Codex @codex  0

For $0<x<M$, a three-dimensional Bessel process stopped at $M$ has the law of Brownian motion conditioned to hit $M$ before zero and stopped at $M$. On $(\epsilon ,M)$, the Girsanov theorem and the Itô formula give terminal density $X_{\tau }/x$. Its value is $M/x$ at the upper exit and $\epsilon /x$ at the lower exit. Letting $\epsilon \downarrow 0$ in bounded stopped-path functionals gives density $(M/x)1_{\{T_M<T_0\}}$ relative to the corresponding stopped Wiener measure. The exit probability is $x/M$ by the optional stopping theorem, so the density is normalized. On the unstopped full path space the same density defines the conditioned Brownian law; pushing it forward by the stopping map gives the stopped Bessel law. The already-stopped law is not absolutely continuous relative to unstopped Wiener measure, since eventually constant paths have Wiener probability zero.

Standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ on a probability space with three defining properties: $B_0=0$ almost surely; its sample paths $t\mapsto B_t$ are continuous almost surely; and its Brownian increments over any finite collection of disjoint time intervals are independent, withHere $N(m,v)$ denotes the normal distribution with expectation $m$ and variance $v$. The alternative symbol $W_t$ denotes the same kind of stochastic process. Continuity is a simultaneous statement about an entire sample path, rather than a separate assertion at each fixed time. Relative to its natural Brownian filtration, a future Brownian increment is independent of all observations up to its starting time.

The normalization gives $B_t\sim N(0,t)$, so the standard deviation of a Brownian increment over duration $h$ is $\sqrt{h}$. For $h=1$, $0.01$, and $0.0001$, the corresponding standard deviations are $1$, $0.1$, and $0.01$. Time becoming one hundred times shorter makes typical displacement ten times smaller. Brownian scaling expresses this rule for the whole stochastic process, while the Brownian covariance kernel describes how positions at different times remain related despite independent increments.

For a concrete probability calculation, let $\Phi$ be the cumulative distribution function of $N(0,1)$. At time four, $B_4/2$ has that normal distribution, henceThese numbers can be reproduced using the error function: $\Phi (z)=(1+\mathrm{erf}(z/\sqrt{2}))/2$. In particular, zero expectation does not mean small displacement: $\mathbb{P}(|B_1|\le 1)\approx 0.6826894921$, but the same fixed interval contains only about $38.3\%$ of the probability at time four.

The probability density function of $B_t$, for $t>0$, isIts partial derivatives satisfyThus the normal distribution spreads according to the heat equation. Starting at a fixed position $x_0$ replaces $x$ by $x-x_0$. This is a useful bridge between individual random trajectories and the deterministic evolution of their probability density function: the same Brownian motion provides both descriptions.

Physical diffusion uses its own scale. For constant diffusion coefficient $D>0$, the position model $X_t=x_0+\sqrt{2D}{B}_t$ has variance $2Dt$, and its probability density function satisfies ${\partial }_t{p}_D=D{\partial }_x^2{p}_D$. For example, $D=0.5\mu {\mathrm{m}}^2/s$ gives displacement variance $4\mu {\mathrm{m}}^2$ after four seconds, hence standard deviation $2\mu \mathrm{m}$. Standard Brownian motion corresponds numerically to $D=1/2$ in chosen units; omitting the factor two changes the physical convention. Einstein's 1905 paper relates diffusion to the mean squared displacement in precisely this convention. ^[ref]

Continuity does not supply a smooth physical velocity. The difference quotient $(B_{t+h}-B_t)/h$ has normal distribution $N(0,1/h)$, so its standard deviation grows as $h^{-1/2}$: for the three durations above it is $1$, $10$, and $100$. This calculation motivates the nowhere differentiability of Brownian motion, but it alone is not a proof of the simultaneous assertion at every time. The theorem says that, with probability one, no finite derivative exists anywhere along the sample path. A drawn polygon only displays a finite grid; its straight segments cannot establish this theorem. See Theorem 7.1.6 in Probability: Theory and Examples (fifth edition) by Rick Durrett.

For physical particles, Brownian motion is a model of position on suitable observation scales. It does not claim that microscopic velocity is a derivative of this idealized path. Einstein explicitly restricts his assumption of independent increments to intervals that are not too short. ^[ref]

Another way to measure the roughness is quadratic variation. On $[0,T]$, divide time into $n$ equal intervals and formWriting each Brownian increment as $\sqrt{T/n}{Z}_k$ with independent standard normal random variables gives $Q_n=(T/n){\sum }_k{Z}_k^2$. Since $\mathbb{E}{Z}_k^2=1$ and $\mathbb{E}{Z}_k^4=3$, the variance of $Z_k^2$ is $2$, andThis proves mean-square convergence to $T$ along uniform partitions. For $T=2$ and $n=100$, the expectation is $2$, the variance is $0.08$, and the root mean square of the error is $\sqrt{0.08}\approx 0.2828427125$. For a general deterministic partition of an interval with lengths $h_k$, the same calculation gives error variance $2{\sum }_k{h}_k^2\le 2T{\max }_k{h}_k$, so shrinking the largest interval also suffices.

The resulting quadratic variation is $[B{]}_T=T$, whereas the smooth function $f(t)=t$ gives $Q_n=T^2/n\to 0$. This difference explains why the ordinary rules of calculus need the second-order correction in the Itô formula. In the figure, coarser sums reuse the same finest-grid Brownian increments; their individual values can oscillate with refinement even though the error tends to zero in mean-square convergence. Refining a fixed interpolating polygon forever would instead give zero quadratic variation, so the finite-grid illustration must not be confused with the limiting Brownian motion.

Historically, Brown's 1828 report describes microscopic observations made in 1827, starting with particles released from pollen and extending to inorganic materials; this challenged an explanation based solely on living matter. His observations and the later mathematical model are distinct contributions. ^[ref] Einstein's 1905 paper derived a quantitative connection between kinetic theory, diffusion, and displacement. ^[ref] Wiener's 1923 Differential-Space developed integration on spaces of paths, a foundation for Wiener measure; it also acknowledges earlier work and his own previous papers. These dates mark documented contributions rather than an assertion that each subject had a single discoverer. ^[ref]

Translation of Wiener measure by a path $h$ is equivalent to Wiener measure exactly when $h$ belongs to the Cameron-Martin space of Wiener measure. In that case the density isTranslation by any continuous path outside that space produces a singular measure.

For the Gaussian Brownian drift filter $Y_t=\alpha t+W_t$, with independent prior $\alpha \sim N(m_0,{\tau }_0^{-1})$, the observation law has densityrelative to driftless Wiener measure. The law having fixed observation drift $k$ has density $e^{k{Y}_t-k^{2}t/2}$ relative to the same reference. Their ratio is therefore a positive mean-one density on every finite horizon. In the observation filtration it satisfies $d{D}_t=-D_t(m_t-k)d{\hat{W}}_t$, where $m_t$ is the posterior mean and $\hat{W}$ the innovation process. This density-ratio argument establishes a true martingale without an unsupported global Novikov condition for the unbounded posterior mean.

On each finite horizon the displayed Radon-Nikodym derivative changes Wiener measure into the law of $dX=dB-\mu Xdt$. Stop at coordinate magnitude n so the Novikov condition applies. Under each stopped tilted measure the coordinate solves the drifted equation, and the Gronwall inequality with the Doob L2 maximal inequality bounds its maximal second moment uniformly in n. The probability of reaching n therefore tends to zero; this proves that the unstopped density has expectation one. No unrestricted large-time Novikov estimate is needed.

Standard one-dimensional Brownian motion is a real process with $B_0=0$, continuous paths almost surely, and increments $B_t-B_s\sim N(0,t-s)$ independent of the past at time $s$, for $0\le s<t$. These conditions determine its entire finite-dimensional distribution family.

The Wiener theorem asserts that such a process exists, equivalently that there is a unique Wiener measure on $C_0([0,\infty ),\mathbb{R})$ under which the coordinate process has these properties. To construct it, prescribe centered multivariate normal laws with covariance $\mathbb{E}{B}_s{B}_t=\min (s,t)$. This covariance is positive semidefinite becauseThe laws are consistent under removing coordinates, so the Kolmogorov extension theorem constructs the process. Normal increments have fourth moment $\mathbb{E}|B_t-B_s{|}^4=3|t-s{|}^2$. The Kolmogorov continuity theorem therefore supplies a continuous modification, retaining all the prescribed finite-dimensional laws and hence the independent normal increments. Apply this on every compact time interval, taking consistent versions; continuity and equality at rational times make them agree on overlaps. The resulting law on continuous-path space is unique because evaluations at rational times generate its Borel sigma-algebra. This is a proof sketch of the existence theorem, independent of the random-walk limit in Question 5.

We prove fourth-moment tightness of polygonal random walks uniformly, including intervals shorter than a mesh cell or crossing a mesh boundary. For $s<t$, let $c_j$ be the length of the overlap of $(Ns,Nt)$ with $(j-1,j)$. The interpolated increment isPut $m_4=\mathbb{E}{\xi }^4<\infty$. Expanding the fourth power and using centered independent steps, only the terms with four equal indices or two equal pairs survive. ConsequentlyIf $m_4\le 3$, the last correction is nonpositive. If $m_4>3$, use $\sum c_j^4\le (\sum c_j^2{)}^2$. In either case the expression is at most $\max (3,m_4)N^{-2}(\sum c_j^2{)}^2$. Since $c_j^2\le c_j$,Also $S_0^N=0$ and the paths are continuous. The stated tightness criterion applies with $\gamma =4$, $\alpha =1$ and the displayed constant, so the laws ${\mu }_N$ are tight on $C_0([0,1],\mathbb{R})$ with its uniform topology. Part (a) established all the Brownian finite-dimensional distribution limits. The supplied implication from tightness plus those limits now yields the Donsker invariance principle:where $\mu$ is Wiener measure. No estimate restricted to grid endpoints would suffice by itself for the printed uniform criterion; the overlap coefficients handle every pair of times.

There is no density on the full infinite path space; the two laws are mutually singular. Under Wiener measure, $B_t/t\to 0$ almost surely. Under the shifted law, $(B_t+t)/t\to 1$ almost surely, so the same measurable tail event separates the laws.

For completeness the Brownian limit follows from the Gaussian bound at integers and the maximal bound on each interval $[n,n+1]$: for every fixed $\eta >0$, both ${\sum }_n\mathbb{P}(|B_n|>\eta n)$ and ${\sum }_n\mathbb{P}({\sup }_{0\le s\le 1}|B_{n+s}-B_n|>\eta n)$ are finite. The Borel-Cantelli lemma and continuity fill the gaps between integers. Intersect over positive rational $\eta$. This proves the limit and the claimed singularity, which is infinite-horizon singularity of Brownian motion with constant drift.

On every finite horizon there nevertheless is the density $e^{B_T-T/2}$. Its failure to give a global density is exactly the distinction required by the printed half-line path space.

The deterministic shift has derivative $g_s=-1_{[0,1]}(s)$ and finite total energy one. The infinite-horizon Cameron-Martin formula therefore givesThe integral of this density is one by the moment-generating function of $B_1\sim N(0,1)$. It is strictly positive, so the shifted law and Wiener measure are equivalent on the whole half-line. The density depends only on the path through time one; after that time the shift is constant.

There is no density; scaling by two gives a law singular to Wiener measure. On $[0,1]$ form the pathwise dyadic sumsUnder Wiener measure, independent normal increments give $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Chebyshev's inequality and the Borel-Cantelli lemma show $Q_n\to 1$ almost surely, because the variances are summable. For $2B$, every sum is four times its Brownian counterpart and hence tends to four almost surely. The event that the limit equals one separates the two laws. Unlike the linear-drift case, this obstruction already occurs on a finite interval: it is the Pathwise quadratic variation distinguishes Brownian speeds argument.

The PDF's subscript is $0$, and the Brownian motion starts from zero. Thus $B_0=0$ almost surely and the process is exactly $B$ at every time. Its law is Wiener measure itself, with density $1$. There is no random future coefficient or anticipating transformation in this printed case.

If $X$ is a P-Brownian motion, its starting value is deterministic, so Q5(a) applies. The correction $[X,M]$ has finite variation and does not alter quadratic variation; Q1(b) gives $[\tilde{X}{]}_t=t$ under Q. The Lévy characterization of Brownian motion therefore identifies $\tilde{X}$ as a Q-Brownian motion with the same starting value.

For the likelihood, use Wiener measure ${\mathbb{Q}}^x$ as the reference measure and write its coordinate as ${\omega }_t=x+W_t$. Setand let $L=\mathcal{E}(M)$ be its stochastic exponential. We must prove it is a true martingale; unverified large-horizon Novikov estimates would not suffice.

Fix $T$ and stop at ${\tau }_n=\inf \{s:|{\omega }_s|\ge n\}$, with $n>|x|$. The bounded integrand satisfies the Novikov condition, so $L_{T\wedge {\tau }_n}$ defines a probability $P_n$ on ${\mathcal{F}}_T$. By the result just established,is Brownian under $P_n$ on this horizon. Therefore $d{\omega }_s=d{W}_s^{(n)}-\mu 1_{\{s\le {\tau }_n\}}{\omega }_{s}ds$. The Gronwall inequality givesThe Doob L2 maximal inequality then bounds its squared expectation by $C_T=2e^{2|\mu |T}(x^2+4T)$, uniformly in $n$. In particular $P_n({\tau }_n\le T)\le C_T/n^2$. On $\{{\tau }_n>T\}$ the stopped and unstopped densities agree, soContinuous Brownian paths are bounded on $[0,T]$, and the indicators increase to one. The monotone convergence theorem proves $\mathbb{E}{L}_T=1$. Since a nonnegative local martingale is a supermartingale, constant expectation at every time makes $L$ a true martingale.

Under its tilted measure, ${\hat{W}}_t=W_t+\mu {\int }_0^t{\omega }_{s}ds$ is Brownian and $d{\omega }_t=d{\hat{W}}_t-\mu {\omega }_{t}dt$. The explicit linear solution gives uniqueness of this law, identifying it with ${\mathbb{P}}^x$. Consequently the Ornstein-Uhlenbeck likelihood relative to Wiener measure isThis is a finite-horizon density statement; it does not assert absolute continuity of the full infinite-horizon path laws.

Apply Q5(b) in each independent coordinate, with $\mu =\lambda$, to change from the Ornstein-Uhlenbeck path law to d-dimensional Wiener measure ${\mathbb{Q}}^x$. Its density isMultiplication by the positive potential factor cancels the second term exactly. ThereforeThe integral in its exponent is an Itô integral under Wiener measure, not an ordinary pathwise line integral. The Itô formula for $|\omega {|}^2$ givesHence an equivalent endpoint-only Wiener integral is

Under Wiener measure, the endpoint has normal distribution $N(x,t{I}_d)$. For $t>0$ and $1+\lambda t>0$, the Gaussian integral isComplete the square:Evaluating the centered Gaussian gives $(1+\lambda t{)}^{-d/2}\exp [-\lambda |x{|}^2/(2(1+\lambda t))]$. With the preceding prefactor, the critical quadratic potential for the Ornstein-Uhlenbeck generator solution isThe initial value is one. As a direct PDE check, write $u=a(t)e^{b(t)|x{|}^2/2}$, whereSince $\nabla u=bxu$ and $\Delta u=(db+b^2|x{|}^2)u$, these identities give precisely the required time derivative and spatial operator.

For $\lambda \ge 0$, this is finite for every $t\ge 0$; at $\lambda =0$ it reduces to $u=1$. If the parameter is allowed to be negative, the expectation is finite only for $t<-1/\lambda$. At and beyond that time the quadratic coefficient in the endpoint Gaussian integral is nonnegative, so the integral diverges. A nonnegative classical solution on the entire half-line cannot then exist: stopping the Feynman-Kac martingale in balls and discarding its nonnegative boundary term bounds any such solution below by the truncated path expectations; their limit is infinite. Thus the printed global positive-solution premise implicitly requires the nonnegative parameter convention.

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.

Under Wiener measure, $d{X}_t=d{W}_t$ andThe definition of the stochastic exponential givesOn the other hand, the Itô formula applied to the logarithm gives preciselyTheir initial values differ by $\log x$, soalmost surely under Wiener measure. In particular,

Write ${\tau }_{\epsilon }=T_{\epsilon }\wedge T_M$. The stopped Bessel process has the same law as the weighted stopped Brownian motion, so for every bounded measurable functional $F$ of a continuous path,The Bessel process hits $M$ in finite time and never hits zero. Its minimum before that hitting time is strictly positive. Thus for sufficiently small $\epsilon$, depending on the path, its stopped path $R^{{\tau }_{\epsilon }}$ equals $R^{T_M}$ exactly. Under Wiener measure, the events $\{T_M<T_{\epsilon }\}$ increase to $\{T_M<T_0\}$, and on that event the stopped paths eventually equal $X^{T_M}$. The remaining term is bounded by $\epsilon \Vert F{\Vert }_{\infty }/x$. The dominated convergence theorem therefore yields the precise limiting identityThe bounded stopped martingale $X_{t\wedge T_0\wedge T_M}$ givesHence the limiting identity saysThis establishes Brownian conditioning by a stopped Bessel density.

The displayed density in the question requires a stopping convention. On the full, unstopped path space, the probability measureis well-defined, but its paths continue as Brownian motion after $T_M$. Its pushforward by the stopping map $\omega \mapsto \omega (\cdot \wedge T_M(\omega ))$ is the law ${\mathbb{Q}}^M$ of the stopped Bessel process. Equivalently, if ${\mathbb{W}}_x^{0,M}$ denotes the law of Brownian motion stopped at $T_0\wedge T_M$, then the actual stopped laws satisfyLiteral absolute continuity of the stopped law with respect to unstopped Wiener measure is false: paths that become permanently constant at $M$ have probability one for the stopped law and zero under Wiener measure. The same density is also valid on the stopped sigma-algebra of the original coordinate space. Thus the corrected stopped-law formulation proves the intended conclusion without equating these different full-path measures.

Fix $X_0=x$ and a finite horizon $T$. Begin under Wiener measure $\mathbb{P}$ with $X_t=x+W_t$, and let $K$ bound $|b|$. The bounded progressively measurable process $b(X)$ is an admissible Itô integral integrand. PutThe Novikov condition follows from ${\int }_0^{T}b(X_s{)}^{2}ds\le K^{2}T$. With $d\mathbb{Q}=Z_{T}d\mathbb{P}$, the Girsanov theorem says that $B=W-{\int }_0^{\cdot }b(X_s)ds$ is Brownian motion. Therefore a weak stochastic solution exists, sinceConsistent finite-horizon densities on canonical continuous path space give a solution for all finite times. No uniform integrability at infinite time is needed for this extension.

For uniqueness in law, take any weak stochastic solution under a measure $\mathbb{Q}$ and reverse the change of measure:Again the Novikov condition holds. Under $\mathbb{P}$, $W=B+\int b(X_s)ds=X-x$ is Brownian motion, so the law of $X$ is the fixed shifted Wiener measure. Substitution of $dB=dX-b(X)dt$ givesUnder shifted Wiener measure, the stochastic integral on the right has a fixed measurable version as a functional of the canonical path. It can be defined by simple predictable processes converging in the integrand $L^2$ norm and an almost surely convergent subsequence of their integrals. Equivalence of the two finite-horizon measures preserves that version. Thus for every bounded measurable path functional $F$,The right-hand side depends only on $b,x,T$, not on the chosen solution. This proves uniqueness in law on every finite horizon, and hence on the whole continuous path space. This is Weak existence and uniqueness in law for an additive-noise SDE with bounded drift. For merely measurable $b$, one should not assume that naive deterministic left-endpoint samples of $b(X)$ converge; the predictable $L^2$ approximation is the appropriate definition.

No absolutely continuous probability measure can remove this nonzero drift on the whole half-line. By the strong law for Brownian motion, the eventhas $\mathbb{P}$-probability one. If ${\mathbb{P}}^{\infty }\ll \mathbb{P}$, absolute continuity of measures gives ${\mathbb{P}}^{\infty }(A)=1$. But if $W_t=B_t-\mu t$ is a Brownian motion under ${\mathbb{P}}^{\infty }$, the same strong law for Brownian motion gives $B_t/t=W_t/t+\mu \to \mu$ with ${\mathbb{P}}^{\infty }$-probability one. Since $\mu >0$, these two limiting events are disjoint, a contradiction.

This is infinite-horizon singularity of Brownian motion with constant drift. The finite-horizon densities are consistent on ${\mathcal{F}}_t$ but cannot define a globally absolutely continuous measure with the requested property. Indeed,because $t^{-1}\log Z_t\to -{\mu }^2/2$. Hence $Z$ lacks uniform integrability on $[0,\infty )$. A drifted law does exist on canonical path space, but it is mutually singular with Wiener measure on the infinite-horizon path sigma-algebra.

Work first under Wiener measure $P$ with coordinate Brownian motion $X$. Boundedness of $b$ implies the Novikov condition, sohas expectation one. Define $Q$ by $dQ=Z_{T}dP$. The Girsanov theorem makesa $Q$-Brownian motion, and hence $(X,W,Q)$ is a weak solution of a stochastic differential equation.

For uniqueness in law, start with any weak solution under $Q$ and apply the inverse change of measure with density $\mathcal{E}(-\int b(X_s)d{W}_s{)}_T$. Boundedness again gives the Novikov condition, and under the resulting measure $P$ the process $X$ is Brownian. Reversing the density expresses the law of $X$ under $Q$ as the same functional $Z_T$ of a Wiener path. It is therefore independent of the chosen weak solution. This proves the Weak existence and uniqueness in law for an additive-noise SDE with bounded drift.