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Measurability is the property that inverse images of measurable target sets are measurable in the source space. For a stochastic process, joint measurability in time and sample is a different condition from the separate adaptedness condition at each fixed time. The predictable sigma-algebra and progressive measurability encode particular joint measurability requirements.

A real Lévy process is a real-valued stochastic process $(X_t{)}_{t\ge 0}$ with the following properties:

One convention also includes càdlàg paths in the definition. Equivalently, under the intrinsic definition above one chooses the càdlàg modification, which exists for such a stochastic process. Thus the usual working version of a Lévy process has right-continuous paths with left limits. There is no assumption of finite moments or continuous paths.

Use the independent increments of Brownian motion, rather than merely checking that an Itô formula drift vanishes. For $0\le s<t$, put $h=t-s$ and write $B_t=B_s+Z$, where $Z$ is independent of ${\mathcal{F}}_s$ and has normal distribution $N(0,h)$. Its first four moments are $0,h,0,3h^2$. ThereforeFor the cubic expression, the coefficient of $B_s$ after conditioning is $3(t-s)+\alpha (t)$, so $\alpha (t)=-3t$ makes it $\alpha (s)$. For the quartic expression, choose $\beta (t)=-6t$. Its conditioned coefficient of $B_s^2$ is then $6(t-s)-6t=-6s$. The constant term becomeswhich equals $3s^2$ when $\gamma (t)=3t^2$. Thus a standard choice isThe resulting stochastic processes are Hermite polynomial martingales $H_3(B_t,t)$ and $H_4(B_t,t)$. They are genuine integrable martingales, since Gaussian moments are finite at every finite time and the displayed conditional identities establish the martingale property directly.

The choice is not unique. Constants $c,d,e\in \mathbb{R}$ give the valid family $\alpha (t)=c-3t$, $\beta (t)=d-6t$, and $\gamma (t)=3t^2-dt+e$: these add $c{B}_t$ to the cubic martingale and $d(B_t^2-t)+e$ to the quartic one. The boxed choice sets these harmless additions to zero.

A version of a stochastic process means a stochastic process on the same probability space such that, for every fixed $t$,The exceptional null set may depend on $t$. Indistinguishability of stochastic processes means that there is one null set outside which $X_t=Y_t$ for all $t\ge 0$ simultaneously.

For an example separating the definitions, let $U$ have uniform distribution on $(0,1)$, and setFor every fixed $t$, $\mathbb{P}(U=t)=0$, so $Y$ is a version of a stochastic process with original stochastic process $X$. But for every sample outcome the stochastic processes differ at its time $t=U$. HenceThe spike path of $Y$ is not right-continuous at $U$, which explains why the next part's regularity assumption rules out this example.

For each nonnegative rational $q$, equality of the two versions gives $\mathbb{P}(X_q=Y_q)=1$. Intersect these countably many full-probability events with the full-probability event on which both paths are càdlàg. Call the resulting event $A$; then $\mathbb{P}(A)=1$.

Fix $\omega \in A$ and any real $t\ge 0$. Choose rational numbers $q_n>t$ decreasing to $t$. Right continuity givesThe same event $A$ works for every $t$, because the argument is pathwise after $\omega$ is fixed. The stochastic processes are therefore indistinguishable. This proves that càdlàg versions are indistinguishable; the left limits are not needed for this implication, since right continuity alone suffices.

Relative to a filtration $({\mathcal{F}}_t)$, progressive measurability means that, for every $T<\infty$, the mapis measurable for the product sigma-algebra $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$ and the Borel sigma-algebra on $\mathbb{R}$.

Fix $T>0$ and divide $[0,T]$ into $2^m$ equal subintervals with mesh $h_m=T/2^m$. DefineSince $X$ is adapted, every random variable $X_{k{h}_m}$ is ${\mathcal{F}}_{k{h}_m}$-measurable, hence ${\mathcal{F}}_T$-measurable. Each approximation is consequently $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$-measurable. For $t<T$, its sampling time lies strictly to the right of $t$, tends to $t$, and never exceeds $T$. Right continuity implies $X_t^{(m)}\to X_t$; at $T$ equality is exact. Thus $X$ is the pointwise limit of measurable functions on this product space. As $T$ was arbitrary, $X$ is progressively measurable. This is the theorem that right-continuous adapted processes are progressively measurable.

The right-endpoint approximations need not themselves be adapted at their intermediate times. What the proof requires is their joint measurability with respect to the single terminal sigma-algebra ${\mathcal{F}}_T$. The proof uses the pathwise càdlàg convention. If path regularity is assumed only almost surely, under a completed filtration setting the stochastic process to zero on its common exceptional null event gives an indistinguishable progressively measurable version. Arbitrary values on that null event need not make the original stochastic process progressively measurable: even with a complete filtration, a null sample point may be assigned a non-Borel time function. This is why almost sure path regularity does not ensure progressive measurability.

Fix $t$ and put $\sigma =t\wedge \tau$. The stopping time property implies that $\sigma$ is an ${\mathcal{F}}_t$-measurable random variable with values in $[0,t]$: for $s<t$,and for $s\ge t$ the event is the whole space. By part (c), the restriction of $X$ to $[0,t]\times \Omega$ is $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$-measurable.

The evaluation map $\omega \mapsto (\sigma (\omega ),\omega )$ is measurable from $(\Omega ,{\mathcal{F}}_t)$ to this product space: the inverse image of a measurable rectangle $A\times C$ is $\{\sigma \in A\}\cap C$. Composing it with the jointly measurable stochastic process gives an ${\mathcal{F}}_t$-measurable random variableSince this holds for every fixed $t$, the stopped process is adapted. If path regularity holds only almost surely, first apply the proof to its pathwise regular representative; with a completed filtration, the original stopped variable differs only on a null event and is also ${\mathcal{F}}_t$-measurable. The adaptedness of a stopped right-continuous process requires neither boundedness of $\tau$ nor a martingale assumption; $\tau =\infty$ is harmless because $t\wedge \tau \le t$.

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 $\Psi =-\psi$.

Work on a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_n{)}_{n\ge 0},\mathbb{P})$. A submartingale is a real-valued stochastic process satisfying the following three conditions: $Y_n$ is ${\mathcal{F}}_n$-measurable, $\mathbb{E}|Y_n|<\infty$, andThus it is adapted, has integrable random variables at every time, and its conditional expected next value is at least its current value. Repeated conditioning gives $\mathbb{E}[Y_m\mid {\mathcal{F}}_n]\ge Y_n$ for every $m\ge n$. The inequality concerns conditional expectation; individual sample paths need not increase. If no filtration is specified, the definition can be applied to the natural filtration generated by the process.

A standard real Brownian motion relative to a filtration $({\mathcal{F}}_t)$ is an adapted stochastic process with $B_0=0$ almost surely, continuous paths almost surely, andEquivalently, it has independent increments and stationary increments, each with a normal distribution of mean zero and variance equal to the elapsed time, together with continuous paths and the stated initial value. The filtration may be its natural filtration. A Brownian motion started at $z$ is $z+W_t$ for a standard process $W$; multidimensional standard Brownian motion has independent standard real coordinate processes. The normalization is variance $t$ at time $t$.

A real Lévy process is a stochastic process $(X_t{)}_{t\ge 0}$ with $X_0=0$ almost surely, independent increments, stationary increments, and stochastic continuity. The latter means $X_t-X_s\to 0$ in probability as $t\to s$. Its usual path convention is càdlàg: paths are right-continuous and have left limits. Equivalently one starts with the increment and continuity-in-probability conditions and then chooses the càdlàg version.

The Lévy characteristic exponent is specified here bySome conventions use $e^{t\tilde{\psi }(\theta )}$ instead, with $\tilde{\psi }=-\psi$. The sign convention must be fixed before computing the exponent. The zero process satisfies this definition; excluding it requires an additional nondegeneracy condition.

Relative to a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a filtration $({\mathcal{F}}_n)$, the real stochastic process $(M_n)$ is a martingale if it is adapted, each $M_n$ is an integrable random variable, andHere adapted means $M_n$ is ${\mathcal{F}}_n$-measurable, and integrability means $\mathbb{E}|M_n|<\infty$. By the tower property of conditional expectation, the equivalent multi-step condition is $\mathbb{E}[M_j\mid {\mathcal{F}}_i]=M_i$ for all $i\le j$. The filtration is part of the definition; specifying only the marginal means is insufficient.

Let $\tau =\inf \{t\ge 0:B_t=-2\sqrt{x_0}\}$, the first time $W$ reaches zero. Recurrence of one-dimensional Brownian motion implies $\tau <\infty$ almost surely. On the common event where the comparison from part (c) holds, $\tau <T$ would imply $0<Y_{\tau }\le W_{\tau }=0$, a contradiction. ThereforeThe argument only uses the comparison at a time strictly before $T$; it does not assume that the singular drift integral is finite at $T$. The given lifetime characterization identifies this finite endpoint with approach to zero. Any extension that absorbs at zero would be a different, globally defined stochastic process, not the positive-domain maximal local solution of a stochastic differential equation specified here.