]<\infty\), which has not been established.

Girsanov’s Drift and Its Sign

Proof. On \([0,u]\), \(Z\) is a strictly positive true martingale with expectation one, and \(dZ_t=Z_tH_t\,dB_t\), where \(H\) is progressively measurable and square-integrable under \(\mathbb{P}\) on this interval. Girsanov’s theorem therefore applies to the measure with density \(Z_u\). It says that \(B-\int_0^\cdot H_s\,ds\) is Brownian under that measure on \([0,u]\). This measure is precisely \(\mathbb{Q}|_{\mathcal{F}_u}\) by the density formula. Substituting \(H_s=-B_s/(T-s)\) gives the stated sign and the bridge SDE. Since \(u<T\) was arbitrary, the description holds on every interval strictly before \(T\). \(\square\)

The integral in this statement is well-defined on each such interval under \(\mathbb{Q}\) as well. Indeed, the bridge covariance gives \(\mathbb{E}^{\mathbb{Q}}[B_s^2]=s(T-s)/T\), so \(\mathbb{E}^{\mathbb{Q}}\int_0^u (B_s/(T-s))^2\,ds=\int_0^u s/[T(T-s)]\,ds<\infty\). The singularity is therefore not a problem at any fixed time strictly before \(T\); it is the approach to \(T\) that changes the situation.

Why Equivalence Fails at the Terminal Time

On the full interval, the two laws are not equivalent. Under \(\mathbb{Q}\), the endpoint is exactly zero; under \(\mathbb{P}\), a normal random variable with variance \(T>0\) equals zero with probability zero. The endpoint event therefore distinguishes the measures.

Proof. Let \(A=\{B_T=0\}\). The construction of the bridge gives \(\mathbb{Q}(A)=1\), whereas \(B_T\sim N(0,T)\) under \(\mathbb{P}\), so \(\mathbb{P}(A)=0\). Equivalent measures must agree on which events have probability zero; hence \(\mathbb{Q}\) and \(\mathbb{P}\) are not equivalent on \(\mathcal{F}_T\). Also, \(\mathbb{P}(B_T\neq0)=1\). On this event, continuity gives \(B_t\to B_T\neq0\), and therefore \(B_t^2/(2(T-t))\to\infty\). The logarithmic term \(\tfrac12\log(T/(T-t))\) grows more slowly than \(1/(T-t)\), so the explicit density formula implies \(Z_t\to0\). But \(\mathbb{E}^{\mathbb{P}}[Z_t]=1\) for every \(t<T\). If the family were uniformly integrable, its almost-sure limit would also be its \(L^1\) limit, forcing the expectations to converge to the expectation of zero. That contradicts their constant value one. Thus the densities are not uniformly integrable. \(\square\)

A Common Girsanov Pitfall

A tempting but incorrect argument is to take the density formula for \(t<T\), substitute \(t=T\), and treat the result as a terminal Radon–Nikodym density. The limiting calculation instead gives zero \(\mathbb{P}\)-almost surely, while every finite-time density has expectation one. A zero terminal density cannot define a probability measure by integration: its integral is zero, not one. The failure is not a removable endpoint value; it records the loss of uniform integrability and the fact that the bridge concentrates on a Brownian-null event.

Likewise, the finite-time Girsanov argument cannot simply be extended by replacing \(u<T\) with \(u=T\). Girsanov’s theorem on a closed interval requires a valid density process through that interval. Here the density tends to zero under \(\mathbb{P}\), and the bridge drift’s expected squared integral diverges at the endpoint. The useful conclusion is precise: on every \([0,u]\) with \(u<T\), the bridge is a Girsanov drift change with equivalent measures; on \([0,T]\), the bridge and Brownian laws are singular with respect to each other.