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Let $X,Y$ be two continuous solutions on the same filtered probability space, driven by the same Brownian motion and with $X_0=Y_0$ almost surely. Write $D=X-Y$ and choose a common Lipschitz constant $L$ for $b,\sigma$. Introduce the stopping time ${\tau }_n=\inf \{t\ge 0:|X_t|+|Y_t|\ge n\}$. On $\{{\tau }_n=0\}$ the stopped difference is zero; before ${\tau }_n$, both coefficients and the difference are bounded.

The Itô formula and the zero expectation of the stopped square-integrable Itô integral yieldThe Gronwall inequality implies $\mathbb{E}{D}_{t\wedge {\tau }_n}^2=0$. Continuous solution paths are bounded on each compact time interval, so ${\tau }_n\uparrow \infty$ almost surely. Letting $n\to \infty$ proves $X_t=Y_t$ almost surely at each fixed time, and a countable dense set of times together with continuity gives indistinguishability of stochastic processes. Therefore the stochastic differential equation has pathwise uniqueness.

Let $U_t=(1,0,0)+(B_t^1,B_t^2,B_t^3)$, with three independent standard Brownian motions, and set $R_t=|U_t|$ and $M_t=R_t^{-1}$. The radial stochastic process $R$ is a three-dimensional Bessel process, which never hits zero by the Hitting-zero classification for a Bessel process. In three dimensions $u\mapsto |u{|}^{-1}$ is a harmonic function away from zero: direct differentiation gives $\Delta (|u{|}^{-1})=0$.

For the stopping times $T_n=\inf \{t:R_t\notin (1/n,n)\}$, $n\ge 2$, the Itô formula givesThe coefficients are bounded by $n^2$, so each stopped Itô integral is a true martingale on every finite horizon. The stopping times form a localizing sequence; as permitted, its divergence need not be proved here. Thus $M$ is a nonnegative continuous local martingale.

To see that $M$ is not a true martingale, integrate the three-dimensional multivariate normal density of $U_t$ in spherical coordinates. The angular integral yieldswhere $\Phi$ is the standard normal distribution function. Therefore the reciprocal of a three-dimensional Bessel process is a strict local martingale.

The limit is in probability, or more strongly in uniform convergence on compacts in probability; a pathwise limit for every continuous local martingale is not implicit. The original PDF uses $\lceil t{2}^n\rceil$, whereas the supplied TeX changes this to $\lfloor t{2}^n\rfloor$; both versions have the same limit by continuity.

Put $\delta =2^{-n}$ and $m=\lfloor t/\delta \rfloor$. For the completed intervals, the algebraic identityseparates a left-point sum from a quadratic covariation sum.

The results from stochastic calculus used are these: predictable left-step approximations to a continuous adapted integrand converge in their Itô integrals in uniform convergence on compacts in probability; and products of increments of continuous semimartingales on deterministic partitions of an interval with vanishing mesh converge to their quadratic covariation in the same sense. The first follows after localization from the Itô isometry and the Doob L2 maximal inequality, since the integrand approximations converge uniformly on compact intervals; the second is the deterministic-partition characterization of quadratic covariation.

Thus the two sums converge to ${\int }_0^t{Y}_{s}d{X}_s$ and $\frac{1}{2}⟨Y,X{⟩}_t$. For the ceiling convention, at most one additional full interval extends past $t$; on $[0,T+1]$ its absolute contribution is at most ${\sup }_{s\le T+1}|Y_s|$ times the modulus of continuity of $X$ at scale $\delta$, which tends to zero almost surely. Therefore

Interpret the Stratonovich integral in its standard extension to continuous semimartingale integrands:This extension is needed because $f^{\prime }(X)$ is generally a semimartingale rather than a local martingale. Also the introductory prose in the PDF reverses integrand and integrator relative to its displayed definition; the displayed definition fixes the convention used here.

Apply the Itô formula to $f^{\prime }(X)$, using $f\in C^3$:The continuous finite-variation process in the second term has zero quadratic covariation with $X$, while the quadratic covariation rule for an Itô integral givesA second application of the Itô formula, now to $f(X)$, therefore yieldsAll unbounded coefficients are handled by stopping $X$ on compact spatial intervals, then removing the localizing sequence.

The space-time Hermite polynomials obtained by scaling the Probabilists' Hermite polynomials are genuine polynomials in both variables, including at $t=0$:This expansion follows by differentiating the defining Gaussian exponential, and shows that $H_n(x,0)=x^n$ extends smoothly through $t=0$. Consequently the Itô formula is applicable from time zero. The given backward heat equation cancels the drift, and the derivative identity givesThe integrand is continuous and adapted, hence previsible, and stopping the Brownian motion and time makes it bounded. Thus $H_n(B_t,t)$ is a local martingale for every $n\ge 0$. The $n=0$ case is the constant $H_0=1$; in fact Gaussian moments make the displayed Itô integral square-integrable on every finite horizon, so these are true martingales as well.

The first three Probabilists' Hermite polynomials are $h_1(x)=x$, $h_2(x)=x^2-1$ and $h_3(x)=x^3-3x$. Hence

Use the time-harmonic wave convention $\psi e^{-i\omega t}$ and write $\mathbf{z}=(y,z)$, $E(0,\mathbf{z})=E_0(\mathbf{z})$, with a deterministic incident wave envelope. The printed speed ratio is inconsistent with $k=k_{0}n$: the refractive index used below is $n=c_0/c$. Substituting $\psi =e^{i{k}_{0}x}E$ in the Helmholtz equation givesThe paraxial approximation discards ${\partial }_x^{2}E$. The usual weak-fluctuation model also linearizes $n^2-1=2\mu W+O({\mu }^2{W}^2)$, givingDropping the quadratic contrast is an additional weak-fluctuation assumption, not a consequence of small propagation angles. Although the linearized random potential generates attenuation of order ${\mu }^2$, it does not retain every effect of that order in the literal finite-correlation index: the discarded quadratic contrast can also produce a mean wave phase shift. This linearization must precede a Gaussian white noise limit: the square of ideal white noise has no ordinary pointwise meaning.

The split-step Fourier method alternates free-space diffraction, $U_h=e^{hD}$, with a random phase screen,For jointly Gaussian random fields, Gaussian phase averaging gives $\mathbb{E}{e}^{i{\varphi }_h}=e^{-\mathrm{Var}({\varphi }_h)/2}$. More generally, a product of $p$ fields and $q$ conjugate fields picks up $\exp (i{\sum }_a{ϵ}_a{\varphi }_h({\mathbf{z}}_a))$, with signs ${ϵ}_a=+1$ or $-1$. Its screen average is determined entirely by the wave phase covariance matrix. The deterministic diffraction step acts on each coordinate, with opposite signs on factors formed by complex conjugation. This is the basis of the field-moment equations.

There is an important closure qualification. A stationary Gaussian random field need not have independent longitudinal increments. For finite-correlation fluctuations, the unlinearized parabolic wave equation instead givesIn the weak-fluctuation model, the exact first-moment equation isThe last term cannot in general be replaced by a constant times $m$. An exact generic solution of the linearized model iswhere $\mathcal{T}$ denotes time ordering of the propagation operators. For the unlinearized model, add $i{k}_0{\mu }^2{W}^2/2$ inside the propagation generator. The covariance function of the medium is needed to evaluate this expression; One-point Gaussian distributions alone would not even determine the joint wave phase statistics.

For example, omit diffraction and take a longitudinal autocovariance function $R(s)=\mathbb{E}[W(s,\mathbf{z})W(0,\mathbf{z})]$. Direct Gaussian phase averaging givesEven here, unit-variance stationary Gaussian random fields with different $R$ give different answers. In the general problem the diffraction and multiplication operators do not commute, so this scalar attenuation cannot simply be multiplied by $U_x$ without an additional approximation.

The standard closed answer uses the Markov approximation for a random medium, made explicit in part ii. Let the longitudinally integrated covariance kernel beReplace the medium by longitudinal Gaussian white noise with this strength. A screen of thickness $h$ then has $\mathbb{E}[{\varphi }_h({\mathbf{z}}_a){\varphi }_h({\mathbf{z}}_b)]=k_0^2{\mu }^{2}hB({\mathbf{z}}_a-{\mathbf{z}}_b)$ and is independent of the incoming field. For the moment $M=\mathbb{E}{\prod }_a{E}_a^{({ϵ}_a)}$, expanding both steps to order $h$ givesIn particular, the coherent attenuation in a white-noise random medium and its Fresnel propagator solution areEquivalently, write $Wdx=d{\beta }_x$, where ${\beta }_x(\mathbf{z})$ is Brownian motion in $x$ with transverse covariance kernel $\mathbb{E}[d{\beta }_x(\mathbf{z})d{\beta }_x({\mathbf{z}}^{\prime })]=B(\mathbf{z}-{\mathbf{z}}^{\prime })dx$. The Stratonovich integral formulation is $dE=DEdx+i{k}_0\mu E\circ d{\beta }_x$. Its Itô integral form isThe mean of the Itô integral vanishes, independently confirming the attenuation drift. For $x>0$ in two transverse dimensions,Thus a unit plane wave has $m=e^{-\gamma x}$. For a general incident wave envelope, the Fresnel propagator supplies its spreading. The attenuation is redistribution between the coherent and diffuse wave fields, rather than wave absorption: for a field and its conjugate at the same point, the screen contribution in the second-moment equation cancels.

A process $H$ is progressively measurable when, for every $T$, its restriction to $[0,T]\times \Omega$ is measurable for $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$. Continuous adapted processes are progressively measurable, and a Borel function of such a process remains so. A progressively measurable square-integrable Brownian integrand has an equivalent predictable integrand for the Itô integral, up to equality for $dtd\mathbb{P}$.

The spot variance is the instantaneous squared noise coefficient $\sigma (t{)}^2$ of an Itô integral. It can be estimated by averaging the weighted realized variance over a shrinking time window. A Hölder continuity assumption bounds the smoothing bias.

For a zero-starting L2-bounded continuous martingale $M$, the space $L^2(M)$ consists of predictable $H$ with $\mathbb{E}{\int }_0^{\infty }{H}_s^{2}d[M{]}_s<\infty$, identifying processes equal almost everywhere for the quadratic-variation measure. The finite-horizon version stops the integral at $T$. The Itô integral is the isometric extension from elementary predictable integrands.

// Target: probability-and-statistics.bigb

A theorem interchanging a parameter integral with an Itô integral, under appropriate measurability and integrability hypotheses. For a bounded deterministic integrand on a finite parameter-time rectangle, the square-integrability conditions hold; triangular domains can be handled by an indicator of the domain.

For continuous semimartingales, the Stratonovich integral is defined by ${\int }_0^t{Y}_s\circ d{X}_s={\int }_0^t{Y}_{s}d{X}_s+\frac{1}{2}[Y,X{]}_t$, where the first term is an Itô integral and the correction is quadratic covariation. It obeys the ordinary chain rule and is the limit in probability of symmetric endpoint sums.

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.