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Let a probability space have two sample points, one of probability zero and one of probability one, and give it the complete power-set sigma-algebra at every time. Let $A\subseteq [0,1]$ be non-Borel. Assign the stochastic process the path $1_A(t)$ at the null sample point and the zero path at the other point. Each fixed-time variable is measurable and the paths are càdlàg almost surely, but the stochastic process is not progressively measurable: a section in the time variable of a product-measurable function must be Borel measurable at every sample point. The zero stochastic process is an indistinguishable progressively measurable version. Completing the filtration does not complete the time-product sigma-algebra or remove this distinction.

A discrete-time stochastic process has bounded increments if a single deterministic $C<\infty$ satisfies $|X_{n+1}-X_n|\le C$ almost surely for every $n$. The constant is uniform in time and sample outcome; bounding each increment by a different constant does not suffice. This property applies to differences, so it does not require the process itself to stay bounded.

A stochastic process in which individuals independently produce offspring according to specified distributions. It models growing populations and gives an approximation to sparse percolation clusters when collisions between branches can be neglected. On a lattice, loops and overlapping branches prevent this approximation from being an exact identity.

For standard Brownian motion, the Brownian bridge $U_t=B_t-t{B}_1$, $0\le t\le 1$, is independent of $B_1$. Every finite vector of bridge coordinates is jointly multivariate normal with $B_1$, and its cross-covariances with $B_1$ vanish. The independence of uncorrelated jointly normal variables and independence extended from generating pi-systems prove independence of the whole stochastic process.

A càdlàg modification of a stochastic process is a modification of a stochastic process whose paths are càdlàg on one event of probability one. Equality almost surely at every separately fixed time does not imply indistinguishability of stochastic processes when the original process lacks path regularity. A stochastic process starting at zero, with independent increments, stationary increments, and stochastic continuity, has a càdlàg modification and is a Lévy process under the intrinsic definition.

If two versions of a stochastic process both have càdlàg paths on an event of probability one, equality at every rational time holds simultaneously outside one null event. Approaching an arbitrary time by rationals from the right then gives equality at every time on the same event. Thus the stochastic processes are indistinguishable. Right continuity suffices; left limits are unnecessary for this conclusion.

For a Lévy process, stochastic continuity gives continuity of its characteristic function in time at every fixed frequency. The elementary estimate $|{\phi }_{X_s}(u)-{\phi }_{X_t}(u)|\le |u|\delta +2\mathbb{P}(|X_s-X_t|>\delta )$ proves this by first taking $s\to t$ and then $\delta \downarrow 0$. The argument works for any stochastically continuous stochastic process and needs no finite-moment assumption.

A continuous modification of a stochastic process is a modification of a stochastic process whose paths are continuous outside a single null event. For a process originally given only on a countable dense set, the analogous construction gives a continuous extension agreeing simultaneously at all original indices. The Kolmogorov continuity theorem and dyadic increment chaining are standard ways to obtain it.

The covariance function of a second-order stochastic process $(X_t)$ isIt is a positive-semidefinite kernel and determines the finite-dimensional distributions of a centered Gaussian process.

For a function on the dyadic rationals in $[0,1]$, let $A_n$ be its largest absolute adjacent increment on the level-$n$ grid. If $K_{\alpha }<\infty$ for $\alpha >0$, the function extends uniquely to a Hölder continuous function withChoose $n$ with $2^{-n}\le |t-s|<2^{1-n}$. The level-$n$ left approximations differ by at most two grid steps, and each finer approximation adds at most one level increment. Thus the difference is bounded by $2{\sum }_{j\ge n}{A}_j\le 2^{-n\alpha }{K}_{\alpha }$. For a stochastic process with $\Vert {\xi }_t-{\xi }_s{\Vert }_p\le C|t-s{|}^{\beta }$, the bound $\Vert A_n{\Vert }_p\le C{2}^{-n(\beta -1/p)}$ makes this series converge in the Lp norm for $0<\alpha <\beta -1/p$.

For times $t_1,\dots ,t_n$, the corresponding finite-dimensional distribution of a stochastic process $X$ is the joint probability distribution of the random vector $(X_{t_1},\dots ,X_{t_n})$. The collection of these distributions records every finite set of coordinates of the process.

A fluid limit is a scaling limit of a stochastic process obtained by scaling space and time so that random fluctuations vanish and a deterministic trajectory remains. It can justify a fluid model when the required convergence conditions hold.

A Gaussian process is a stochastic process whose every finite vector of values has a multivariate normal distribution. Its law is determined by its mean and covariance functions.

An interacting particle system is a stochastic process whose local states evolve through random transitions whose rates depend on nearby states.

If a stochastic process indexed by a subset of ${\mathbb{R}}^d$ satisfiesfor some $p,\beta >0$, then it has a continuous modification. More precisely, that modification is locally $\gamma$-Hölder continuous for every $\gamma <\beta /p$.

For standard Brownian motion started at zero and first exit ${\tau }_x$ from $(-x,x)$, the Laplace transform is $1/\cosh (x\sqrt{2\lambda })$ for $\lambda >0$. The stochastic process $e^{-\lambda t}\cosh (\sqrt{2\lambda }{B}_t)$ is the average of two copies of the Exponential martingale for Brownian motion. Its values stopped at $t\wedge {\tau }_x$ are bounded, so bounded-time application of the optional stopping theorem followed by dominated convergence evaluates the transform without assuming uniform integrability of an unstopped exponential martingale on the infinite horizon.

A Markov process is a stochastic process for which, conditional on the present state, the future is independent of the past. Its transition probabilities therefore determine every finite-dimensional distribution once the initial distribution is known.