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A Bessel process of dimension $d$ is the radial part of $d$-dimensional Brownian motion. Away from zero it solves the stochastic differential equationThis equation also defines noninteger dimensions through its weak solutions.

For a maturity-differentiable Heath-Jarrow-Morton model forward field with the integrability needed for stochastic differentiation, the short rate $r_t=f(t,t)$ has the displayed dynamics. Indeed changing observation time uses the forward-rate stochastic differential equation, while changing maturity adds ${\partial }_{T}f(t,t)dt$. Under a risk-neutral measure, $\alpha (t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so $\alpha (t,t)=0$. The remaining drift depends on the diagonal slope of the whole forward curve, which need not be determined by $r_t$; a Markov process for the short rate is an additional modelling restriction. Without sufficient maturity regularity, the diagonal need not admit this classical Itô process formula.

The coefficient $\sigma$ multiplying the Brownian motion increment in a scalar stochastic differential equation. The local variance rate is ${\sigma }^2$, and the second-order coefficient in the backward Kolmogorov backward equation is ${\sigma }^2/2$. This distinguishes the amplitude from the physical diffusion coefficient convention $D={\sigma }^2/2$.

The scalar stochastic differential equation $d{X}_t=\tanh X_{t}dt+d{W}_t$ has a pathwise unique global strong solution, since its coefficients are globally Lipschitz. The positive martingale $e^{t/2}/\cosh X_t$ gives a Girsanov theorem change of measure under which $X_t-X_0$ is Brownian motion. For $X_0=x$, its transition density isThis is a Doob h-transform with $h=\cosh$, and a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$.

The drift coefficient is the coefficient $b$ of $dt$ in a stochastic differential equation. It describes the local deterministic tendency of the process.

For any step $\Delta >0$, set $a=e^{-\theta \Delta }$. The Ornstein-Uhlenbeck transition distribution gives the exact updateusing a fresh independent standard normal random variable for each interval. The update reproduces the joint probability distribution on the selected time grid. It can start from a fixed value or use Ornstein-Uhlenbeck stationary initialization. On an equally spaced grid it is an autoregressive process of order one after subtracting $\mu$. Its stationary variance follows directly from $v=a^{2}v+v_{\infty }(1-a^2)$.

For the stipulated example with $\Delta =0.1$, $a=e^{-0.1}\approx 0.90484$ and the innovation variance is $1-e^{-0.2}\approx 0.18127$. Given $X_t=6$, the next conditional expected value is about $5.61935$ and the conditional standard deviation is about $0.42576$.

The Euler-Maruyama method instead approximates the same stochastic differential equation byFor $0<\theta \Delta <2$ its stationary variance solves $v=(1-\theta \Delta {)}^{2}v+{\sigma }^2\Delta$, givingAt $\Delta =0.1$ in the example, this is $1/0.95\approx 1.05263$, about $5.26\%$ above the continuous model's stationary variance. If $\theta \Delta \ge 2$, this Euler recurrence has no finite stationary variance despite the stable continuous model. Exact transitions avoid this discretization error at sampled times; neither update alone determines the unresolved motion between samples.

A local solution is a locally defined stochastic process taking values in a specified open domain $U$ and satisfying the stochastic differential equation on every stopped interval before its lifetime. For $dX=b(X)dt+\sigma (X)dB$, the integral equation holds after each announcing stop, with the requisite local drift and noise integrability. A maximal local solution of a stochastic differential equation cannot be extended while remaining in $U$; a finite lifetime can be a boundary hit rather than divergence to infinity.

For a scalar stochastic differential equation, the Milstein method adds the quadratic-variation correctionUnder standard smoothness assumptions it has strong order of convergence one and weak order of convergence one.

An Ornstein-Uhlenbeck process combines mean reversion with continuing random fluctuations. In the scalar model, its stochastic differential equation iswhere $W$ is a standard Brownian motion. The drift points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of mean reversion, while $\sigma$ sets the strength of the random forcing. We consider a fixed or normal initial value independent of future Brownian increments, giving a Gaussian process and a Markov process.

For a stipulated example, use $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the expected value relaxes as $2+4e^{-t}$ and the variance grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting variance is one. The sample paths keep fluctuating even after the expected value is close to two. The Ornstein-Uhlenbeck transition distribution describes this evolving uncertainty; Ornstein-Uhlenbeck stationary initialization instead starts with that limiting normal distribution already in place.

A physical motivation is the velocity of a particle undergoing Brownian motion: linear drag removes a persistent initial velocity, while random forcing continually renews it. The original calculation gives a normal distribution for the velocity after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's mass.^[ref] A position coordinate in an overdamped harmonic optical trap gives another interpretation of the same scalar equation, with $\theta =k/\zeta$ for spring constant $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in thermal calibration of an optical trap.

The linear stochastic differential equation has the explicit solution $Y_t=e^{-\lambda t}{Y}_0+\sigma {\int }_0^t{e}^{-\lambda (t-s)}d{W}_s$. Its integrating factor is $e^{\lambda t}$. For $\lambda >0$, an independent initial normal variable of variance ${\sigma }^2/(2\lambda )$ makes its zero-mean Gaussian process stationary.

On a filtered probability space carrying a standard Brownian motion $W$ with independent increments relative to the filtration, a strictly positive security price follows geometric Brownian motion with constant drift $\mu$ and volatility $\sigma$ if it solves the stochastic differential equationHere $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and has the normal distribution $N(0,t-s)$ for $s<t$. The solution isIndeed applying the Itô formula to this exponential gives the specified drift and volatility. Consequently the price is positive and continuous, andThus disjoint log-return increments are independent and stationary. The parameter $\mu$ is the instantaneous expected relative return under the physical probability measure, not the mean log-return. Under the pricing risk-neutral measure it becomes the risk-free rate for a non-dividend-paying stock.

Separate $L$ into a diffusion generator and the constant killing term $-1/2$. The appropriate stochastic differential equation and its explicit geometric Brownian motion solution areThe second-order term in the Itô formula cancels the drift of $\log |X|$ when $x\ne 0$; if $x=0$, the solution stays zero.

Fix a terminal time $t$. Apply the Itô formula to $e^{-s/2}u(t-s,X_s)$ for $0\le s\le t$. Its drift isAfter localization this is a local martingale, and it is bounded because $u$ is bounded. It is therefore a true martingale. Taking expectations at its endpoints gives the Feynman-Kac formula

For $t>0$ and $x\ne 0$, the change of variable $y=x{e}^z$ in the normal density produces the killed geometric Brownian heat kernel with respect to Lebesgue measure:For $xy>0$, an equivalent form isIn particular,The apparent missing $1/y$ in this last expression is absorbed into the completed square, together with the killing factor; it is not a transcription error. The total mass is $e^{-t/2}$, as expected for killing at rate $1/2$. At $x=0$ the fundamental kernel is instead the measure $e^{-t/2}{\delta }_0(dy)$; it has no Radon-Nikodym derivative with respect to Lebesgue measure. Thus $u(t,0)=e^{-t/2}g(0)$. At $t=0$ the kernel is ${\delta }_x$, and continuity of bounded $g$ gives the initial condition. This also specifies the fundamental solution at the degenerate point omitted by a density-only formula.

Differentiate the material derivative of the passive scalar component by component. With ${\sigma }_{ij}={\partial }_j{u}_i$,The chain rule along a Lagrangian trajectory $\dot{X}=u(X,t)$ therefore gives the covector evolution of a material scalar gradientThe transpose matters: a gradient is transported as a covector, rather than as a material separation vector. When $u_3=0$, the third row of $\sigma$ vanishes, andLet $k_h=(k_1,k_2{)}^T$ and setFor $c>0$, the matrix exponential gives the exact horizontal solutionThe two projections are onto the eigenspaces with eigenvalues $-c$ and $+c$. If $p_{-}\ne 0$, write $p_{-}=k_0(\cos \varphi ,\sin \varphi {)}^T$ with $k_0=|p_{-}|>0$. ThenThe stated growing asymptotic requires this nonzero projection. An initial gradient entirely in the other eigenspace decays, and when $c=0$ the horizontal gradient is constant.

For constant vertical shear, write $h=({\sigma }_{13},{\sigma }_{23}{)}^T$. Integrating the third component using the exact horizontal solution yieldsConsequently, if $D=h\cdot (\cos \varphi ,\sin \varphi )$ is nonzero, $k_3\sim -k_{0}D{e}^{ct}/c$, andIf $D=0$, exponential growth of $k_3$ is absent, but the ratio still tends to zero on the growing horizontal branch. Thus exponential vertical growth is generic, not unconditional. The scalar-gradient inclination in a steady planar strain is the tangent of the angle between the scalar's normal and the horizontal plane. Equivalently, it is the cotangent of the inclination of an isoconcentration surface to that plane. Large $\alpha$ corresponds to nearly horizontal scalar surfaces; small $\alpha$ corresponds to nearly vertical ones.

For the probability calculation, take $\Gamma ,\Lambda >0$. The independent random variables $a,b$ with normal distributions of variance ${\Gamma }^2$ make $c$ distributed according to the Rayleigh distribution, withThe advertised Gaussian-over-Rayleigh inclination distribution also needs $D$ to be independent of $(a,b)$; its Gaussian marginal alone does not imply this. Under that intended assumption, for $A>0$,The cumulative distribution function is zero for $A\le 0$. Differentiating givesand the probability density function is zero for negative $A$. Its integral is one because the cumulative distribution function tends to one at infinity. Independence is a real hypothesis: $D=(\Lambda /\Gamma )a$ has exactly the stated Gaussian marginal but makes $\alpha \le \Lambda /\Gamma$ almost surely, contradicting the unbounded support of this density. A sufficient physical model is isotropic Gaussian vertical shear, independent of $a,b$, whose projection in any horizontal direction has variance ${\Lambda }^2$.

If the strain varies slowly, the horizontal gradient must align with the current growing direction, and the vertical-to-horizontal ratio must adjust before that strain changes. Both adjustment times are of order $c^{-1}$. Thus the local condition is $c{\tau }_{\sigma }\gg 1$, and the typical condition for the specified statistics isThis is an adiabatic approximation, not a uniform statement about arbitrarily weak instantaneous strains: samples with $c{\tau }_{\sigma }\lesssim 1$ retain memory of earlier strain. The instantaneous eigenvector must likewise turn slowly compared with $c$.

For rapidly fluctuating vertical shear, insert the large-time horizontal solution into its stochastic differential equation:The linear combination $W=\cos \varphi W^{(1)}+\sin \varphi W^{(2)}$ is a Brownian motion: it has independent Gaussian increments and increment variance $({\cos }^2\varphi +{\sin }^2\varphi )dt=dt$. For the signed inclination $\beta =k_3{e}^{-ct}/k_0$, the Itô product rule introduces no extra cross term because the exponential factor is deterministic. It follows thatThis is an Ornstein-Uhlenbeck process, explaining why white-noise vertical shear gives an Ornstein-Uhlenbeck inclination. Its Fokker-Planck equation isIn a stationary state the Fokker-Planck probability current $j=-cBp-(g^2/2)p^{\prime }$ is constant. A nonzero constant current cannot give an integrable, nonnegative density at both ends of the real line, so $j=0$. Integrating $p^{\prime }/p=-2cB/g^2$ and normalizing with the Gaussian integral givesThe stationary variance is $g^2/(2c)$. More explicitly the explicit Ornstein-Uhlenbeck solution has mean $\beta (0)e^{-ct}$ and variance $g^2(1-e^{-2ct})/(2c)$, confirming convergence to this density. Here $c>0$ and $g\ne 0$ are needed for the stated smooth stationary density; at $c=0$ there is no confinement, while $g=0$ gives a point mass at zero.

Set $F_t=e^{\alpha B_t-{\alpha }^{2}t/2}$, a positive stochastic exponential with $d{F}_t=\alpha F_{t}d{B}_t$. The Itô formula givesFor $Y=X/F$, the Itô product rule and $d[X,F^{-1}{]}_t=-{\alpha }^2{X}_t{F}_t^{-1}dt$ cancel both the Brownian terms and the extra drift. ConsequentlyThis is an ordinary Bernoulli differential equation with continuous random coefficients. For $\delta \ne 1$, separate variables to obtainTherefore the Bernoulli stochastic differential equation has the explicit maximal positive solutionvalid while its bracket remains strictly positive. When $\delta =1$, the reduced equation is $d{Y}_t=Y_{t}dt$, andSubstituting these expressions back through the product rule verifies the original stochastic differential equation, not just the reduced equation.

Uniqueness in law means that, for a fixed initial value, every two weak stochastic solutions of the stochastic differential equation have the same distribution as entire solution paths, even if their filtered probability spaces and driving Brownian motions differ. For an equation on $(0,\infty )$, initially consider the stopped path and lifetime at the boundary.

We first justify that the boundary cannot intervene when $\gamma \ge 2$. For any solution of the displayed Bessel equation, let ${\tau }_{\epsilon ,L}$ be its first exit from $(\epsilon ,L)$, where $0<\epsilon <r<L$. Applying Itô to $R^2$ givesThus the annulus exit has finite expectation and is almost surely finite. The one-dimensional generator is $\frac{1}{2}{\partial }_{rr}+\frac{\gamma -1}{2r}{\partial }_r$. Its harmonic scale functions are $\log r$ for $\gamma =2$ and $r^{2-\gamma }$ for $\gamma >2$, as direct differentiation verifies. They are bounded on the stopped annulus, so optional sampling givesBoth tend to zero as $\epsilon \downarrow 0$. A continuous path hitting zero before ${\tau }_L$ would hit every inner level first, so that event has probability zero. Exhausting the upper levels proves that zero is not hit at any finite time. Nor can an upper explosion occur: localization of $R^2$ gives $\mathbb{P}({\tau }_L\le t)\le (r^2+\gamma t)/L^2\to 0$. These are the Hitting-zero classification for a Bessel process arguments needed for the dimension-at-least-two case.

Now suppose $\gamma$ is an integer and take a $\gamma$-dimensional Brownian motion $U$ from any deterministic vector $u$ with $|u|=r$. For $\rho =|U|>0$, differentiation of the Euclidean norm givesDefine the radial noise $\beta ={\sum }_i\int U_i/\rho d{W}_i$, filling the integrand by a fixed unit vector on the zero set, as in part (b). It has bracket $t$ and is a Brownian motion. the Itô formula, initially before a possible zero, givesThe boundary argument just given makes this a global positive weak stochastic solution. The assumed uniqueness in law therefore identifies every solution's path law with that of $|U|$. Orthogonal invariance of Brownian motion makes this radial law independent of the selected point $u$ on the sphere. ConsequentlyThis proves equality of process laws, not merely equality of one-time marginal distributions.

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.

Apply the Itô formula separately to cosine and sine, using $[B{]}_t=t$:In vector standard form $d{Z}_t=\sigma (Z_t)d{B}_t+b(Z_t)dt$, with one-dimensional driving noise and two-dimensional state, the coefficient functions areThe original trigonometric process starts at $(1,0)$, but these coefficient functions define a stochastic differential equation on all of ${\mathbb{R}}^2$.

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.

Use the usual stochastic-integrability assumptions implicit in the price stochastic differential equation: the coefficients have progressively measurable versions, the volatility is locally square integrable, and initial prices are finite deterministic positive numbers. A no-consumption admissible trading strategy consists of predictable risky share holdings $\pi$ for which the price stochastic integral exists and the portfolio wealth of the self-financing portfolio isThe remainder $X_t-{\pi }_t^T{S}_t$ is held in the unit cash asset. Admissibility requires $X_t\ge -a$ throughout the horizon for some deterministic finite $a$; the more restrictive nonnegative-wealth convention also works. A finite-horizon arbitrage is such a strategy with $x=0$, $X_T\ge 0$ almost surely and $P(X_T>0)>0$. The lower bound is essential to exclude doubling strategies.