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A standard one-dimensional Brownian motion is a real-valued stochastic process $(B_t{)}_{t\ge 0}$ satisfying $B_0=0$ almost surely, having continuous paths almost surely, and having independent increments withHere $N(0,t-s)$ is the normal distribution with zero expected value and variance $t-s$. Thus the increments are also stationary increments.

If Brownian motion is defined relative to a specified filtration $({\mathcal{F}}_t)$, additionally require that it is an adapted process and that $B_t-B_s$ is independent of ${\mathcal{F}}_s$. For its natural filtration, this follows from independent increments. These are the defining properties of standard Brownian motion.

For a stopping time $T$ of a Brownian filtration, define the reflected stochastic processThe Brownian reflection at a stopping time form of the Brownian reflection principle states that $\hat{B}$ is again standard Brownian motion. If $T=\infty$, leave the path unchanged; the second branch is used only when $T<t$.

For an almost surely finite $T$, the Strong Markov property says that $(B_{T+s}-B_T{)}_{s\ge 0}$ is a fresh Brownian motion independent of ${\mathcal{F}}_T$. Its negative has the same probability distribution, so reflecting the future preserves the full path probability distribution. Allowing $T=\infty$ follows by applying this argument to $T\wedge n$ and restricting to each fixed finite time interval.

Let $\mathcal{P}$ be the predictable sigma-algebra on $\Omega \times [0,\infty )$, generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,u]$ with $A\in {\mathcal{F}}_s$. A stochastic process $H$ is previsible precisely when $(\omega ,t)\mapsto H_t(\omega )$ is $\mathcal{P}$-measurable.

Every left-continuous adapted process is previsible. Indeed, on intervals $(k{2}^{-n},(k+1)2^{-n}]$, put $H_t^{(n)}=H_{k{2}^{-n}}$, and put $H_0^{(n)}=H_0$. Each approximation is a predictable process, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable; left continuity gives $H_t^{(n)}\to H_t$ for every $t$, so measurability passes to the limit.

An adapted càdlàg process need not be previsible. For a concrete example, let $\xi$ take values $0,1$ with equal probabilities, let ${\mathcal{F}}_t$ be trivial for $t<1$ and ${\mathcal{F}}_t=\sigma (\xi )$ for $t\ge 1$, and set $H_t=\xi 1_{[1,\infty )}(t)$. This is an adapted càdlàg process, but $H_1=\xi$ is not ${\mathcal{F}}_{1-}$-measurable, where ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{s<1}{\mathcal{F}}_s)$. Sections at time $1$ of the generators of $\mathcal{P}$ all belong to ${\mathcal{F}}_{1-}$, so every predictable process has an ${\mathcal{F}}_{1-}$-measurable value there; this proves the counterexample directly.

The Itô formula applied to $F(t,x)=e^{t/2}\cos x$ gives exact cancellation of the drift:On every finite interval $[0,T]$, the Itô integral is square-integrable since $\mathbb{E}{\int }_0^T{e}^s{\sin }^2{B}_{s}ds\le e^T-1$. Therefore $e^{t/2}\cos B_t$ is a true martingale, and hence a local martingale.

For the other stochastic process, $d(B_t-t^2)=d{B}_t-2tdt$. If $B_t-t^2$ were a local martingale, its difference with $B$ would make the continuous finite-variation process $-t^2$ a local martingale, contradicting the fact that a continuous finite-variation local martingale is constant. Also $\mathbb{E}(B_t-t^2)=-t^2$ is not constant. Thus $B_t-t^2$ is neither a local martingale nor a martingale.

The stochastic process $W$ starts from zero and has continuous paths. Its increments on disjoint intervals are independent, since the corresponding pairs of Brownian motion increments are independent, andThus $W$ is a standard Brownian motion, including $\rho =\pm 1$. Here the intended filtration is the joint natural filtration, or another filtration relative to which the pair is a two-dimensional Brownian motion; independence of their path laws alone would not justify arbitrary extra information in a larger filtration.

By bilinearity of quadratic covariation and part (i),

Let $U_t=(1,0,0)+(B_t^1,B_t^2,B_t^3)$, with three independent standard Brownian motions, and set $R_t=|U_t|$ and $M_t=R_t^{-1}$. The radial stochastic process $R$ is a three-dimensional Bessel process, which never hits zero by the Hitting-zero classification for a Bessel process. In three dimensions $u\mapsto |u{|}^{-1}$ is a harmonic function away from zero: direct differentiation gives $\Delta (|u{|}^{-1})=0$.

For the stopping times $T_n=\inf \{t:R_t\notin (1/n,n)\}$, $n\ge 2$, the Itô formula givesThe coefficients are bounded by $n^2$, so each stopped Itô integral is a true martingale on every finite horizon. The stopping times form a localizing sequence; as permitted, its divergence need not be proved here. Thus $M$ is a nonnegative continuous local martingale.

To see that $M$ is not a true martingale, integrate the three-dimensional multivariate normal density of $U_t$ in spherical coordinates. The angular integral yieldswhere $\Phi$ is the standard normal distribution function. Therefore the reciprocal of a three-dimensional Bessel process is a strict local martingale.

Take $X=Y=B$, a standard Brownian motion with $B_0=0$. The Itô formula and $⟨B{⟩}_t=t$ giveIf this nonnegative stochastic process, which starts from zero, were a local martingale, question 4(i) would make it a supermartingale and force its expectation to be at most zero. But $\mathbb{E}(B_t^2/2)=t/2>0$ for $t>0$. Hence a Stratonovich integral is in general not a local martingale.

A path probability is a probability measure, or a density relative to a specified reference measure, on histories of a stochastic process. A formal Onsager--Machlup path probability must be interpreted with its discretization and initial-state conditioning specified; conditioning on both endpoints produces a separately normalized bridge measure.

For a stochastic process $X_t$ and a fixed sample outcome $\omega$, its sample path is the function $t\mapsto X_t(\omega )$. Statements about continuous, càdlàg or Hölder continuous function sample paths concern these functions simultaneously in time; they differ from assertions made only at each fixed deterministic time. A probability-one path property can be checked using countable dense times when the appropriate continuity is available.

Spot volatility is the instantaneous diffusion coefficient multiplying relative price increments in a stochastic differential equation for an asset. In a stochastic volatility model it is itself a stochastic process.

A stock is a traded ownership interest in a company. In a mathematical finance model, its price is a stochastic process and a self-financing portfolio may hold positive or negative numbers of its shares.

The stopped stochastic process follows $X$ until the stopping time $\tau$ and thereafter keeps the value $X_{\tau }$ when $\tau$ is finite. For a pathwise right-continuous adapted process, the stopped process is adapted, by progressive measurability and measurable evaluation at $t\wedge \tau$. When $X$ is a martingale, further stopping and integrability conditions determine whether its stopped process is a martingale.