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Suppose a finite-horizon market has a strictly positive numéraire $B$ and an equivalent measure under which all prices $S^i/B$ are local martingales. The discounted wealth of a self-financing strategy is a stochastic integral against those prices. If it is bounded below by a deterministic constant, that wealth is a supermartingale: add the lower-bound constant to obtain a nonnegative local martingale, then apply conditional Fatou to a localizing sequence. A zero-cost admissible portfolio therefore has expected discounted terminal wealth at most zero. Nonnegative terminal wealth with positive probability of a strict gain would have strictly positive expectation under the equivalent measure, a contradiction. Thus there is no admissible arbitrage. True martingales are not needed for this implication.

For $Z_t=e^{B_t+at}$, use the clock $C_t={\int }_0^t{Z}_s^{2}ds$. After the inverse time change, the Dambis-Dubins-Schwarz theorem givesso the time-changed process is a Bessel process of dimension $d=2a+2$. Its lifetime is $C_{\infty }$, which is finite almost surely exactly when $d<2$.

Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are local martingales, their logarithms first recover adaptation of $X,A$ and the semimartingale property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.

For a positive function $g$ in the domain of a Markov jump-process generator, the generator martingale for $g(X_t)$ and semimartingale integration by parts give the displayed local martingale. Suitable finite-horizon deterministic bounds on $g$ and $Lg/g$ make it a true martingale. For a single marked exponential jump process, take $g(x)=e^{\theta x}$. Then the exponent's compensating process is $r_{\theta }(t\wedge T)$, where $r_{\theta }=(e^{\theta }-1)/\theta -1$ for $\theta \ne 0$ and $r_0=0$; it is continuous adapted and hence predictable.

For a pure birth process with fixed initial state and finite statewise rates, the jump-count compensator gives $dM=(e^{\theta }-1)M_{-}[dN-\lambda (X_{-})dt]$, so $M$ is a local martingale up to explosion. If $\lambda \le C$, a rate-$C$ Poisson candidate clock with accepted jumps of probability $\lambda (X_{-})/C$ prevents explosion and dominates the jump count. Its finite exponential moments dominate every stopped $M$ on each finite horizon, upgrading the local martingale to a true martingale. A random initial state requires $\mathbb{E}{e}^{\theta X_0}<\infty$; uniformly bounded rates do not supply this missing initial integrability.

For the displayed stochastic differential equation, the infinitesimal generator is $\mathcal{L}g=(f^2/2)(g^{\prime \prime }-g^{\prime })$, so $\mathcal{L}s=0$ and the Itô formula makes $e^{X_t}$ a positive local martingale. If $f$ is bounded, strictly positive, and globally Lipschitz, the equation has a unique global strong solution of a stochastic differential equation. For $a<x<b$ and $T_y=\inf \{t:X_t=y\}$, the Itô isometry for the bounded stopped exponential gives finite mean exit time: the bracket density $e^{2z}f(z{)}^2$ has a positive minimum on $[a,b]$. The optional stopping theorem then yields the boundary hitting probability from a diffusion scale functionTaking $a\to -\infty$ gives ${\mathbb{P}}_x(T_b<\infty )=e^{x-b}$ for $b>x$. Passing to strict exceedances and then to their limits shows that ${\sup }_{t\ge 0}{X}_t-x$ has the exponential distribution of rate one. The probability distribution of the maximum is independent of the particular $f$; changing the speed of motion preserves this scale function.

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 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.

Suppose $d{r}_t=a(r_t)dt+b(r_t)d{W}_t$ under a risk-neutral measure and $f_t(T)=F(T-t,r_t)$. By Itô formula, the forward-rate volatility is ${\sigma }_t(T)=b(r_t)F_r(T-t,r_t)$. The Heath-Jarrow-Morton model drift restriction is equivalent toTo verify it directly for bonds, put $G(\theta ,r)={\int }_0^{\theta }F(s,r)ds$. Integration gives $F-r=a{G}_r+\frac{1}{2}{b}^2{G}_{rr}-\frac{1}{2}{b}^2{G}_r^2$. The zero-coupon bond price $P(t,T)=e^{-G(T-t,r_t)}$ therefore satisfiesIts bank-account-discounted price is a positive local martingale. A common equivalent local martingale measure excludes admissible arbitrage for any finite collection of these traded maturities.

The polynomial $X^4-6X^2⟨X⟩+3⟨X{⟩}^2$ has zero Itô drift. Finite fourth maximal and second bracket moments make it a true martingale. If $\mathbb{E}{X}_t^2=t$ and $\mathrm{Cov}(X_t^2,⟨X{⟩}_t)=0$, then $\mathbb{E}{X}_t^4=3t^2-3\mathrm{Var}(⟨X{⟩}_t)$. Equality at every time forces a deterministic clock and hence Brownian motion by the Lévy characterization of Brownian motion.

A bounded entire harmonic function satisfies $u(x)={\mathbb{E}}_{x}u(B_t)$ because $u(B)$ is a bounded local martingale. The Gaussian heat kernel has a directional derivative with $L^1$ norm $\sqrt{2/\pi }/\sqrt{t}$. Consequently $|u(x)-u(y)|\le \Vert u{\Vert }_{\infty }\sqrt{2/\pi }|x-y|/\sqrt{t}$; letting $t\to \infty$ proves the harmonic Liouville theorem in every dimension.

Let $M$ be a zero-starting continuous local martingale and let $T<\infty$ be a stopping time with $[M{]}_{t\wedge T}\le a$ and $[M{]}_T=a$ for a deterministic $a$. The complex exponential $\exp (iu{M}_{t\wedge T}+u^2[M{]}_{t\wedge T}/2)$ is a bounded local martingale, hence a true martingale. Its terminal expectation is one, giving the characteristic function $\mathbb{E}{e}^{iu{M}_T}=e^{-u^{2}a/2}$. Conditional optional sampling between two bracket levels also gives independent Gaussian increments in inverse-bracket time; Gaussian one-time marginals alone do not establish that independence.

Girsanov's theorem describes how a change of probability measure changes the drift of a semimartingale. If the appropriate stochastic exponential is a true martingale, weighting by it turns $W_t+{\int }_0^t{\theta }_{s}ds$ into a Brownian motion under the new measure.

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.

With the local martingale definition requiring each stopped process itself to be a martingale, the initial value is integrable. A normalized continuous semimartingale decomposition $X=M+A$ with $A_0=0$ can therefore have a local-martingale part only if $X_0$ is integrable. A smooth transformation need not preserve this condition: take $M=0$, $A_t=G$ for an initially known standard normal random variable, and $f(x,a)=e^{a^2}$. Then $f(M_0,A_0)$ is finite almost surely but has infinite expectation. The always-valid decomposition is instead $X=X_0+N+V$ with a zero-starting continuous local martingale $N$ and a continuous finite-variation process $V$ starting at zero.

An integrable discrete-time local martingale is a true martingale. Its stopped increments have the form $1_{\{\tau \ge t\}}(X_t-X_{t-1})$, and the dominated convergence theorem removes a localizing sequence from the conditional increment identity.

If a local martingale has an integrable running absolute supremum on each fixed finite horizon, localization and conditional dominated convergence make it a true martingale. This upgrades polynomial Itô local martingales when maximal moments and bracket moments control all terms.

Let $M_1,\dots ,M_N$ be continuous local martingales starting at zero, with $⟨M_i,M_j⟩=0$ for $i\ne j$ and $⟨M_j{⟩}_{\infty }=\infty$ almost surely. The individual Dambis-Dubins-Schwarz theorem time changes produce independent standard Brownian motions $B_j$ with $M_j(t)=B_j(⟨M_j{⟩}_t)$. Independence is between the Brownian motions; each may still depend on its own clock.

The law of one price requires traded assets or portfolios with the same terminal payoff to have the same earlier price. In a model where every discounted traded price is a true martingale under a common equivalent measure, it follows by taking the conditional expectation of that terminal payoff. With only local martingale prices and a lower-bound restriction on admissible trading strategies, it can fail: the seemingly profitable short position may have unbounded possible losses before maturity.

A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion exactly when its quadratic variation satisfies $[M{]}_t=t$.

The characterization asserts that a continuous local martingale starting at zero with quadratic variation $t$ is Brownian motion.