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Conditionally on an ${\mathrm{SLE}}_{\kappa }$ initial segment through time $t$, mapping out that segment by $g_t-U_t$ turns the future into an independent ${\mathrm{SLE}}_{\kappa }$ in $(\mathbb{H},0,\infty )$. This follows because its driving function is $U_{t+s}-U_t$, and Brownian motion has stationary increments and independent increments.

When the market price of risk is constant and nonzero spot volatility gives access to the driving Brownian motion, changes in volatility only rescale the dollar holding needed for fixed Brownian portfolio exposures. With unrestricted holdings and constant interest and discount rates, the CRRA utility value is consequently independent of the factor state. Zero volatility requires separate treatment.

A real Lévy process with continuous paths is Brownian motion with deterministic linear drift: $X_t=bt+\sigma W_t$. Continuity forces the jump measure in the Lévy–Khintchine formula to vanish, leaving characteristic function $\exp (t(ibu-{\sigma }^2{u}^2/2))$. If its law is also invariant under $X_t\mapsto {\lambda }^{-1}{X}_{{\lambda }^{2}t}$ for every $\lambda >0$, then $b=0$. The zero-variance case is included.

Let $M$ be a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$. Forthe process $B_s=M_{{\tau }_s}$ is Brownian motion andWhen $[M{]}_{\infty }<\infty$, one can enlarge the probability space and continue $B$ independently beyond that time.

The Dickey–Fuller test detects a unit-root autoregressive process against a stationary causal alternative. Regress $\Delta X_t$ on $X_{t-1}$, with deterministic terms appropriate to the model. Under the null, the usual regression statistic has a nonnormal Brownian motion functional limit, so ordinary normal critical values are inappropriate.

The coefficient $\sigma$ multiplying the Brownian motion increment in a scalar stochastic differential equation. The local variance rate is ${\sigma }^2$, and the second-order coefficient in the backward Kolmogorov backward equation is ${\sigma }^2/2$. This distinguishes the amplitude from the physical diffusion coefficient convention $D={\sigma }^2/2$.

The scalar stochastic differential equation $d{X}_t=\tanh X_{t}dt+d{W}_t$ has a pathwise unique global strong solution, since its coefficients are globally Lipschitz. The positive martingale $e^{t/2}/\cosh X_t$ gives a Girsanov theorem change of measure under which $X_t-X_0$ is Brownian motion. For $X_0=x$, its transition density isThis is a Doob h-transform with $h=\cosh$, and a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$.

If $f$ is a continuous probability density on ${\mathbb{R}}^2$ and $B$ is planar Brownian motion, thenalmost surely. The density is bounded below on some disc, recurrence supplies infinitely many visits to a smaller concentric disc, and the Strong Markov property gives infinitely many visits of a fixed positive duration.

Conditionally on the entire past up to $t$, the future domains mapped and centered by $g_t-U_t$ have the original chain law and are independent of that past. For Schramm–Loewner evolution this follows from independent stationary Brownian motion increments and the composition rule for chordal Loewner driving functions. The Strong Markov property gives the version at almost surely finite stopping times.

A dyadic interval is one cell in a dyadic partition. Its children are the left and right half intervals. Disjoint dyadic intervals at a fixed level produce orthogonal indicator functions and independent Brownian motion increments.

On $[0,1]$, the sum $Q_n$ of squared adjacent dyadic increments of standard Brownian motion has $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Therefore $Q_n\to 1$ in $L^2$. On an interval of length $t$, scaling gives mean $t$ and variance $2t^2{2}^{-n}$. Independence is used within each grid, not between different grids.

For a suitable twice differentiable function $u$ and an integrable stopping time $T$,

For every $\theta \in \mathbb{R}$ and standard Brownian motion $B$,is a positive martingale. Applying Itô formula cancels the drift, and its expectation is one by the moment-generating function of the normal distribution.

For $dX=\mu dt+\sigma dB$ on $[0,T]$, take $\theta =(\nu -\mu )/\sigma$ and use density $\mathcal{E}(\int \theta dB{)}_T$. If $\mu ,\nu$ are bounded and $\sigma$ is continuous and bounded below by a positive constant, the exponential has bounded bracket and defines an equivalent measure. Under it, $\tilde{B}=B-\int \theta dt$ is Brownian and $dX=\nu dt+\sigma d\tilde{B}$. Set $\theta =0$ after $T$ to define the density on the entire original sigma-algebra and the new Brownian motion on the whole time axis.

For standard Brownian motion, set $H_a=\inf \{t:B_t\ge a\}$ and $T_a=\inf \{t:B_t>a\}$. The Strong Markov property and immediate positive excursions give $T_a=H_a$ almost surely for each fixed $a\ge 0$. However, these level-indexed processes are not indistinguishable stochastic processes. Almost surely the random level $A={\sup }_{t\le 1}{B}_t$ is positive and exceeds $B_1$. It is first attained before time one but is not exceeded until after time one, so $H_A<1<T_A$. The simultaneous-equality event therefore has probability zero. The strict passage process $T_a$ is the right-continuous choice needed for a subordinator.

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.

Every centered continuous Gaussian martingale starting at zero is a deterministic time change of Brownian motion. Its clock is the continuous increasing function $f(t)=\mathbb{E}[M_t^2]$, and $[M{]}_t=f(t)$.

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

A centered Gaussian process with almost surely continuous paths and covariance $\mathbb{E}[B_s{B}_t]=\min (s,t)$ is a standard Brownian motion.

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a Dirac delta function in time, and possibly in space. It is the formal time derivative of Brownian motion and has independent fluctuations on disjoint time intervals.