Stochastic differential equation ($d{X}_t=b(t,X_t)dt+\sigma (t,X_t)d{W}_t$)

A stochastic differential equation specifies infinitesimal drift and random diffusion through a stochastic integral equation.

For a zero-starting solution with $|\sigma |\le C$, the Itô isometry gives $\mathbb{E}{X}_t^2\le C^{2}t$, so $X$ is a true martingale on every finite horizon. The Itô formula represents the displayed process as $2{\int }_0^t{X}_s\sigma (X_s)d{B}_s$, whose expected squared integrand integral is at most $2C^4{t}^2$. Thus it too is a true square-integrable martingale. No continuity or Lipschitz condition on the coefficient is needed once a solution is given.

On the positive half-line, divide by the stochastic exponential $F_t=e^{\alpha B_t-{\alpha }^{2}t/2}$. The Itô product rule reduces the equation to the random Bernoulli differential equation $d(X/F)/dt=F^{\delta -1}(X/F{)}^{\delta }$. For $\delta \ne 1$ this gives $X_t=F_t[x_0^{1-\delta }+(1-\delta ){\int }_0^t{F}_s^{\delta -1}ds{]}^{1/(1-\delta )}$, until the bracket vanishes; for $\delta =1$, $X_t=x_0{F}_t{e}^t$.

The explicit Bernoulli stochastic differential equation solution remains positive and finite on every finite interval when $0<\delta \le 1$. For $\delta >1$ it diverges when the displayed integral reaches its threshold. At $\alpha =0$ this occurs deterministically. At $\alpha \ne 0$, Brownian motion has positive probability of making the integral exceed any threshold on a fixed interval, so global existence is not almost sure. The distinction between positive explosion probability and certain explosion matters: the negative linear drift of $\log F$ makes its total positive-power exponential integral finite almost surely when $\alpha \ne 0$.

The SDE $dX=-YdB-Xdt/2$, $dY=XdB-Ydt/2$ has the unique strong solution obtained by rotating $(X_0,Y_0)$ through angle $B_t$. Its coefficients are globally Lipschitz with linear growth, and $X_t^2+Y_t^2=X_0^2+Y_0^2$. The origin is absorbing and every other solution remains on its initial circle.

For smooth bounded vector fields, the support of a Stratonovich diffusion in uniform path topology is the closure of solutions to the controlled ordinary differential equation with the same drift and controls in the Cameron-Martin space of Wiener measure. The support of enhanced Brownian motion and the universal limit theorem prove this through continuity of the enhanced solution map. For an equation written in Itô form, the controlled drift is the Itô drift minus $\frac{1}{2}{\sum }_{i}D{V}_i{V}_i$.

A nonnegative diffusion with zero absorbing, often called the critical Feller branching diffusion; other conventions multiply the coefficient by a fixed positive constant. Its diffusion coefficient is not Lipschitz at zero. Independent copies add according to the addition law for square-root branching diffusions, and a cutoff construction of an absorbed square-root diffusion gives strong existence without applying the global Lipschitz theorem to the original coefficient.

Replace the square-root coefficient near zero by smooth globally Lipschitz cutoffs that vanish below half the cutoff level. With a common Brownian driver, local pathwise uniqueness makes the solutions agree until the larger cutoff level. The exit times increase as the levels decrease. The compatible integrands $1_{\{s\le T_n\}}\sqrt{Z_s}$ have increasing squares and expected time-integrals at most $zt$ by the nonnegative local martingale bound. Their limit is square-integrable on every finite horizon. The Itô isometry gives a continuous global limiting stochastic integral; at a finite limiting exit time its value is zero, and it remains absorbed afterward.

For solutions with independent Brownian drivers in a common filtration, combine the drivers with weights $\sqrt{Z/(Z+Z^{\prime })}$ and $\sqrt{Z^{\prime }/(Z+Z^{\prime })}$ where the sum is positive. At zero use one original driver. The resulting continuous local martingale has bracket $t$ and is Brownian motion by the Lévy characterization of Brownian motion. Multiplying its differential by $\sqrt{Z+Z^{\prime }}$ gives exactly the sum of the original stochastic differentials.

Two scalar SDEs driven by the same Brownian motion, with the same globally Lipschitz diffusion coefficient and ordered globally Lipschitz drifts, preserve a strict initial order at every finite time. If $Z=Y-X$ starts positive, apply a reciprocal barrier to the first hit of zero. A Gronwall bound on $(Z+ϵ{)}^{-1}$ makes the hitting probability at most $ϵe^{(L+K^2)t}/(Z_0+ϵ)$, which tends to zero. Common noise and the Lipschitz bound on the diffusion difference are essential to this argument.

Before the difference $Z$ hits zero, its diffusion coefficient has magnitude at most $KZ$ and its drift is at least $-LZ$. Itô's formula bounds the drift of $f_{ϵ}(Z)$ by $(L+K^2)f_{ϵ}(Z)$. Its stochastic integrand is bounded by $K/(4ϵ)$, making expectation legitimate on finite horizons. The Gronwall inequality and the value $f_{ϵ}(0)=1/ϵ$ then exclude finite-time contact from a strictly positive starting difference.

Globally Lipschitz drift and diffusion coefficients have linear growth and give a nonexplosive strong stochastic solution for a prescribed initial state and Brownian motion. The solution satisfies pathwise uniqueness. Local versions apply after stopping inside compact regions.

With $s(x)=1$ for $x\ge 0$ and $s(x)=-1$ for $x<0$, the equation starting at zero has uniqueness in law but not pathwise uniqueness. For existence, take a Brownian motion $W$ and set $B=\int s(W)dW$; the Lévy characterization of Brownian motion and associativity of stochastic integration give $W=\int s(W)dB$. Every solution has quadratic variation $t$ and hence the Brownian motion law. Both $W$ and $-W$ are solutions driven by the same $B$, since the time spent at zero is zero and $s(-W)=-s(W)$ away from zero. Using $s(0)=0$ instead would allow the constant solution and invalidate this uniqueness-in-law example.

An Itô diffusion solves a stochastic differential equation $d{X}_t=b(X_t)dt+\sigma (X_t)d{B}_t$ driven by Brownian motion.

Couple $d{X}_t^{\epsilon }=b(X_t^{\epsilon })dt+\epsilon d{W}_t$ with ${\dot{x}}_t=b(x_t)$ and the same initial value. If $b$ has Lipschitz constant $L$, the Gronwall inequality gives ${\sup }_{s\le t}|X_s^{\epsilon }-x_s|\le \epsilon e^{Lt}{\sup }_{s\le t}|W_s|$. Applying the Gaussian maximal bound for Brownian motion coordinatewise in dimension $d$ bounds the deviation probability by $2d\exp [-{\delta }^2{e}^{-2Lt}/(2dt{\epsilon }^2)]$ for $t>0$. Matching initial values is essential.

The Itô formula identifies the operator of an SDE with drift $b$ and noise vector fields ${\sigma }_k$ as the displayed second-order operator. With bounded measurable coefficients, every solution satisfies the true martingale identity for $C_b^2$ test functions, hence is an L-diffusion in the martingale-problem sense.

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For deterministic $X_0=x$ and bounded continuous drift and volatility, the initial right derivative of the mean is $b(x)$ and that of the variance is $\sigma (x{)}^2$. The stochastic integral has mean zero and second moment ${\int }_0^t\mathbb{E}\sigma (X_s{)}^{2}ds$; the bounded drift contributes only higher-order terms to the variance.

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The neutral one-dimensional Wright–Fisher diffusion records a proportion in $[0,1]$, with absorbing endpoints, zero drift and generator $Lf(x)=x(1-x)f^{\prime \prime }(x)/2$. Its conditional variance rate is largest near one-half and vanishes at fixation. Mutation or selection variants add drift and must be specified separately.

An urn with $j$ red balls among $n$ is sampled independently $n$ times with replacement, and the next generation's red count is the number of red draws. Its proportion is a bounded discrete-time martingale. Accelerating generation time by $n$ gives a Wright–Fisher diffusion limit: the one-step variance of the proportion is $x(1-x)/n$, and the higher moments make Taylor remainders negligible.

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 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 time spent by a Itô diffusion in a measurable set $I$ up to time $T$ is $A_T(I)={\int }_0^T{1}_I(X_t)dt$. Under stationarity its expected value is $T\mu (I)$, where $\mu$ is the Invariant distribution of an Itô diffusion. Occupation times form the denominator of a local drift estimator.

A power diffusion has a power of its positive state as noise coefficient. Here the normalized zero-drift family is considered on $U=(0,\infty )$ with $X_0>0$. Its coefficients are locally Lipschitz continuous on that domain, so a maximal local solution of a stochastic differential equation is pathwise unique before its lifetime. The Lamperti transform reduces its noise coefficient to a constant; the case $\alpha =1$ is geometric Brownian motion. This normalized family is part of the constant-elasticity-of-variance models, but is not the whole model with arbitrary drift and scale.

For the positive-domain power diffusion with $0<\alpha \le 1$, assuming the lifetime is the limit of the hitting times of $1/n$, it is finite almost surely exactly when $\alpha <1$. For $\alpha <1$, the Itô formula givesThe negative drift bounds this positive process above by $X_0^{1-\alpha }+(1-\alpha )B$, whose first hit of zero is finite by recurrence of one-dimensional Brownian motion. The lifetime must precede that hit. At $\alpha =1$, $X_t=X_0{e}^{B_t-t/2}$ is positive and finite on every compact time interval, so the lifetime is infinite. Approaching zero only as $t\to \infty$ is not a finite boundary lifetime.

On an interval where $\sigma >0$ is continuously differentiable, define $F(x)={\int }^x\sigma (y{)}^{-1}dy$. The Itô formula transforms $dX=b(X)dt+\sigma (X)dB$ intoThe transformed Itô diffusion has constant noise coefficient. For a power diffusion $\sigma (x)=x^{\alpha }$, $F(x)=x^{1-\alpha }/(1-\alpha )$ when $\alpha \ne 1$, and $F(x)=\log x$ when $\alpha =1$. This simplifies comparison with Brownian motion and exposes the boundary drift.

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

From a continuously observed Itô diffusion, a local drift estimator divides ${\int }_0^T{1}_{[x-h,x+h]}(X_t)d{X}_t$ by the corresponding diffusion occupation time. Its error separates into a local Hölder continuity bias and an Itô integral divided by occupation time. The Itô isometry gives a pointwise noise scale $(Th{)}^{-1/2}$ when the invariant density is locally positive.

Geometric Brownian motion has the explicit strong solution

The differential operator $L=\frac{1}{2}{x}^2{\partial }_{xx}+\frac{1}{2}x{\partial }_x-\frac{1}{2}$ has solution operator $P_{t}g(x)=e^{-t/2}\mathbb{E}[g(x{e}^{B_t})]$. For $x\ne 0$ its integral kernel with respect to Lebesgue measure is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t}{)}^{-1}\exp [-{\log }^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the measure $e^{-t/2}{\delta }_0$, not a density. The Itô formula applied to $e^{-s/2}u(t-s,X_s)$ proves this Feynman-Kac formula.

For $D=\sigma {\sigma }^T/2$, a density $\pi$ is invariant for $dX=bdt+\sigma dB$ exactly when the stationary Fokker-Planck equation $\nabla \cdot (b\pi -\nabla \cdot (D\pi ))=0$ holds with suitable boundary decay.

Underdamped Langevin dynamics augments position by momentum and combines Hamiltonian transport with friction and matching Gaussian noise. For potential $U$, momentum variance $\eta$, and friction $\gamma$, its invariant density is proportional to $\exp [-U(x)-|p{|}^2/(2\eta )]$.

A free particle following Langevin dynamics initialized at zero position and velocity has relaxation time $\tau =m/\zeta$ and thermal diffusivity $D=k_{B}T/\zeta$. Its position is the convolution of the random force with $[1-e^{-t/\tau }]/\zeta$. The fluctuation-dissipation relation for a Langevin particle gives force covariance amplitude $2\zeta k_{B}T$, and integrating the squared kernel yields the displayed variance. At short times it grows as $2D{t}^3/(3{\tau }^2)$ because the initial velocity is cold; at long times it grows as $2Dt$. A Maxwellian initial velocity produces a different short-time transient.

Neglecting inertia in Langevin dynamics gives the overdamped limit.

For potential $F$ and mobility $\mathcal{M}$, overdamped Langevin dynamics isThe matching noise amplitude is the fluctuation-dissipation relation that makes the Boltzmann density proportional to $e^{-F/(k_{B}T)}$ invariant.

With local thermal noise matched to spatially varying drag, the overdamped probability density obeys the displayed diffusion equation. Its equivalent Itô diffusion is $dX=D^{\prime }(X)dt+\sqrt{2D(X)}dW$, since the two terms in the Fokker-Planck equation combine to ${\partial }_x(D{P}_x)$. Thus $d⟨X⟩/dt=⟨D^{\prime }(X)⟩$, even with no applied force. The drift points toward greater diffusivity, but a uniform density has zero flux: the drift is balanced by the spatial variation of the noise. Holding the force-noise amplitude fixed instead would not produce this isothermal Fickian equation.

A periodic corrugation plus a uniform tilt. For the Adler phase equation, $V(\theta )=-\omega \theta -ϵ\cos \theta$ gives force $-V^{\prime }=\omega -ϵ\sin \theta$ and $V(\theta +2\pi )-V(\theta )=-2\pi \omega$. It has wells when $|\omega |<ϵ$ and no extrema when $|\omega |>ϵ$. It is a potential on the unwrapped phase, not a single-valued periodic potential on the circle.

For weak noise $D$ in a one-dimensional potential $\varphi$, escape from a minimum $m$ across a neighboring maximum $M$ occurs at leading rate

For $d{X}_t=a(X_t)dt+\sigma (X_t)d{W}_t$, the Euler-Maruyama method advanceswhere $\Delta W_n\sim N(0,\Delta t)$.

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.

A stochastic numerical method has strong order $p$ when, under a coupling using the same driving noise, its expected pathwise error is $O(\Delta t^p)$. Strong convergence measures approximation of individual sample paths.

A stochastic numerical method has weak order $p$ when its error in expected sufficiently smooth observables is $O(\Delta t^p)$. Weak convergence measures approximation of distributions rather than individual sample paths.

A Markov diffusion is a continuous-path Markov process whose local evolution is described by a drift vector and diffusion matrix. For sufficiently regular coefficients, its infinitesimal generator is a second-order differential operator and its conditional expectations satisfy a Kolmogorov backward equation.

A driftless square-root diffusion process is the nonnegative solution of $d{X}_t=\sqrt{X_t}d{W}_t$, with zero an absorbing boundary. For elapsed time $h>0$, starting from $x\ge 0$, its Laplace transform isTo prove this, fix the final time $h$ and set $F(t,x)=\exp [-xu/(1+u(h-t)/2)]$. Differentiation gives $F_t+\frac{1}{2}x{F}_{xx}=0$, while $F(h,x)=e^{-ux}$. By Itô formula, $F(t,X_t)$ is a local martingale; it takes values in $[0,1]$, so localization and bounded convergence make it a true martingale. Taking expectations at the two endpoints proves the formula. The Markov property gives the same formula conditionally at any starting time.

The elapsed-time $h>0$ transition distribution of a driftless square-root diffusion starting at $x$ is the compound Poisson distributionwhere all variables on the right are independent and the empty sum is zero. Here the exponential distribution parameter is its rate. The Laplace transform of this sum iswhich is the Laplace transform of the driftless square-root diffusion; uniqueness of the Laplace transform proves the distributional identity. In particular, for $0\le v<2/h$, the exponential moment isFor $x>0$ this exponential moment is infinite when $v\ge 2/h$, since the event $N=1$ has positive probability and one exponential jump already has an infinite moment there. At $x=0$ the process remains zero and every such moment is one.

A diffusion limit rescales a sequence of stochastic processes so that its limit is a diffusion process. For suitably rescaled independent small jumps, the first two jump moments determine the limiting drift and diffusion coefficient. Brownian motion arises when the limiting drift is zero and the variance grows linearly in time.

For a diffusion with generator $({\sigma }^2/2){\partial }_{xx}+b{\partial }_x$ and increasing scale function of a one-dimensional diffusion $s$, its speed density is $m=2/({\sigma }^2{s}^{\prime })$. Scale records hitting probabilities; speed records expected occupation and exit times. Multiplying scale by a positive constant divides speed by the same constant, leaving their Green-kernel product unchanged.

The expected exit time from $(\ell ,r)$ for a one-dimensional diffusion is ${\int }_{\ell }^{r}K(x,v)m(v)dv$ when finite. Differentiating this expression on each side of $v=x$ gives the equation $\mathcal{L}u=-1$ with zero endpoint values. For the SLE two-boundary-point ratio diffusion, $s(v)-s(1)$ behaves as $(v-1{)}^{1-4/\kappa }$ and the speed density behaves as $(v-1{)}^{4/\kappa }$, so the product is integrable at $1$ for $\kappa >4$.

A maximal local solution solves a stochastic differential equation until its lifetime, usually the first exit from every compact subset of the region where its coefficients are locally Lipschitz. If the lifetime is finite, the path approaches the boundary of that region or becomes unbounded.

If the drift and diffusion coefficients are locally Lipschitz on an open domain, then for every initial point there is a pathwise unique strong solution up to a maximal lifetime. That lifetime is the limit of the exit times from an increasing sequence of compact subsets of the domain.

The linear growth condition bounds the drift norm and diffusion-matrix norm by $C(1+|x|)$ for one constant $C$. Together with globally Lipschitz coefficients it gives global strong existence and pathwise uniqueness, preventing finite-time explosion. Linear coefficient maps satisfy it even when they are not bounded.

For $dX=\sigma (X)dB+b(X)dt$, $X_0=0$, and ${\sigma }^2+b^2\le A(1+X^2)$, the Doob L2 maximal inequality and the Cauchy-Schwarz inequality imply, for $t\le 1$, $F(t)\le 8A{\int }_0^t(1+F(s))ds$, where $F(t)=\mathbb{E}{\sup }_{s\le t}|X_s{|}^2$. The Gronwall inequality yields $F(t)\le e^{8At}-1$. Apply the calculation first to stopped paths; truncating locally Lipschitz functions and using this uniform bound gives exit probabilities at most $(e^{8A}-1)/n^2$, proving nonexplosion on the unit time interval.

For the one-dimensional diffusiona scale function is a strictly increasing function $s$ satisfyingThe Itô formula then makes $s(X_t)$ a local martingale before the diffusion reaches a boundary.

For independent Brownian motions $W,B$, the Itô formula cancels the drift after the arctangent transform. Normalizing the two noise coefficients gives a continuous local martingale $Z$ with quadratic variation $t$, hence another Brownian motion by the Lévy characterization of Brownian motion.

A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.

For $dX=b(X)dt+dW$ with continuous drift, integrate the positive derivative displayed above. The resulting scale function of a one-dimensional diffusion solves $g^{\prime \prime }=-2b{g}^{\prime }$. The Itô formula removes the drift of $g(X)$, giving $dY=h(Y)dW$, where $h=g^{\prime }\circ g^{-1}$ on the interval $g(\mathbb{R})$. If the drift is bounded, $h^{\prime }=-2b\circ g^{-1}$ is bounded, so $h$ is Lipschitz on that interval.

The zero extension of a scale diffusion coefficient at finite endpoints makes the transformed equation globally Lipschitz. Inverting its solution gives the original additive-noise equation. The pathwise bound $|X_t|\le |x|+{\sup }_{s\le T}|W_s|+\Vert b{\Vert }_{\infty }T$ prevents the inverse scale from reaching infinity at any finite time. This proves global strong existence and pathwise uniqueness even when the original drift is not Lipschitz.

The diffusion coefficient obtained from a scale transform for an additive-noise diffusion need only be defined on an open interval. If it is Lipschitz, its limit at a finite endpoint is zero: a positive limit would bound the derivative $1/h$ of the inverse scale and prevent the inverse from diverging. Extending it by zero beyond finite endpoints therefore preserves global Lipschitz continuity.

If $a<x<b$ and the diffusion exits $(a,b)$ almost surely, optional stopping of the bounded local martingale $s(X_{t\wedge \tau })$ gives

The equation $\frac{1}{2}{\sigma }^2{s}^{\prime \prime }+b{s}^{\prime }=0$ makes the scale transform a local martingale but does not alone prove the usual two-boundary hitting ratio. Stopped on $[a,b]$, it is bounded, so optional stopping gives that ratio when exit is almost surely finite and the endpoint scale values differ. If exit can be infinite, the terminal martingale has a further contribution on the non-exit event. For example, $\sigma =b=0$, $X_t=x\in (a,b)$ and $s(y)=y$ satisfy the differential equation but never hit either endpoint; the claimed ratio would nevertheless be positive. A constant solution for $s$ also gives an undefined denominator and is excluded by the usual strict-monotonicity convention for scale.

A weak solution may choose its probability space and driving Brownian motion as part of the solution. It is weaker than a strong solution, which must be adapted to a prescribed Brownian motion.

If $b:\mathbb{R}\to \mathbb{R}$ is bounded and measurable, then on every finite time intervalhas a weak solution and uniqueness in law. Starting with Wiener measure, the Novikov condition and Girsanov theorem add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.

A strong solution is adapted to the filtration of a prescribed driving Brownian motion on a prescribed probability space and satisfies the stochastic integral equation almost surely.

A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.

Pathwise uniqueness holds when any two solutions with the same initial value and driven by the same Brownian motion on the same filtered probability space are indistinguishable.

Uniqueness in law holds when any two weak solutions with the same initial distribution have the same distribution on path space, even though they may live on different probability spaces.

For a diffusion with infinitesimal generator $L$, the Kolmogorov backward equation is ${\partial }_{t}u=Lu$. Its solutions propagate terminal observables backward through the transition semigroup.

For a bounded smooth solution of $u_t=Lu$ with initial function $f$, apply the Itô formula to $u(t-s,X_s)$. Its drift vanishes and its boundedness upgrades the localized stochastic integral to a true martingale. Taking endpoint expectations yields $u(t,x)={\mathbb{E}}_{x}f(X_t)$ without a global bound on $u_x$.

The Feynman-Kac formula represents solutions of certain parabolic partial differential equations as conditional expectations of functionals of a diffusion. It follows by applying Itô formula to the solution along the diffusion and taking expectations.

For $\mathcal{L}=\frac{1}{2}\Delta -\lambda x\cdot \nabla$ and potential $V={\lambda }^2|x{|}^2/2$, the Feynman-Kac formula with initial value one gives $u={\mathbb{E}}_x^{OU}{e}^{{\int }_0^{t}V(X_s)ds}$. The Ornstein-Uhlenbeck likelihood relative to Wiener measure cancels the quadratic time integral, leaving a Gaussian endpoint integral. Completing the square gives the displayed answer for $1+\lambda t>0$. It is finite for all nonnegative times when $\lambda \ge 0$; when $\lambda <0$ the path expectation diverges at and beyond $t=-1/\lambda$.

For a bounded domain, a bounded classical solution with boundary data $g$, nonnegative bounded $V$, bounded $f$ and integrable diffusion exit time $\tau$ satisfiesApply the Itô product rule to the discounted stopped solution and add its source integral. The stopped martingale is dominated by $\Vert u{\Vert }_{\infty }+\Vert f{\Vert }_{\infty }\tau$, so dominated convergence permits passage to the exit. For $V=f=0$, this is a boundary-value averaging principle for functions harmonic under $L$.

For a nonexplosive diffusion process, bounded source $f$, terminal data $g$ and nonnegative bounded potential $V$, a bounded classical solution with $u(T,x)=g(x)$ isThe Itô product rule shows that the discounted solution plus its accumulated discounted source is a local martingale. Its deterministic finite-horizon bound upgrades it to a true martingale, giving the formula and uniqueness without a global derivative bound.

For a continuous Itô diffusion with diffusion generator $\mathcal{L}$, let $u$ solve $\mathcal{L}u=\lambda u$ with $\lambda >0$, be bounded on an open domain and its boundary, and equal $g$ on the boundary. Stop at the first boundary hit $\tau$. The Itô formula makes $e^{-\lambda (t\wedge \tau )}u(X_{t\wedge \tau })$ a bounded martingale. Its limit is zero on $\{\tau =\infty \}$ and equals $e^{-\lambda \tau }g(X_{\tau })$ on $\{\tau <\infty \}$. ThusBoundedness justifies passage to the terminal expectation even if hitting is not almost surely finite.

For bounded $V$ and bounded initial data $f$, a sufficiently regular classical solution bounded on every finite time slab satisfiesTo prove this, the Itô formula and Itô product rule make $u(T-t,B_t)e^{{\int }_0^{t}V(B_s)ds}$ a local martingale. Its deterministic bound on $[0,T]$ makes it a true martingale; taking expectations at the two endpoints gives the formula. This also gives uniqueness in the finite-horizon bounded class. The exponent has the same sign as the potential in the equation.

Let $\varphi$ be continuous, nonnegative, zero on $[0,\infty )$, and strictly positive on $(-\infty ,0)$. By continuity of Brownian paths, its occupation integral vanishes exactly on nonnegative paths. Dominated convergence gives the displayed limit. The Brownian reflection principle makes it $1-2{\mathbb{P}}_x(B_t\le 0)$ for $x\ge 0$ and zero for $x<0$. In particular the limit at $x=0$ is zero for every positive time. The choice $\varphi (x)=\min (1,\max (0,-x))$ is a bounded soft killing potential.

For globally Lipschitz drift $b$, couple $d{X}^{\epsilon }=b(X^{\epsilon })dt+\epsilon dW$ and $\dot{x}=b(x)$ from the same point. The Gronwall inequality gives ${\sup }_{s\le t}|X_s^{\epsilon }-x_s|\le \epsilon e^{Lt}{\sup }_{s\le t}|W_s|$. Thus convergence is uniform on each finite horizon. The Feynman-Kac formula for generator ${\epsilon }^2\Delta /2+b\cdot \nabla +c$ has weight $\exp \int c$, with a positive sign. Bounded continuous $c$ is uniformly continuous on a compact neighbourhood of the deterministic trajectory, and bounded continuous $f$ supplies a deterministic integrable bound. The dominated convergence theorem proves the displayed limit without requiring global uniform continuity of $c$.

Suppose bounded $V\ge 0$ is bounded below by a positive constant on some nonempty open interval, and $f\ge C>0$. The Feynman-Kac formula with a bounded potential gives $u(t,x)\ge C{\mathbb{E}}_x{e}^{{\int }_0^{t}V(B_s)ds}$. Infinite occupation time of one-dimensional Brownian motion makes that integral tend to infinity almost surely. The monotone convergence theorem then gives $u(t,x)\to \infty$ for every fixed $x$. Thus no globally bounded classical solution exists under these hypotheses, although boundedness on every finite time slab is compatible with the conclusion.

For a differential operator $L$, the martingale problem asks for a process $X$ such thatis a local martingale for every test function $f$ in a suitable domain.