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For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up probability is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm \sigma \sqrt{h}$; expanding $q_h$ gives mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[ref]

The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price drift loses the $\rho$ term, as required for the discounted price to be a martingale. Convergence of claim prices also needs the payoff functional to be continuous at almost every limiting path, together with Uniform integrability when payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.

A birth-death process with unit upward and downward jumps and rates $b(x)$ and $d(x)$ has infinitesimal generatorIf $f$ varies slowly over one jump, Taylor expansion through second order gives $Lf\simeq (b-d)f^{\prime }+(b+d)f^{\prime \prime }/2$. Thus its diffusion approximation has drift coefficient $u=b-d$ and, in the conventiondiffusion coefficient $D=(b+d)/2$. Equivalently it solves $d{X}_t=u(X_t)dt+\sqrt{b(X_t)+d(X_t)}d{B}_t$, where $B$ is Brownian motion. The sum measures the variance of the short-time increment: births and deaths both add noise even when their mean drifts cancel.

For a population density $x=n/N$ with jump size $1/N$ and birth/death rates $N\beta (x),N\delta (x)$, the drift is $\beta -\delta$ and the diffusion coefficient is $(\beta +\delta )/(2N)$. Fluctuations therefore shrink as $N^{-1/2}$. A useful approximation requires many reaction events while the rates change little; it can fail near an absorbing state or when only a few individuals remain. A continuous diffusion can even visit negative values unless its boundary treatment is specified. The approximation replaces the discrete jump model rather than asserting equality of their laws.^[ref]

For jumps $r$ from state $x$ at nonnegative rates $W(x,r)$, the Fokker-Planck diffusion matrix in the infinitesimal-covariance convention isAssume the second jump moments are finite. Over a short interval $dt$, the conditional covariance of the state increment is $B(x)dt+o(dt)$. Subtracting the product of the mean increments changes only order-$d{t}^2$ terms. The matrix is symmetric and positive semidefinite because $v^{\mathsf{T}}Bv={\sum }_{r}W(x,r)(v\cdot r{)}^2\ge 0$ for every real vector $v$.

With this definition, its term in the Fokker-Planck equation is $\frac{1}{2}{\sum }_{i,j}{\partial }_i{\partial }_j(B_{ij}p)$. It is therefore $B/2$, not $B$, that corresponds to a scalar diffusion coefficient multiplying ${\partial }_x^{2}p$ in one dimension. Unit births and deaths at rates $b,d$ give drift $b-d$ and increment variance $B=b+d$: opposite mean displacements cancel in the drift but both contribute positively to the variance. The chemical diffusion matrix applies the same outer-product construction to reaction channels.

The drift vector is the expected infinitesimal displacement per unit time. For jumps $r$ with rates $W(x,r)$, it is $A_i(x)={\sum }_r{r}_{i}W(x,r)$. It is the first jump moment entering the Fokker-Planck equation.

A Fokker-Planck equation describes how a probability density evolves under drift and diffusion. For an Itô diffusion $d{X}_t=b(X_t,t)dt+\sigma (X_t,t)d{W}_t$, put $a=\sigma {\sigma }^{\mathsf{T}}$. When a density exists and the required regularity holds, its equation isThe first term transports probability with the drift; the second spreads it through the local increment covariance. It is the forward equation adjoint to the backward infinitesimal generator $Lf=b\cdot \nabla f+\frac{1}{2}{\sum }_{i,j}{a}_{ij}{\partial }_i{\partial }_{j}f$.

Writing $J_i=b_{i}p-\frac{1}{2}{\sum }_j{\partial }_j(a_{ij}p)$ gives the continuity equation ${\partial }_{t}p+\nabla \cdot J=0$. Boundary conditions specify whether probability is reflected, absorbed or exchanged with the exterior. An invariant density must satisfy both the stationary equation and the boundary conditions. A zero divergence of current need not mean zero current.

For constant drift zero and $a=2DI$, the equation reduces to the heat equation ${\partial }_{t}p=D\Delta p$; a point source spreads into a Normal distribution with covariance $2DtI$. For one-dimensional drift $b(x)=-\gamma x$ with $\gamma >0$ and constant $a=2D>0$, the stationary density is normal with variance $D/\gamma$. Some texts call $a$ the diffusion matrix, while others use $a/2$; stating the displayed convention avoids a factor-of-two ambiguity.

The Brownian transition semigroup identity proves that this positive density process has mean one for $t<1$. The backward heat equation and Itô's formula give $d{M}_t=M_t(y-B_t)(1-t{)}^{-1}d{B}_t$. On every fixed horizon $T<1$, Girsanov theorem under $dQ=M_{T}dP$ makes $B_t-{\int }_0^t(y-B_s)(1-s{)}^{-1}ds$ Brownian motion. Thus the original coordinate acquires the bridge drift coefficient toward $y$ at time one, without defining an equivalent endpoint density at the singular time.

For fixed independent and identically distributed arrivals of mean $\mu$ and positive finite variance ${\sigma }^2$, put service at $\mu +\epsilon d$. Rescaling net input by space $\epsilon$ and time ${\epsilon }^{-2}$ gives Brownian motion with drift $-d$ by the functional central limit theorem. The continuous Skorokhod reflection map on the half-line gives the displayed transient limit, with a correspondingly scaled initial workload. A stationary approximation additionally requires justification of limit interchange. When valid, the stationary law of negatively drifted reflected Brownian motion predicts $\mathbb{P}(Q_{\epsilon }>b/\epsilon )\approx e^{-2db/{\sigma }^2}$.

For a finite-valued subordinator $T$ starting at zero, the Laplace exponent of a subordinator is the function $\Phi$ determined byIts Lévy–Khintchine formula takes the formwhere $d$ is the deterministic drift coefficient and $\nu$ is the Lévy measure. This representation separates steady motion of the clock from its positive jumps.^[ref]

For a deterministic clock $T_t=dt$, $\Phi (\lambda )=d\lambda$. For a Poisson process of rate $a$ with unit jumps, $\Phi (\lambda )=a(1-e^{-\lambda })$. For a sum of independent subordinators, the Laplace exponent of a subordinator is the sum of the individual functions, because the Laplace transforms multiply. The transform at $\lambda =0$ gives $\Phi (0)=0$.

The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and SDE noise amplitude $\sqrt{2}$, corresponding to diffusion coefficient one in the displayed Fokker-Planck equation. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

Use the intrinsic definition of a Lévy process: $X_0=0$ almost surely, its increments over disjoint time intervals are independent, their probability distributions depend only on interval length, and $X$ has stochastic continuity. In symbols,with times restricted to the half-line. Such a stochastic process has a càdlàg modification, and is usually represented by that version. Requiring càdlàg paths in the definition is a common equivalent convention at the level of modifications; it is important to distinguish this from a claim about the paths of an arbitrary supplied version.

For the characteristic function, writeHere $\psi (0)=0$ and $\psi$ is the positive-time-sign characteristic exponent of a Lévy process. The independent increments and stationary increments give ${\varphi }_{s+t}(\theta )={\varphi }_s(\theta ){\varphi }_t(\theta )$; stochastic continuity gives continuity in time and ${\varphi }_0=1$. This continuous multiplicative semigroup has the stated exponential form. Its exponent has the Lévy–Khintchine formulawhere $b\in \mathbb{R}$, ${\sigma }^2\ge 0$, and the Lévy measure $\nu$ satisfies $\int (1\wedge y^2)\nu (dy)<\infty$. The truncation convention fixes the drift coefficient $b$; the displayed sign convention agrees with $\Psi =-\psi$.

The printed assertion is false as a supremum bound uniform over shrinking bandwidths. The supplied hint proves a pointwise stochastic order with constants uniform in $x$. Taking the supremum inside the probability changes the problem: a growing number of spatial windows produces a logarithmic cost. We first establish the valid consequence of the hint, then give a counterexample satisfying the printed assumptions.

For $I=I_{x,h}=[x-h,x+h]$, let $p=\mu (I)$, $A_T={\int }_0^T{1}_I(X_t)dt$ and $N_T={\int }_0^T{1}_I(X_t)\sigma (X_t)d{W}_t$. The Invariant distribution of an Itô diffusion has a probability density function bounded above and bounded away from zero on the required compact interval. Choosing $h_0<M/4$ therefore gives constants $c_0,C_0>0$ such that $c_{0}h\le p\le C_{0}h$, uniformly for $|x|\le M/2$ and $0<h<h_0$.

For the centered function $f=1_I-p$, $\int |f|d\mu =2p(1-p)\le 2p$. Outside $[-M,M]$, $f=-p$. The given second-moment estimate and the Chebyshev inequality yieldOn the event $A_T\ge Tp/2$, the drift coefficient contributes an average whose error is at most $R{h}^{\alpha }$, by Hölder continuity. The Itô isometry and stationarity give $\mathbb{E}{N}_T^2\le \Vert \sigma {\Vert }_{\infty }^{2}Tp$. Another use of the Chebyshev inequality provesThis also accounts for the zero-denominator convention through the first exceptional event. The bound proves pointwise $R{h}^{\alpha }+O_{\mathbb{P}}((Th{)}^{-1/2})$, uniformly in the location's tail probabilities. It supplies no bound for the probability of a supremum over all locations.

For an explicit counterexample, take $\sigma \equiv 1$ and the bounded globally Lipschitz functionIt has Lipschitz constant $R=1$, exponent $\alpha =1$, and satisfies the inward-drift conditions with $M=2$, $\gamma =2$. The stochastic differential equation has a unique strong solution of a stochastic differential equation. Its Invariant distribution of an Itô diffusion has densityThe exponent is zero on $[-1,1]$, equals $-(|x|-1{)}^2$ when $1<|x|<2$, and equals $3-2|x|$ outside $[-2,2]$. This verifies integrability, positivity and a constant density ${\rho }_0=Z^{-1}$ on the central interval. Start the Itô diffusion in this Invariant distribution of an Itô diffusion to obtain the required stationary process.

Set $h=T^{-1/3}$ and choose $N=\lfloor (4h{)}^{-1}\rfloor$ windows with centers $x_j=-1/2+(4j-2)h$, $1\le j\le N$. Their intervals $I_j=[x_j-h,x_j+h]$ are disjoint and lie in $[-1,1]$, where the drift is zero. WriteThen ${\hat{b}}_T(x_j,h)=M_j(T)/A_j(T)$ whenever $A_j(T)>0$. The same occupation bound as above gives $\mathbb{E}(A_j(T)-q{)}^2\le C_{1}T{h}^2$ for $T\ge 1$. With $\delta =T^{-1/6}$, a union bound and the Chebyshev inequality implyThe quadratic variations are $⟨M_j⟩=A_j$, and the cross quadratic covariations vanish since the windows are disjoint. Each clock tends to infinity almost surely: for any fixed window, the occupation estimate at times $2^k$, followed by the Borel-Cantelli lemmas, gives $A_j(2^k)/(2^k\mu (I_j))\to 1$. The Knight theorem for orthogonal martingales therefore represents $M_j(t)=B_j(A_j(t))$ using independent standard Brownian motions $B_1,\dots ,B_N$. This theorem is applied to the finite collection for each $T$; no growing-dimensional central limit theorem is being assumed.

On the preceding clock event, compare each time-changed Brownian motion with its value at deterministic time $q$. The Brownian reflection principle and a union bound show that, for every fixed $\epsilon >0$,This bound does not require the Brownian motions to be independent of their clocks. In particular, ${\max }_j|M_j(T)-B_j(q)|/\sqrt{q}\to 0$ in probability. The variables $B_j(q)/\sqrt{q}$ are independent $N(0,1)$ random variables. Their Gaussian maximum exceeds $\sqrt{\log N}$ with probability tending to one: for $r\ge 1$, integration of the normal density over $[r,r+1/r]$ gives $\mathbb{P}(|Z|>r)\ge c{r}^{-1}{e}^{-r^2/2}$, and henceSince $A_j(T)\le (1+\delta )q$ on the clock event, it follows that, for some $c_3>0$,But $\sqrt{Th}Rh=1$ for this bandwidth. Thus the error after subtraction of $Rh$ and multiplication by $\sqrt{Th}$ is not bounded in probability, contradicting the printed stochastic order. The valid pointwise bound above is provable; the uniform shrinking-bandwidth assertion is not. This counterexample shows why a uniform rate needs at least a $\sqrt{\log (1/h)}$ enlargement of the noise scale. If $h$ is instead held fixed and constants may depend on $h$, this counterexample does not contradict that different interpretation.

The drift coefficient $b(z)=rz$ and diffusion coefficient $a(z)=\sigma z$ are Lipschitz continuity and have linear growth. The standard existence and pathwise uniqueness theorem for stochastic differential equations therefore supplies a unique global strong solution of a stochastic differential equation, adapted to the driving Brownian filtration.

Apply Itô's formula to $x\exp ((r-{\sigma }^2/2)t+\sigma B_t)$. Its time derivative contributes $(r-{\sigma }^2/2)X_t$ and its Itô correction contributes ${\sigma }^2{X}_t/2$, giving the required equation and initial value. HenceIn particular the solution stays positive. For $t>s$, factor out $X_s$ and use independence and the normal law of $B_t-B_s$:The exponential moment exists, so this conditional expectation is well-defined for every finite $s,t$.

Itô's formula identifies the infinitesimal generator on twice continuously differentiable functions:Subtract the drift coefficient integral from that formula to obtainStopping when $|X|$ leaves a bounded interval makes the integrand bounded on each finite time interval. Therefore $M^f$ is a continuous local martingale. In fact $f^{\prime }\in L^{\infty }$ and the lognormal distribution solution has finite second moments on finite intervals, so here it is square integrable on each such interval as well.

The quadratic covariation of two stochastic integrals against the same Brownian motion is the integral of the product of their integrands. ConsequentlyThis is the quadratic covariation, which agrees with the pathwise quadratic covariation for these continuous local martingales.

A finite-horizon form of Girsanov theorem is as follows. If $\theta$ is predictable, ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely, and its stochastic exponential $M$ is a true martingale of mean one, then under the probability $d\mathbb{Q}=M_{T}d\mathbb{P}$ the processis Brownian motion. The same conclusion holds on the completed natural filtration, since a strictly positive density gives equivalent probabilities.

Part (b) establishes precisely the required density martingale and integrand ${\theta }_s={\partial }_x\log p(1-s,B_s,y)$. ThereforeIn particular it is the requested local martingale. Equivalently, under $\mathbb{Q}$, $B$ solves $d{B}_t=(y-B_t)(1-t{)}^{-1}dt+d{X}_t$, the Brownian bridge drift coefficient toward $y$ at time one. Keeping $T<1$ avoids the singular endpoint in this density change.

Write the diffusion and drift coefficients asBoth are continuously differentiable, withThese derivatives are bounded. For example $2|x|\le 1+x^2$ bounds both absolute values by one. The mean value theorem therefore makes both coefficients globally Lipschitz. They are also bounded, so satisfy the linear growth condition for an SDE.

The global existence theorem for stochastic differential equations with Lipschitz coefficients consequently givesIn particular, there is no finite explosion time. The solution has continuous paths, so all subsequent level-hitting and stopping arguments apply.

Let $X$ be a Bessel process of dimension $\delta$, started at $x>0$, and let $\alpha >0$. Work before the first hit of zero. The Itô formula givesUse the strictly increasing clock $A_t={\alpha }^2{\int }_0^t{X}_s^{2\alpha -2}ds$ and its inverse $\tau$. Then $Y_u=X_{{\tau }_u}^{\alpha }$ has unit noise coefficient and drift coefficient $(\delta +\alpha -2)/(2\alpha Y_u)$. It is therefore a Bessel process of dimension ${\delta }^{\prime }=2+(\delta -2)/\alpha$ on this changed stochastic interval. If ${\delta }^{\prime }$ is negative, the assertion means the positive diffusion killed at zero. No continuation through zero follows from this computation, where the clock or the power derivative can become singular.

For linear drift $A(x)=ax$ and constant symmetric positive-definite diffusion matrix $b$, a zero-current stationary Gaussian $P(x)\propto \exp (-x^{T}Dx/2)$ obeysThis reversible solution exists when $D=-2b^{-1}a$ is symmetric positive definite. Its covariance $C=D^{-1}$ satisfies the Continuous Lyapunov equation

A stochastic differential equation specifies a stochastic process through a combination of ordinary drift and random increments. In the scalar Itô integral convention, the notationmeans the stochastic integral equation $X_t=X_0+{\int }_0^{t}b(s,X_s)ds+{\int }_0^t\sigma (s,X_s)d{W}_s$, with the required measurability and local integrability. The driver $W$ is a Brownian motion; $b$ is the drift coefficient and $\sigma$ the diffusion amplitude.

For example, the Ornstein-Uhlenbeck process satisfies $d{X}_t=-\lambda X_{t}dt+\sigma d{W}_t$: linear restoring drift competes with persistent noise. A strong solution of a stochastic differential equation uses a prescribed driver, whereas a weak solution of a stochastic differential equation may construct the probability space and driver. Pathwise uniqueness, uniqueness in law and the lifetime of a maximal local solution of a stochastic differential equation are separate questions. If the equation uses a Stratonovich integral, its convention changes the drift conversion and must be stated.