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The test rejects for large values of this tail-weighted empirical discrepancy. For a specified continuous null, its standard limiting statistic is ${\int }_0^1{B}^0(u{)}^2/[u(1-u)]du$, using a Brownian bridge. The reciprocal variance weight gives tail deviations more influence than the Cramér–von Mises criterion. If null parameters are estimated, fitting must be included when computing or simulating its reference law.

The covariance kernel of a standard Brownian bridge on $[0,1]$ is $c(s,t)=\min (s,t)-st$. Its integral operator has eigenfunctions $\sqrt{2}\sin (k\pi t)$ and eigenvalues $(k\pi {)}^{-2}$ for $k\ge 1$.

For standard Brownian motion, the Brownian bridge $U_t=B_t-t{B}_1$, $0\le t\le 1$, is independent of $B_1$. Every finite vector of bridge coordinates is jointly multivariate normal with $B_1$, and its cross-covariances with $B_1$ vanish. The independence of uncorrelated jointly normal variables and independence extended from generating pi-systems prove independence of the whole stochastic process.

For $Y_s=\lambda s+W_s$ and fixed $t>0$, subtracting $(s/t)Y_t$ leaves the Brownian bridge $W_s-(s/t)W_t$. It is independent of $(\lambda ,Y_t)$, since it is independent of $\lambda$ and has zero covariance with $W_t$ in their jointly Gaussian law. The observed path is determined by this bridge and its endpoint, so the bridge gives no further information about $\lambda$. Hence conditioning on the full observation filtration at $t$ gives the same posterior as conditioning on $Y_t$ alone.

A confidence band provides simultaneous bounds on an unknown function over its entire index set. Its coverage event is the intersection over all indices, unlike separate pointwise confidence intervals. For a continuous cumulative distribution function, the Kolmogorov-Smirnov theorem gives an asymptotic band $F_n\pm c_{\alpha }/\sqrt{n}$, clipped to $[0,1]$, where $c_{\alpha }$ is a $(1-\alpha )$-quantile of the absolute Brownian bridge maximum. The simultaneous event equals $\sqrt{n}\Vert F_n-F{\Vert }_{\infty }\le c_{\alpha }$, so convergence in distribution at this continuity point proves coverage tending to $1-\alpha$.

This goodness-of-fit criterion integrates the squared difference between the empirical distribution function and a specified continuous null distribution. Under a fully specified null, the probability integral transform makes its law distribution-free; its limiting law is ${\int }_0^1{B}^0(u{)}^{2}du$ for a Brownian bridge. Parameter fitting changes that calibration.

For a cumulative distribution function $F$ and a standard Brownian bridge $\mathbb{G}$, the F-Brownian bridge is the centred Gaussian process ${\mathbb{G}}_F=\mathbb{G}\circ F$. Its covariance is $F(\min (s,t))-F(s)F(t)$, by substituting $F(s),F(t)$ into the bridge covariance. It can jump when $F$ jumps. Its law is tight in the supremum norm, since composition with $F$ is a continuous map from the separable space of continuous bridge paths.

For fixed $x_1,\dots ,x_d$, the vector $({\alpha }_n(x_j){)}_{j=1}^d$ converges in distribution to $(B(G(x_j)){)}_{j=1}^d$, where $B$ is a standard Brownian bridge. To prove it, write the vector as $n^{-1/2}$ times the sum of independent centred bounded indicator vectors. Their covariance matrix is $G(\min (x_i,x_j))-G(x_i)G(x_j)$. The multivariate central limit theorem gives a centred multivariate normal distribution with this covariance, exactly the bridge vector. Singular covariance matrices, including endpoint times or repeated values of $G$, cause no problem. This is a finite-dimensional assertion; a full function-space limit additionally requires tightness.

For an empirical distribution function based on an independent sample from a continuous distribution $F$,where $B$ is a Brownian bridge.

This limit calibrates the Kolmogorov-Smirnov test; the test compares an empirical distribution with its null distribution using the supremum discrepancy.

The Lévy construction builds Brownian motion on $[0,1]$ by successive independent midpoint displacements on dyadic intervals. Start with $B_0=0$ and $B_1=Z_0$, where $Z_0$ has the standard normal distribution. On each dyadic interval $[u,v]$, choose a fresh independent standard normal random variable $Z$ and use the displayed midpoint rule. Conditional on the endpoints, this is the correct Brownian bridge midpoint law.

To verify the increment law, if $D=B_v-B_u\sim N(0,\ell )$ with $\ell =v-u$, the new increments are $D/2\pm \sqrt{\ell }Z/2$. They are Jointly Gaussian, each has variance $\ell /2$, and their covariance is zero, hence they are independent. Induction gives all dyadic Brownian increments. The change between consecutive linear interpolants has uniform norm $2^{-n/2-1}{\max }_{k<2^n}|Z_{n,k}|$. The Gaussian tail bound and union bound bound the chance that this maximum exceeds $n$ by $2^{n+1}{e}^{-n^2/2}$. The First Borel-Cantelli lemma and ${\sum }_{n}n{2}^{-n/2}<\infty$ prove uniform convergence almost surely. The continuous limit has the required Brownian increments at arbitrary times by dyadic approximation.

The Brownian motion displacement has a normal distribution with variance $\Gamma t$. Its transition probability density function is thereforeComparison with the free particle heat kernel obtained in part (c) gives the required matching:This matches whole path distributions after a linear reparametrization, not merely endpoint variances. To see this, divide the Brownian duration $t$ into $M$ slices of size $\delta t$. The independent increments have normal distributions with variance $\Gamma \delta t$, so their joint density contributesFor the Euclidean path integral, the corresponding expression isThe matching condition makes $\Gamma \delta t=\hslash \delta \tau /m$, so both the exponents and all normalization factors coincide. In particular, identifying Brownian time with imaginary time, $t=\beta \hslash$, requires $\Gamma =\hslash /m$, or diffusion coefficient $D=\hslash /(2m)$.

With both endpoints fixed, the normalized path law is a Brownian bridge. The Euclidean path integral itself is the endpoint density times that conditional path law. In the normalized time coordinate $s=\tau /(\beta \hslash )$, its expected value and covariance areThe classical path is therefore the bridge's expected value, while individual Brownian bridge paths fluctuate about it. Typical Brownian motion paths are continuous but nowhere differentiable: the kinetic Euclidean action is interpreted through the time-sliced diffusion path integral, rather than by assuming that a typical path has an ordinary velocity.

For times $t_1,\dots ,t_n<1$ and real coefficients $a_i$, the linear combination of coordinates isThe Gaussianity of deterministic Brownian stochastic integrals makes this a centered normal random variable. Indeed step-function approximations give linear combinations of independent normal increments, and the Itô isometry gives convergence in $L^2$; their characteristic functions converge to that of a normal variable with the limiting variance. Since every linear combination is normal, the coordinate vector is a Gaussian random vector. Appending the deterministic coordinate $X_1=0$ preserves that property, so $X$ is a centered Gaussian process on $[0,1]$.

For $0\le s\le t<1$, the covariance version of the Itô isometry givesAt $t=1$ the covariance is zero because $X_1=0$, agreeing with the formula. Thus the Brownian bridge covariance kernel isIts finite-dimensional distributions identify it as a Brownian bridge; continuity at the terminal endpoint is established next.

For a fully specified continuous null distribution $F_0$, compare the empirical distribution function $F_n$ with $F_0$. The probability integral transform makes $F_0(Y_i)$ iid uniform under the null, giving distribution-free calibration. Three common discrepancies areThese define the Kolmogorov-Smirnov test, Cramér–von Mises criterion and Anderson–Darling test, respectively. Reject for large values. The supremum detects the largest discrepancy, the unweighted integral combines discrepancies throughout the distribution, and the reciprocal weight emphasizes the tails.

The Donsker theorem for empirical distribution functions gives $\sqrt{n}(F_n-F_0)\Rightarrow B^0\circ F_0$ for a Brownian bridge $B^0$, with covariance $\min (u,v)-uv$. Consequently the standard continuous-null limits are ${\sup }_{0\le u\le 1}|B^0(u)|$, ${\int }_0^1{B}^0(u{)}^{2}du$, and ${\int }_0^1{B}^0(u{)}^2/[u(1-u)]du$ for $\sqrt{n}{D}_n,C_n,A_n$, respectively; the weighted last limit also requires control near the endpoints. Exact finite-sample or asymptotic null quantiles provide critical values.

If parameters in $F_{\theta }$ are estimated from the same observations, these fully specified-null critical values generally fail. A first-order expansion isso the estimated parameter changes the limiting process. One must use the fitted-model null law, adjusted asymptotic theory, or a parametric bootstrap that re-estimates the parameters in each replicate.

In statistical decision theory, the statistical invariance principle says that a procedure should respect transformations that leave the inferential problem unchanged. If a group action preserves the model and carries each of the null and alternative parameter sets into itself, an invariant test has rejection probability function $\phi (gy)=\phi (y)$. A maximal invariant identifies its orbits, so every invariant test factors through that statistic: define its value at an orbit using any representative; invariance makes the definition independent of the representative. If the group is transitive on the null parameter set, the null law of any invariant statistic is free of that nuisance parameter.

For example, testing a normal mean $\mu ={\mu }_0$ with unknown scale, positive transformations $y_i-{\mu }_0\mapsto b(y_i-{\mu }_0)$ preserve a one-sided mean alternative. The statistic $\sqrt{n}(\bar{Y}-{\mu }_0)/s$, with the usual unbiased sample-variance scale $s$, is invariant and has the Student t-distribution with $n-1$ degrees of freedom under the null. For estimation the corresponding requirement is an equivariant estimator, together with a loss function invariant under simultaneous transformation of decision and parameter. Invariance alone does not prove unrestricted optimality or supply a unique best procedure.

There is a distinct probabilistic use of the term: the Donsker invariance principle states that centred iid partial sums, normalized by $\sigma \sqrt{n}$, converge as processes on $[0,1]$ to Brownian motion. Its empirical-process version gives a Brownian bridge, used by goodness-of-fit tests in the next note. These functional limit theorems should be distinguished from the group-based inferential principle.

Assume, as in the Bayesian model, that the unknown drift $\alpha$ is independent of the driving Brownian motion, with known positive prior precision ${\tau }_0$. Recover the observed signal from the positive stock price:For a fixed time $t$, the centered observed path $Y_s-(s/t)Y_t$ is a Brownian bridge independent of $(\alpha ,Y_t)$: it is independent of the prior and has zero covariance with $W_t$ in their joint Gaussian law. Thus the whole observed history has the same posterior information about $\alpha$ as its endpoint. This proves Brownian endpoint sufficiency for a constant drift rather than assuming it.

The likelihood of that endpoint, as a function of $\alpha$, is proportional to $\exp (\alpha Y_t-{\alpha }^{2}t/2)$. Multiplying by the normal distribution prior and completing the square gives the Gaussian Brownian drift filterDefine the observed innovation process ${\hat{W}}_t=Y_t-{\int }_0^t{\hat{\alpha }}_{s}ds$. It is a continuous observed-filtration martingale: for $s<u$, independence of future noise and the tower property of conditional expectation give $\mathbb{E}[\alpha -{\hat{\alpha }}_u\mid {\mathcal{F}}_s^S]=0$, so the conditional mean of its increment vanishes. Its quadratic variation is $t$, since subtracting the finite-variation drift from $Y$ does not change quadratic variation. The Lévy characterization of Brownian motion makes $\hat{W}$ an observed-filtration Brownian motion. Differentiating the posterior mean givesConversely, the last equation determines $\hat{\alpha }$ from $\hat{W}$ with deterministic coefficients and known initial value, and $Y=\hat{W}+\int \hat{\alpha }dt$. Thus the observed and innovation filtrations coincide, supplying the natural Brownian information needed for replication. Consequently the observed stock dynamics are $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma {\hat{\alpha }}_{t}dt$. Its observed market price of risk is ${\kappa }_t={\hat{\alpha }}_t-r/\sigma$, and its normalized state-price density solvesThe posterior mean is unbounded, so bounded-coefficient Novikov condition reasoning from Question 2 is not available automatically. A direct finite-horizon argument supplies true pricing. Relative to driftless Wiener measure for $Y$, the Bayesian mixture has observation densityA law with observation drift $k=r/\sigma$ instead has density $e^{k{Y}_t-k^{2}t/2}$. ThereforeThis is the finite-horizon pricing density for Gaussian drift learning. The ratio changes the physical observed law into a genuine probability law, hence has mean one. Indeed, differentiation of the explicit likelihood, using $d{Y}_t=d{\hat{W}}_t+{\hat{\alpha }}_{t}dt$, givesThe Itô formula therefore gives $d{D}_t=-D_t{\kappa }_{t}d{\hat{W}}_t$, agreeing with the stochastic exponential above. Under this pricing law the stock drift is $\sigma k=r$.

For logarithmic utility, terminal marginal utility is $1/w_T$. The state-price budget constraint and the marginal condition imply $w_T^{\ast }=1/(y{\zeta }_T)$ and $1/y=w_0$. Since ${\zeta }_T{w}_T^{\ast }=w_0$ is constant, its conditional-expectation pricing process is constant as well. Thus the entire optimal wealth process isIndeed, for any other nonnegative admissible terminal wealth $X$, the elementary inequality $\log (X/w_T^{\ast })\le X/w_T^{\ast }-1$ givesThis directly verifies optimality. Expected candidate log wealth is finite because $\mathbb{E}{\hat{\alpha }}_t^2={\hat{\alpha }}_0^2+{\tau }_0^{-1}-({\tau }_0+t{)}^{-1}$ and its time integral is finite on every fixed horizon. Applying the Itô formula to $1/\zeta$ givesThe observed self-financing portfolio with risky dollar fraction ${\pi }_t$ has diffusion coefficient ${\pi }_t\sigma$. Matching the coefficients proves the log-optimal investment with Gaussian drift learning ruleThe remaining dollars are held in the bank account; the fraction may be negative or exceed one because the question imposes neither short-sale nor borrowing restrictions.

Fix $T<\infty$ and a standard $d$-dimensional Brownian motion $B$ starting at zero. Let $D_n$ be deterministic nested finite partitions of $[0,T]$ with mesh tending to zero; dyadic partitions are the basic example. Let $B^n$ be the polygonal interpolation through the partition values and put ${\mathbf{B}}^n=S_2(B^n)$. The limit is enhanced Brownian motion, with incrementsHere the integral is a Stratonovich integral. In terms of Itô integrals,Its symmetric part is $\frac{1}{2}{B}_{s,t}^{\otimes 2}$, while its antisymmetric part is Lévy area,These identities and splitting the integral at an intermediate time give the Chen identity, so $\mathbf{B}$ has values in $G^2({\mathbb{R}}^d)$.

A precise nested polygonal approximation of Brownian rough paths statement is: for every $0<\alpha <1/2$,where the rough path metric used here isIn particular, for each $2<p<3$, choosing $1/p<\alpha <1/2$ gives convergence in the inhomogeneous $p$-variation rough path metric. The limiting signed area is the Lévy area, and the limiting full second level is Stratonovich, not Itô. This distinction is necessary: a polygonal lift has the prescribed symmetric second level, whereas an Itô lift would lack half the quadratic-variation correction.

Here is a proof sketch emphasizing the nested martingale structure. SetA Brownian bridge between two prescribed endpoints has the affine interpolation as conditional mean. ConsequentlyFor $i\ne j$ the coordinate processes remain independent when their respective partition values are conditioned on. Approximating the Itô integral by its left sums, conditioning each product, and then taking its $L^2$ limit givesIndeed each conditioned factor is its polygonal interpolation, and their sums converge to the ordinary integral of those piecewise linear functions. For a fixed pair $s,t$, the right side is therefore a martingale in the refinement index $n$. It is not being asserted to be a martingale in physical time; polygonal interpolation uses the right endpoint of its current cell.

The union of the partitions is dense. Continuity of $B$ implies that ${\mathcal{G}}_{\infty }=\sigma ({\bigcup }_n{\mathcal{G}}_n)$ contains the full Brownian path and its stochastic integrals, up to completion. The martingale convergence theorem for conditional expectations therefore yields convergence of $B_t^n$ and all off-diagonal ${\mathbb{B}}_{s,t}^{n,ij}$ at every fixed pair, almost surely and in $L^q$ for finite $q$. Simultaneously it holds for a countable dense collection of pairs. The diagonal terms must be handled separately:They are not conditional expectations of the diagonal Brownian second level: the latter would also contain a conditional variance.

Pointwise convergence alone is insufficient for the rough path metric. The Burkholder-Davis-Gundy inequalities, Brownian scaling and Gaussian increment moments give, for arbitrarily large $q$,Equivalently, using the homogeneous norm on $G^2$, the $q$-th moment of the Carnot-Carathéodory distance of an increment is bounded by $C_q|t-s{|}^{q/2}$. The metric form of the Kolmogorov continuity theorem, with $q$ chosen large, gives for every $\beta <1/2$ a random $K_{\beta }$ having all finite moments such thatConditioning these bounds givesThe diagonal bounds follow by squaring the first bound. The two conditional-expectation processes are nonnegative martingales; the Doob Lp maximal inequality makes their suprema over $n$ finite almost surely, with all needed moments. Thus all the polygonal lifts have a common random bound at exponent $\beta$.

These bounds and the Chen identity give joint equicontinuity of the increments as functions of $(s,t)$. Convergence on the dense collection of pairs consequently gives uniform convergence of both levels on the time triangle. Take $\alpha <\beta <1/2$. For either level $k=1,2$, with $M$ a common bound and ${\epsilon }_n$ its uniform difference, interpolation givesThis proves the stated Hölder rough path metric convergence. Finally, for a partition $D$, the $k$-th level difference satisfiesbecause $\alpha p>1$. Taking the appropriate powers and suprema proves $p$-variation convergence too. Nesting is what permits the conditional-expectation martingale proof; without nesting one instead needs direct approximation estimates.

Work on a finite interval $[0,T]$ with a fixed initial point $y_0\in {\mathbb{R}}^m$. To give a clean global statement, assume $V_0,V_1,\dots ,V_d$ are bounded vector fields with bounded continuous derivatives through order three. Consider the Stratonovich integral form of the stochastic differential equationFor a control in the Cameron-Martin space of Wiener measure,let $y^h$ solve the controlled ordinary differential equationThe Stroock-Varadhan support theorem says, in the uniform metric on continuous paths,Thus a path is in the support of a measure of the diffusion precisely when it can be approximated by these deterministic controlled trajectories. There is no ellipticity assumption: the allowed controls and vector fields already encode any restrictions on reachability.

The central point of the rough-path proof is that ordinary uniform approximation of the driver does not make the solution map continuous. Small planar loops can have small amplitude but accumulated area of fixed size, and vector fields with a nonzero Lie bracket detect that area. The correct driver is enhanced Brownian motion $\mathbf{B}$, including Lévy area. Fix $2<p<3$. The universal limit theorem gives a continuous mapfrom the geometric $p$-variation rough-path space to solution paths. It agrees with the controlled ordinary differential equation when $\mathbf{X}=S_2(h)$. Adding the time coordinate, with its mixed integrals defined by integration against $dt$ and integration by parts, incorporates $V_0$ without changing the continuity argument. This is the role of the universal limit theorem: it turns a support problem for all these nonlinear diffusions into a support problem for one enhanced driving process.

For $\mathbf{X}=\mathbf{B}$, the rough differential equation is the stated Stratonovich equation. Its second-order increment includesReplacing the Stratonovich integral in the second level by its Itô integral expression contributes $\frac{1}{2}{\sum }_{i}D{V}_i{V}_{i}dt$, which is exactly the Stratonovich-to-Itô correction. Polygonal convergence from Question 3 and the universal limit theorem therefore give the Wong-Zakai approximation: the ordinary equations driven by $B^n$ converge to this Stratonovich diffusion.

It remains to establish the support of enhanced Brownian motion:One inclusion follows immediately from Question 3. Every polygonal $B^n$ belongs to $H$, and $S_2(B^n)\to \mathbf{B}$ almost surely in the rough path metric. Since $K$ is closed, $\mathbf{B}\in K$ almost surely, whence its support is contained in $K$. This alone does not prove the reverse inclusion: convergence of random approximations does not automatically give positive mass near every control.

For that inclusion, take dyadic mesh ${\delta }_n=T{2}^{-n}$ and writeOn each grid interval $R^n$ is a pinned Brownian bridge. The bridges are mutually independent and independent of the grid values defining $B^n$, by the Gaussian projection formula. Their canonical Stratonovich enhancement ${\mathbf{R}}^n$ is therefore independent of $B^n$. It is important to enhance the bridge residual itself, including its area, rather than infer area smallness from its uniform size.

The bridge enhancement tends to the zero rough path in probability. Here are the estimates behind this assertion. A bridge on an interval of length ${\delta }_n$ scales as $\sqrt{{\delta }_n}$ times a unit bridge; its full antisymmetric area has mean zero, moments of order ${\delta }_n$, and second moment of order ${\delta }_n^2$. Since its endpoint displacement is zero, at grid points the cumulative second level is just the sum of the independent bridge areas. The Doob L2 maximal inequality givesPartial intervals at the two ends contribute bridge increments and at most two local areas. Bridge scaling, Gaussian moment bounds and the corresponding iterated-integral estimates yield, uniformly in $n$ and for arbitrarily large $q$,For an interval crossing several grid cells, the full-cell second levels are independent centered areas, whose sum has moments bounded by $C_q((t-s){\delta }_n{)}^{q/2}$; the two boundary pieces and their cross product satisfy the displayed bound as well. For an interval inside one cell these are the usual scaled bridge increment and area bounds. At each fixed pair $(s,t)$ both levels converge to zero in probability: the first-level variance is $O({\delta }_n)$, and the second-level second moment is $O((t-s){\delta }_n+{\delta }_n^2)$ for sufficiently fine grids. The Kolmogorov continuity theorem gives uniformly bounded moments of a Hölder rough-path norm at any exponent $\beta <1/2$. Tightness, convergence on a dense set of pairs and interpolation to $\alpha <\beta$ now give ${\rho }_{\alpha }({\mathbf{R}}^n,e)\to 0$ in probability. Taking $1/p<\alpha <1/2$ gives $d_p({\mathbf{R}}^n,e)\to 0$ in probability.

Fix $h\in H$. Its polygonal interpolations $h^n$ converge to $h$ in the total-variation norm: their derivatives are the cell averages of $\dot{h}$, which converge in $L^2$, hence in $L^1$. Moreover $|h^n{|}_{1\text{-var}}\le |h{|}_{1\text{-var}}$. Thus $S_2(h^n)\to S_2(h)$ in the rough path metric.

The rough path translation operation satisfiesFor a bounded variation path $b$, translation adds $b$ at level one and, at level two, the mixed termsThe second mixed term is defined by integration by parts. Their variation bounds show that translation is continuous, uniformly when $|b{|}_{1\text{-var}}$ is bounded, near the zero rough path. More explicitly the mixed second-level norms are bounded by a constant times $\Vert R{\Vert }_{p\text{-var}}|b{|}_{1\text{-var}}$; the other new second-level term is the path signature of $b$.

Given a ball of radius $\epsilon$ around $S_2(h)$, choose a sufficiently large $n$ so that $S_2(h^n)$ is close andfor the small $\eta$ supplied by translation continuity. The relevant variation bounds can be chosen uniformly in $n$, since the variations of the $h^n$ are uniformly bounded. For this fixed $n$, the event $|B^n-h^n{|}_{1\text{-var}}<\eta$ has positive probability: the finite vector of Brownian increments has an everywhere positive Gaussian density, and sufficiently small deviations of its vertices imply this variation bound. These two events are independent. On their intersection, translation continuity gives $d_p(\mathbf{B},S_2(h))<\epsilon$. Every ball around $S_2(h)$ therefore has positive probability. This proves $S_2(h)$ lies in the support, and taking closures gives $K$.

Finally, a continuous map between the separable path spaces in use satisfiesFor one direction the measure is concentrated on its support. For the other, an open neighborhood of $\Phi (x)$ pulls back to an open neighborhood of a support point $x$, which has positive measure. Using continuity of $\Phi$ and the identity $\Phi (S_2(h))=y^h$ proves the boxed Stroock-Varadhan support theorem. This separates the two ingredients: the probabilistic support calculation for Brownian motion and area, and the deterministic continuity supplied by the universal limit theorem.

If the diffusion is instead presented as an Itô integral equation with drift $b$, the controlled equation must useUsing $b$ unchanged in the controlled equation would generally give the wrong support description. The result above states explicit sufficient smoothness and nonexplosion assumptions; extensions to more general coefficients require the usual localization, not an unqualified assertion for arbitrary vector fields.

Fix $t>0$. For $0\le s\le t$, the residual observed path isThis Brownian bridge is independent of $(\lambda ,Y_t)$. Indeed it is independent of $\lambda$, while its covariance with $W_t$ is $sI-(s/t)tI=0$; zero covariance implies independence in the joint Gaussian law. The same argument for every finite collection of bridge times, followed by continuity and generation from rational times, gives independence of the entire bridge. Since the full observed path is determined by its endpoint and this independent bridge, Brownian endpoint sufficiency for a constant drift gives

First suppose the prior covariance matrix $V_0$ is positive definite. The Bayes' theorem multiplies the prior Gaussian density by the conditional observation density, whose dependence on $\lambda$ is $\exp ({\lambda }^T{Y}_t-t|\lambda {|}^2/2)$. Thus the posterior exponent, up to terms not depending on $\lambda$, isCompleting the square gives the Gaussian Brownian drift filter:For invertible $V_0$ these are equivalently $V_t=(V_0^{-1}+tI{)}^{-1}$ and ${\lambda }_t=V_t(V_0^{-1}{\lambda }_0+Y_t)$. At $t=0$ they reduce to the prior parameters.

The boxed formulas remain valid for a singular positive semidefinite matrix $V_0$. To see this directly without inverse prior matrices, the joint Gaussian random vector $(\lambda ,Y_t)$ hasThe latter matrix is invertible for $t>0$. The conditional multivariate normal distribution therefore has meanand covariance matrixAlternatively, subtract this mean from $\lambda$; the resulting Gaussian random vector has zero covariance with $Y_t$ and is therefore independent of it, which proves these conditional parameters.

Write the proposed innovation process in the observable formIt is adapted to the observation filtration and continuous. To prove the martingale property, take $0\le u<v$. Future increments of $W$ are independent of the joint information generated by $\lambda$ and the past of $W$, soAlso the tower property of conditional expectation gives $\mathbb{E}[{\lambda }_s\mid {\mathcal{G}}_u]={\lambda }_u$ for $s\ge u$. Conditional integration is legitimate because $\mathbb{E}|{\lambda }_s{|}^2\le \mathbb{E}|\lambda {|}^2<\infty$. Subtracting the integral proves $\mathbb{E}[{\hat{W}}_v-{\hat{W}}_u\mid {\mathcal{G}}_u]=0$. Removing a continuous finite variation process does not change quadratic covariation, henceThe Lévy characterization of multidimensional Brownian motion now applies: a continuous zero-start martingale with this bracket matrix is a standard $d$-dimensional Brownian motion relative to its given filtration. The observable innovation is a Wiener process in the observation filtration.

Differentiating the posterior formulas yields $V_t^{\prime }=-V_t^2$ andFor the final expression, the printed inverse matrices require the extra assumption that $V_0$ is positive definite. Under that assumption put $A_t=V_t^{-1}=V_0^{-1}+tI$. The Itô formula givessince $A_t{V}_t=I$ and $\mathrm{tr}(A_t{V}_t^2)=\mathrm{tr}{V}_t$. On the other hand,The trace terms cancel in the innovation exponential for Gaussian drift filtering, leavingThus $Z_0=1$ and $Z$ is a positive stochastic exponential, hence a nonnegative local martingale. To obtain the requested supermartingale conclusion explicitly, choose localizing stopping times ${\tau }_n\uparrow \infty$ for which $Z^{{\tau }_n}$ is a true martingale. The Conditional Fatou lemma givesThe same argument at $u=0$ gives $\mathbb{E}{Z}_v\le 1$, supplying integrability. The process is a supermartingale in the observation filtration. No unproved exponential-integrability condition on the random drift is required.

For a singular prior the literal inverse-matrix expression in the PDF is undefined, but there is a natural extension with the same conclusion. Define, for fixed $y$,For positive definite $V_0$, a Gaussian integral shows this reciprocal is exactly the printed expression. For any positive semidefinite $V_0$, the integral is positive and finite, $L_t=-{\Delta }_{y}L/2$, and ${\nabla }_y\log L(t,Y_t)={\lambda }_t$, by the same posterior weighting. The Itô formula consequently gives $d\log L(t,Y_t)={\lambda }_t^{T}d{Y}_t-|{\lambda }_t{|}^{2}dt/2={\lambda }_t^{T}d{\hat{W}}_t+|{\lambda }_t{|}^{2}dt/2$. Its reciprocal satisfies the boxed stochastic differential equation and the same supermartingale proof. This completes the singular-covariance extension while explicitly separating it from the undefined printed formula.