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A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.

A localizing sequence for a local martingale $M$ is an increasing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every stopped process $M^{T_n}$ is a true martingale. One can often choose the times so that the stopped processes are bounded.

A local martingale deflator is a strictly positive process $Y$ such that multiplying each traded asset price by $Y$ produces a local martingale. It can support no-arbitrage arguments even when the deflated prices are not true martingales.

A martingale deflator is a strictly positive adapted process whose product with each cum-dividend asset gain is a local martingale. After normalization by a numéraire, a martingale deflator determines an equivalent martingale measure.

A continuous adapted process started at a fixed point is Brownian motion exactly when $f(X_t)-f(X_0)-\frac{1}{2}{\int }_0^t\Delta f(X_s)ds$ is a martingale for every $f\in C_c^{\infty }({\mathbb{R}}^d)$. Cutoff coordinate and coordinate-product tests give continuous local martingales with quadratic covariations $[X^i,X^j{]}_t={\delta }_{ij}t$. The Itô formula makes $\exp (iu\cdot (X_t-X_0)+|u{|}^{2}t/2)$ a local martingale, whose deterministic modulus bound on bounded horizons makes it a true martingale. Its conditional expectation identifies every increment as centered Gaussian with covariance $(t-s)I$, independent of the past. This proves the converse via the Lévy characterization of multidimensional Brownian motion. The forward implication is the Itô formula.

For continuous local martingales $M$ and $N$, the Itô product rule says thatis a local martingale. If the martingales are square-integrable and converge in $L^2$, then

A market satisfies no-arbitrage when it admits no admissible self-financing strategy with zero initial cost, nonnegative terminal wealth almost surely, and a strictly positive gain with positive probability. The admissibility restriction excludes doubling strategies. Identical attainable cash flows then have the same price.

The Fatou lemma makes every time value of a nonnegative discrete-time local martingale integrable, assuming the usual integrable initial value. The integrable discrete-time result then makes it a true martingale.

Every nonnegative local martingale is a supermartingale. In particular, one that starts from zero is identically zero: its nonnegative value at every time has expectation at most zero.

A continuous local martingale bounded above pathwise throughout a stochastic time interval cannot have infinite quadratic variation at that interval's endpoint. The Dambis-Dubins-Schwarz theorem represents it as Brownian motion at its quadratic-variation clock; an infinite clock would force unbounded oscillations. Its clock therefore has a finite limit, and so does the martingale. A bound below gives the same conclusion by changing sign. The bound may be random; it must hold over the entire interval.

Two continuous local martingales are orthogonal when their quadratic covariation is zero. With zero initial values, the martingale product identity says that their product is a local martingale. For two Brownian motions in a common filtration, orthogonality and the Lévy characterization of multidimensional Brownian motion make the pair a two-dimensional Brownian motion and imply independence. Merely having the two marginal Brownian motion laws on the same space does not imply orthogonality.

Let $f(y)=|y|$ away from zero. Its first and second partial derivatives areThus its Laplace operator is $\Delta f(y)=(n-1)/|y|$. The quadratic covariations of the independent Brownian coordinates satisfy $d[B_j,B_k{]}_s={\delta }_{jk}ds$. The stopped Itô formula therefore givesFor $t<\tau$, this is the required integral representation.

To identify the driving Brownian motion, the stopped stochastic term alone is insufficient: its quadratic variation is $t\wedge \tau$. Extend its unit direction after exit. Define a vector predictable process $u(s)$ byPredictability follows from the continuous adapted direction before stopping and the left-continuous adapted indicator $1_{\{s\le \tau \}}$. This $W$ is a zero-starting continuous martingale, andThe Lévy characterization of Brownian motion identifies $W$ as a one-dimensional Brownian motion. Before $\tau$, the preceding representation becomes the stochastic differential equation of the dimension-$n$ Bessel process:The fully stopped equation has both integrals stopped at $\tau$.

For $n=2$, apply the one-dimensional Itô formula to the logarithm. Before $\tau$, $d[X{]}_t=dt$, so the two drift terms cancel:Consequently,Indeed $X_s\ge r$ before stopping, so the Itô isometry bounds the stochastic term's second moment by $t/r^2$, and $\log X_{t\wedge \tau }\in [\log r,\log R]$. This is the martingale assertion under the paper's standing stopping assumption. Without stopping, the same calculation establishes only a local martingale; the stopped qualification is essential for the stated argument.

Assume the SDE has the required solution and $f$ is sufficiently smooth. Itô's formula along that solution givesThe PDE cancels the first term. Integrating from $t$ to $T$ givesIf this integral is a true martingale, for example if its integrand has finite expected integrated square, its conditional expectation at time $t$ is zero. For the diffusion started at $x$ at time $t$, this yieldsThe notation in the question is the same Markov conditional expectation. The martingale or corresponding uniform-integrability growth hypothesis is part of the usual Feynman-Kac formula justification: a local martingale alone does not automatically have zero expected increment. Standard smooth solutions with the customary integrability bounds satisfy this requirement.

Let $\tau =T-t$. The product rule and Itô's formula givewhere the Black-Scholes equation cancels the drift. Under the usual pricing growth and integrability conditions this local martingale is a martingale. Its terminal value is $f(S_T)$, since the exponential factor becomes one at $T$. Taking a conditional expectation givesThe dynamics already use the risk-neutral measure. This is the Feynman-Kac formula for the given pricing problem; a differential equation alone, without the relevant growth condition, would not justify upgrading a local martingale or uniqueness.

The differential operator in part b is the infinitesimal generator of the auxiliary diffusion process $d{S}_u=r{S}_{u}du+\sigma d{W}_u$, with killing rate $r$. The Feynman-Kac formula therefore givesEquivalently, Itô's formula makes $e^{-r(u-t)}V(S_u,u)$ a local martingale under these dynamics; under appropriate growth and integrability assumptions it is a martingale, whose terminal conditional expectation yields the same formula. The expectation uses the drift-$r$ law, rather than the physical drift-$\alpha$ law in part a. This is risk-neutral pricing.

Here is the Brownian change-of-drift form of the Girsanov theorem. Let $W$ be a standard Brownian motion on $[0,T]$ and $\theta$ predictable with ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely. Suppose the stochastic exponentialis a true martingale with $\mathbb{E}{Z}_T=1$. Define $dQ=Z_{T}dP$. Then $Q$ is equivalent to $P$ andThe same statement holds in $d$ dimensions with scalar products, squared norms and a vector drift. Novikov's condition $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T{\theta }_s^{2}ds)<\infty$ is a standard sufficient condition for the true-martingale hypothesis. Bounded $\theta$ is enough for all applications below.

For the proof, Itô formula gives $d{Z}_t=-Z_t{\theta }_{t}d{W}_t$. The Itô product rule gives the cancellationThe two drift terms cancel because $d[Z,W{]}_t=-Z_t{\theta }_{t}dt$. Stop $W^Q$ when its absolute value reaches $n$, at a time ${\tau }_n$ capped by $T$. The same product calculation makes $Z_t{W}_{t\wedge {\tau }_n}^Q$ a local martingale under $P$. Its absolute value is at most $n{Z}_t$; the density martingale is uniformly integrable over this finite horizon, so the stopped product is a true martingale. Bayes' conditional identity then givesAs $n$ increases, these stopping times exhaust the horizon by continuity. Thus $W^Q$ is a continuous local martingale under $Q$. Adding a finite-variation drift does not change quadratic variation, so $[W^Q{]}_t=t$.

To prove the Brownian conclusion rather than just assert it, for every real $u$ apply Itô formula to $F_t=\exp (iu{W}_t^Q+u^{2}t/2)$. Its differential is $iu{F}_{t}d{W}_t^Q$, and its modulus is bounded on the fixed horizon, so it is a true martingale. ConsequentlyThe conditional characteristic function identifies a centered normal increment with variance $t-s$, independent of the past. Together with continuity and $W_0^Q=0$, this proves the Brownian assertion. For vector $u$, the same argument uses $\Vert u{\Vert }^2$ and gives the multivariate version. If $|\theta |\le K$, stopping $Z$ and applying Itô formula to $Z^2$ gives a uniform second-moment bound $\mathbb{E}{Z}_{t\wedge \tau }^2\le e^{K^{2}t}$ by Gronwall inequality; this proves uniform integrability of the stopped densities and the needed true-martingale property in the bounded case.

In the Black-Scholes model with its usual augmented Brownian filtration, write $d{S}_t=S_t(\alpha dt+\sigma d{W}_t)$, $B_t=e^{\rho t}$, with $\sigma >0$. Take $\theta =(\alpha -\rho )/\sigma$, the market price of risk. Under $Q$, the preceding change of drift givesso $S_t/B_t$ is a martingale. The Martingale representation theorem then makes integrable Brownian-market claims attainable and gives their risk-neutral pricing as discounted conditional expectations. The physical drift $\alpha$ has disappeared from the price dynamics.

For the joint terminal-value and maximum law, first use the Brownian reflection principle at level $a>0$. Reflecting a path after its first hit of $a$ preserves its law as a Brownian motion and sends a terminal value $y<a$ to $2a-y>a$; the symmetry and strong Markov property of the post-hit increments justify this map. Thus, with ${\phi }_t(y)=(2\pi t{)}^{-1/2}{e}^{-y^2/(2t)}$,Take the constant-drift exponential density $e^{\mu W_t-{\mu }^{2}t/2}$. By Girsanov theorem, the coordinate process under this tilted measure has the law of $W_s+\mu s$ under $P$. Consequently, for $t>0$ and $x\le a$,Completing the squares givesIntegrating provesHere $\Phi$ is the standard normal distribution function. The first variable is the terminal Brownian value, distinct from its running maximum.

Set $T=t_0$, $a=\log (b/S_0)/\sigma >0$ and $\mu =(\rho -{\sigma }^2/2)/\sigma$. Under the risk-neutral measure, $\log (S_s/S_0)/\sigma =W_s^Q+\mu s$. Taking $x=a$ in the joint law gives the probability of not hitting the upper level. Complementing it and discounting the unit payment at maturity gives the one-touch option priceThe exponential weight is also $(b/S_0{)}^{2\rho /{\sigma }^2-1}$. The payment occurs at $T$, even if the barrier was hit earlier, so its discount factor is $e^{-\rho T}$, not a hitting-time discount factor.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

For a simple predictable process, define its stochastic integral byCommon refinement of the deterministic partitions makes this definition independent of the representation. Each summand is a martingale increment multiplied by information already available at its left endpoint, so the resulting process is a continuous square-integrable martingale starting at zero. Its terminal value is the same sum with $t$ replaced by $t_n$.

Write ${\Delta }_{k}M=M_{t_{k+1}}-M_{t_k}$. If $j<k$, then $Z_{t_j}{\Delta }_{j}M{Z}_{t_k}$ is ${\mathcal{F}}_{t_k}$-measurable. By the conditional expectation identity for increments of a martingale,The conditional bracket isometry for stopped martingale increments givesFor completeness, this identity follows by applying the optional stopping theorem to $M$ and $M^2-[M]$ and expanding the square. The latter process is a true uniformly integrable martingale here: the Doob L2 maximal inequality bounds $\mathbb{E}{\sup }_t|M_t{|}^2$, stopped square identities and Fatou's lemma give $\mathbb{E}[M{]}_{\infty }<\infty$, and its absolute supremum is bounded by ${\sup }_t|M_t{|}^2+[M{]}_{\infty }$. Consequently all the stopped identities pass to the limit.

Multiplying the conditional identity by $Z_{t_k}^2$, taking expectations, and using the vanished cross terms proves the Itô isometry:

To extend the stochastic integral, use the finite quadratic-variation measure ${\nu }_M(C)=\mathbb{E}\int 1_{C}d[M]$ on the predictable sigma-algebra. The density of simple predictable processes for finite measures gives approximations to every $H\in L^2({\nu }_M)$. The Itô isometry makes the corresponding terminal integrals Cauchy in $L^2$, and the Doob L2 maximal inequality makes the whole processes Cauchy in expected squared supremum. Their limit defines a continuous martingale, and the isometry persists. For a locally bounded predictable process, choose increasing stopping times tending to infinity on which it is bounded, also restrict the time horizon, and perform this construction after stopping. The isometry shows that the stopped definitions agree on overlaps, so they patch into a uniquely defined continuous local martingale $H\cdot M$.

There is an initial-value qualification in the printed conclusion. As written, nothing requires $A_0=0$. For example, $M\equiv 0$ and $A\equiv 1$ satisfy the hypotheses, because $M^2-A\equiv -1$ is a local martingale, but $M_t$ is not $N(0,1)$. The correct variance is $A_t-A_0$; the printed formula holds under the usual normalization $A_0=0$.

Indeed, $M^2-[M]$ is a continuous local martingale. Subtracting it from $M^2-A$ shows that $[M]-A$ is a continuous local martingale with finite variation, hence is its initial constant $-A_0$. ThusIn particular, $C$ is deterministic, continuous, nonnegative and increasing, regardless of the apparent allowance of a general finite-variation process in the premise.

For a real parameter $\theta$, the Itô formula gives the complex local martingaleOn every deterministic interval $[0,T]$, its modulus is at most $e^{{\theta }^2{C}_T/2}$, so its real and imaginary parts are bounded local martingales, hence true martingales. Taking expectations and using $Z_0=1$ yieldsBy the uniqueness theorem for characteristic functions,A zero variance means the point mass at zero. This proves the requested normalized version by a direct characteristic function calculation.

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.