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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 a diffusion process with diffusion generator $L$, the Itô formula makes $\phi (X_t)-\phi (X_0)-{\int }_0^{t}L\phi (X_s)ds$ a local martingale. Under stopping and integrability conditions making it a true martingale, taking expectations gives the displayed identity. For Brownian motion, $L=\frac{1}{2}\Delta$, recovering the usual Brownian formula.

For $Y_t=y+B_t+at$, $y>0$, and $S=\inf \{t\ge 0:Y_t=0\}$,where $e^{-\lambda \infty }=0$. The diffusion generator is $\frac{1}{2}{\partial }_{yy}+a{\partial }_y$. Its bounded eigenfunction on the positive half-line with boundary value one is $u(y)=\exp [-y(a+\sqrt{a^2+2\lambda })]$, giving the formula through the discounted boundary-hitting representation. The zero-discount limit is the hitting probability $e^{-2y\max (a,0)}$.

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.

Write $X_t=x+W_t+\mu t$ with $W$ a zero-starting Brownian motion. A unit-time increment exceeds $2b$ with probabilityIf the process has not exited before an integer time $k$, it is in $(-b,b)$; an increment exceeding $2b$ forces it outside at time $k+1$ and hence forces an exit by then. Independence of Brownian increments gives $\mathbb{P}(T>k)\le (1-p{)}^k$. Thus the geometric tail bound from a uniform escape probability yieldsIn particular $T$ is finite almost surely.

The diffusion generator is $\mathcal{L}=\frac{1}{2}{\partial }_{xx}+\mu {\partial }_x$. The function $f(z)=e^{-2\mu z}$ satisfies $\mathcal{L}f=0$, so the Itô formula gives $df(X_t)=-2\mu f(X_t)d{W}_t$. It is a true martingale, since it is $e^{-2\mu x}$ times the Gaussian exponential martingale $e^{-2\mu W_t-2{\mu }^{2}t}$. The stopped values $f(X_{t\wedge T})$ are uniformly bounded because the stopped path stays in $[-b,b]$. Optional sampling and bounded convergence givewhere continuity ensures that $X_T$ is exactly one of the two endpoints. Solving,This drifted Brownian interval-exit probability tends to $(b-x)/(2b)$ as $\mu \to 0$, as expected.

The differential operator without its potential is the diffusion generator of the d-dimensional Ornstein-Uhlenbeck processIts explicit solution $X_s=e^{-\lambda s}x+{\int }_0^s{e}^{-\lambda (s-r)}d{B}_r$ defines its path law ${\mathbb{P}}^x$ on $C([0,t];{\mathbb{R}}^d)$. For the potential $V(x)={\lambda }^2|x{|}^2/2$ and initial function one, the Feynman-Kac formula gives the nonnegative path integralThe sign in the exponential is positive, because the PDE contains $+Vu$. One can first stop the diffusion in bounded balls, where the usual localized Feynman-Kac argument applies, and then examine finiteness of the resulting expectation. The critical quadratic potential for the Ornstein-Uhlenbeck generator below supplies that finiteness check.

Write the diffusion process as $d{X}_t=b(X_t)dt+\sigma (X_t)d{B}_t$, with $a=\sigma {\sigma }^{\mathsf{T}}$. Its diffusion generator isThe Itô formula says that $\phi (X_t)-\phi (X_0)-{\int }_0^{t}L\phi (X_s)ds$ is a local martingale, and is a true martingale when the stopped terms are integrable. Taking expectations gives Dynkin formula for a diffusion. This is the bridge from infinitesimal stochastic dynamics to differential equations.

For the transition semigroup $P_{t}g(x)={\mathbb{E}}_{x}g(X_t)$, the Markov property gives $P_{t+s}=P_t{P}_s$. On a smooth generator domain where differentiation is valid, differentiating the semigroup identity yields the Kolmogorov backward equation ${\partial }_t{P}_{t}g=L(P_{t}g)$, with initial value $g$. If a transition density $p(t,x,y)$ exists, integration against smooth compactly supported tests and integration by parts give the Fokker-Planck equationThe backward equation acts on the starting point, while this forward equation describes the evolving probability density.

Here is a full representation proof with a source and potential. Suppose $X$ is nonexplosive, $V\ge 0$ is bounded, $f,g$ are bounded, and $u$ is a bounded classical solution on a finite slab ofFor a process started from $x$ at time $t$, put $D_s=\exp (-{\int }_t^{s}V(X_r)dr)$. The Itô product rule givesTherefore $D_{s}u(s,X_s)+{\int }_t^s{D}_{r}f(r,X_r)dr$ is a local martingale. It is bounded in absolute value by $\Vert u{\Vert }_{\infty }+(T-t)\Vert f{\Vert }_{\infty }$, so it is a true martingale, without requiring a global derivative bound. Its endpoint expectations give the parabolic Feynman-Kac formula with a sourceThis proves uniqueness in the bounded classical class. Setting $V=f=0$ recovers the backward heat or diffusion representation; the minus sign in the killing term produces the negative exponential.

For an elliptic problem, stop at the first exit $\tau$ from a bounded regular domain $D$. Suppose ${\mathbb{E}}_x\tau <\infty$, $u\in C^2(D)\cap C(\bar{D})$ is bounded and solves $(L-V)u=-f$ with boundary data $g$. The same calculation, stopped at $t\wedge \tau$, gives a martingale dominated by $\Vert u{\Vert }_{\infty }+\Vert f{\Vert }_{\infty }\tau$. Continuity puts $X_{\tau }$ on the boundary. Taking the limit and using dominated convergence gives the elliptic Feynman-Kac formulaFor $V=f=0$, harmonic functions for $L$ give stopped martingales and are averages of their exit-boundary values. For Brownian motion $L=\frac{1}{2}\Delta$, this is the probabilistic solution of the Dirichlet problem. Suitable smooth coefficients and uniform ellipticity give familiar settings where the required diffusion, exit-time integrability and classical solutions exist. The representation itself follows from the stated regularity and integrability conditions, rather than claiming classical solvability for arbitrary coefficients.

The diffusion generator is $\mathcal{L}f=\frac{1}{2}{\sigma }^2{f}^{\prime \prime }+b{f}^{\prime }$. On a compact subinterval of $I$, the coefficient $s^{\prime }\sigma$ is bounded, and the Itô formula up to its exit time $\rho$ givesbecause $\mathcal{L}s=0$. Increasing compact subintervals proves the local-martingale property up to $T$. The notation $s(X_{t\wedge T})$ at an exit endpoint is understood with its continuous endpoint extension when needed; stopping on $[a,b]\subset I$ below is always well-defined. This is the martingale property of a scale function of a one-dimensional diffusion.

For the hitting ratio there are two necessary qualifications: $s(b)\ne s(a)$ and $\tau =T_a\wedge T_b<\infty$ almost surely. A usual scale function is strictly monotone, which ensures the first condition, but the printed hypotheses alone ensure neither. Under these conditions, $s(X_{t\wedge \tau })$ is a bounded local martingale and hence a uniformly integrable martingale. The optional stopping theorem givesContinuity prevents simultaneous hitting of distinct endpoints, and almost sure exit makes these two probabilities sum to one. Solving yields the boundary hitting probability from a diffusion scale function

The hypotheses for a diffusion scale hitting formula cannot be omitted here. Take $\sigma \equiv 0$, drift $b\equiv 0$, $s(y)=y$ and $X_t\equiv x\in (a,b)$. All printed coefficient and differential conditions hold, but neither endpoint is ever reached, so the left side is zero whereas the displayed ratio is strictly positive. A constant $s$ is another allowed solution of the printed differential equation and makes its denominator zero. In general the bounded stopped martingale has a limit $Y_{\infty }$, and optional stopping instead givesThe following parts have nondegenerate diffusion and strictly monotone powers, and their finite-exit properties are justified explicitly.

Let $a(x)=\sigma (x)\sigma (x{)}^{\mathsf{T}}$. The multidimensional Itô formula gives the second-order diffusion generatorIndeed,The integrand is locally bounded along the continuous path after stopping on compact sets, because the coefficients and derivatives are continuous. Thus the final stochastic integral is a local martingale. No global growth or uniqueness assumption on this already-given weak solution of a stochastic differential equation is needed.

For the drifted Brownian motion $Y_t=y+B_t+at$ on $(0,\infty )$, the diffusion generator is $\mathcal{L}=\frac{1}{2}\frac{d^2}{d{z}^2}+a\frac{d}{dz}$. Seek a bounded solution of $\mathcal{L}u=\lambda u$ with $u(0)=1$. The exponential ansatz $u(z)=e^{rz}$ givesBecause $\lambda >0$, the plus root is positive and the minus root is negative. Boundedness on $[0,\infty )$ therefore selectsIt satisfies the boundary condition and all hypotheses of part (d). HenceThis is the first-passage Laplace transform for Brownian motion with drift, with $e^{-\lambda \infty }=0$. At $a=0$ it reduces to $e^{-y\sqrt{2\lambda }}$. As $\lambda \downarrow 0$, it gives $\mathbb{P}(S<\infty )=e^{-2y\max (a,0)}$, agreeing with certainty of hitting when the drift points towards zero and a possible escape when it points away.

The joint generator. With $\theta$ again measured in dollars, portfolio wealth obeysIts noise and the factor noise are driven by the same Brownian motion. Their quadratic covariation is $\sigma (x)\theta \alpha (x)dt$, so the diffusion generator has a cross derivative. Dynamic programming and the Itô formula giveAll coefficient functions in this formula are evaluated at $x$. The cross derivative is essential: it gives intertemporal hedging demand. Normalizing CRRA utility as $U(c)=c^p/p$, where $p=1-R$, the two optimizations give, at nonzero volatility,and henceHere $p(-1/R)=1-1/R$.

Homogeneity and the reduced equation. Scaling initial portfolio wealth and both controls preserves the portfolio wealth constraint and multiplies the objective by the positive number $b^p$. ThusThis expression is valid for both signs of $p$: $V$ is negative when $R>1$, but its portfolio wealth derivative is positive. Its derivatives areSubstitution yieldsThe resulting feedback isThe first portfolio term is myopic, and the second is intertemporal hedging demand.

Constant market price of risk. If $\mu -r=\sigma \kappa$ and volatility is nonzero, the portfolio term becomes $p(\kappa f+\alpha f^{\prime }{)}^2/(2Rf)$, so the magnitude of stock volatility disappears. Applying the power transformation of a complete-market investment equation $f=g^R$ cancels the squared-gradient terms and gives the further reductionFor ${\gamma }_M>0$ its economic solution is $g=1/{\gamma }_M$. ThusTo see why this solves the investment-consumption problem, optimize directly over Brownian portfolio exposures, writing $y=\sigma (x)\theta$. The portfolio wealth equation becomes $dw=ydW+(rw+\kappa y-c)dt$, which no longer contains $X$. With nonzero volatility the same admissible exposure processes are available for every factor state, so the attainable wealth-consumption pairs, and therefore the value, are exactly those of the Merton consumption-investment problem. This also excludes extraneous solutions of the linear equation without imposing artificial factor boundary data.

The printed boundedness assumptions do not ensure nonzero volatility or a finite value. The unsimplified HJB equation remains the correct control equation at a zero of $\sigma$. There the hedge term vanishes; if $\mu \ne r$, the riskless excess return gives arbitrage with unrestricted holdings. Under $\mu -r=\sigma \kappa$, a zero-volatility state offers only the bank exposure at that instant. For example $\sigma \equiv 0$, $\mu \equiv r$ satisfies this relation for any chosen $\kappa$, but its value uses ${\gamma }_0=[\rho -pr]/R$, not a fictitious nonzero risk premium. The constant-value formula using $\kappa$ presupposes access to the Brownian exposure, with sufficient integrability for the corresponding holdings. Additive utility constants again only shift $V$ by a constant divided by $\rho$.

Define the diffusion generator $L=b{\partial }_x+{\sigma }^2{\partial }_{xx}/2$. Fix a horizon $t$ and start the strong solution from the deterministic state $x$. The Itô formula applied to the time-reversed test function givesThe drift vanishes by the Kolmogorov backward equation. This is initially a local martingale; localization on compact state/time sets justifies the stochastic integral without a global derivative bound. Since $u$ itself is bounded, this local martingale is a true martingale on $[0,t]$. Its two endpoint expectations giveThis is the bounded backward-equation stochastic representation. Starting the strong solution at deterministic $x$ is the precise meaning of the conditional notation at $X_0=x$. Also $f=u(0,\cdot )$ is bounded, even though boundedness was not separately imposed on $f$.

For the stochastic differential equation $d{X}_s=b(s,X_s)ds+\Sigma (s,X_s)d{B}_s$, the diffusion generator on smooth functions isOne form of the Feynman-Kac formula is the following. Suppose the stochastic differential equation is well posed and nonexplosive, $c\ge 0$ is bounded, the terminal data $g$ and source $h$ are bounded, and $F$ is a classical solution bounded on finite time intervals ofWith the regularity needed for the Itô formula, the parabolic Feynman-Kac formula with a source statesHere ${\mathbb{E}}_{t,x}$ means expectation for the diffusion process started at $X_t=x$. Under hypotheses ensuring existence of a classical solution, this representation also identifies that solution uniquely in the stated bounded class. In the remaining parts, $c=h=0$, so $F(t,x)={\mathbb{E}}_{t,x}g(X_T)$ solves the terminal-value Kolmogorov backward equation $F_t+{\mathcal{L}}_{t}F=0$. Complex terminal data are allowed by applying the real formula to their real part and imaginary part separately.

The diffusion generator is $\mathcal{L}=\mu {\partial }_x+\frac{1}{2}{\sigma }^2{\partial }_{xx}$. By the Feynman-Kac formula, the required conditional characteristic function solvesUse separation of variables in the form $F(t,x)=e^{iux}h(t)$. Since $F_x=iuF$ and $F_{xx}=-u^{2}F$, the temporal factor satisfies the ordinary differential equationThus $h(t)=\exp [(iu\mu -\frac{1}{2}{\sigma }^2{u}^2)(T-t)]$, andThis is bounded in absolute value by one, so it lies in the uniqueness class of the Feynman-Kac formula. As an independent check, integrating the stochastic differential equation gives $X_T=x+\mu (T-t)+\sigma (B_T-B_t)$. The independent Brownian increments have the normal distribution with mean zero and variance $T-t$, yielding exactly the same characteristic function. For $\sigma =0$, the conditional distribution is a point mass at $x+\mu (T-t)$, and the same boxed expression still applies.

Here the diffusion generator is $\mathcal{L}=ax{\partial }_x+\frac{1}{2}{\sigma }^2{\partial }_{xx}$. The Feynman-Kac formula givesTry $F(t,x)=\exp [iu\alpha (t)x+\beta (t)]$, where $\alpha (T)=1$ and $\beta (T)=0$. Substitution and collection of the coefficients of $x$ and the constant term giveThe solutions of these ordinary differential equations, with the terminal conditions imposed, areFor $\Delta =T-t$, define the conditional varianceThen the required conditional characteristic function isIn particular, for $a\ne 0$ this exponent is $iu{e}^{a\Delta }x-\frac{{\sigma }^2{u}^2}{4a}(e^{2a\Delta }-1)$. The variance $v_{\Delta }$ is nonnegative for either sign of $a$, and its limit at $a=0$ is ${\sigma }^2\Delta$. Hence the solution is bounded in absolute value by one and the Feynman-Kac formula identifies it with the required expectation. The value $u=0$ gives one directly, including that case without dividing by $u$.

For an independent check, the integrating factor $e^{-as}$ and the Itô product rule giveThe stochastic integral has a normal distribution because its integrand is deterministic; the Itô isometry gives its variance as $v_{\Delta }$. For $a<0$ this is the Ornstein-Uhlenbeck transition distribution. The formula also covers the unstable case $a>0$, and degenerates to deterministic motion when $\sigma =0$.

A second-order nondivergence-form elliptic operator on a domain $D\subseteq {\mathbb{R}}^d$ has the formOnly the symmetric part of $a$ affects the Hessian matrix term, so one may take $a(x)$ symmetric. The operator is a uniformly elliptic operator if there is a constant $\kappa >0$, independent of $x$, such thatA common bounded-coefficient convention additionally requires an upper bound $\sum a_{jk}(x){\xi }_j{\xi }_k\le \Lambda |\xi {|}^2$ with $\Lambda <\infty$. The lower bound says that the principal quadratic form cannot degenerate in any direction or at any location.

For an explicit stochastic example, standard $d$-dimensional Brownian motion has infinitesimal generatorIts principal quadratic form is $|\xi {|}^2/2$, so it is uniformly elliptic with $\kappa =\Lambda =1/2$. More generally, a sufficiently regular stochastic differential equation $d{X}_t=b(X_t)dt+\Sigma (X_t)d{B}_t$ has diffusion generator $b\cdot \nabla +\frac{1}{2}(\Sigma {\Sigma }^{\mathsf{T}}):D^2$; it is uniformly elliptic when $\Sigma {\Sigma }^{\mathsf{T}}$ has a uniformly positive lower bound. A conservative diffusion process has no zeroth-order term, since its infinitesimal generator annihilates constants.

For a radial function $f(r)$ in three dimensions, the Laplacian is $f^{\prime \prime }+2f^{\prime }/r$, while the imposed radial drift contributes $-\alpha f^{\prime }/r$. Thus the radial generator of an SDE diffusion isThe hitting probability for the inner sphere is harmonic for this diffusion generator in the shell, with value one on the inner sphere and zero on the outer sphere. At $\alpha =1$, the boundary value problem is thereforeIntegrating $(r{p}^{\prime }{)}^{\prime }=0$ gives $p(r)=A\log r+B$. The two boundary conditions determineBoth arguments of the logarithms are dimensionless. The result decreases from one to zero across the shell and lies strictly between zero and one for $R_1<r_0<R_2$.

For a direct probability interpretation, stop at the first contact $T$ with either sphere. The stopped process $p(r(t\wedge T))$ is a bounded martingale by Itô's formula. The shell has positive inner radius, so its drift is bounded and its diffusion is nondegenerate; exit occurs almost surely. Optional stopping and dominated convergence therefore giveThe logarithm appears because the inward drift cancels one of the two geometric $f^{\prime }/r$ terms in the three-dimensional radial Laplacian, leaving the same radial diffusion generator as a planar Brownian motion with diffusion coefficient one.