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If the displayed covariance is finite and the mean obeys $m^{\prime }(x)+A(x)=0$, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty )$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary Ornstein-Uhlenbeck processes.

A stochastic convolution propagates random forcing through a linear evolution. Let $S(t)$ be a strongly continuous semigroup on a Hilbert space $H$, let $W$ be a Cylindrical Wiener process on a Hilbert space $U$, and let $Q:U\to H$ be a bounded linear operator. The expressionis an Itô integral whenever its integrand is square integrable in the Hilbert-Schmidt operator norm. In particular,This Itô isometry tests whether the accumulated forcing is actually $H$-valued. The factor $S(t-s)$ evolves the contribution arriving at time $s$ through the remaining time $t-s$. For $du=Audt+Qd{W}_t$ with deterministic initial value $u_0$, the mild solution is $u(t)=S(t)u_0+Z(t)$.^[ref]

For $S(t)=e^{-at}$, $Q=\sigma$ and real Brownian motion $B$, with $a>0$,This is the zero-initial-value Ornstein-Uhlenbeck process: damping limits the variance despite continued forcing. With $a=0$, the limit is ${\sigma }^{2}t$ instead. These formulas follow by integrating ${\sigma }^2{e}^{-2a(t-s)}$ in the Itô isometry.

The word convolution describes the dependence on the time difference $t-s$. It does not mean an ordinary integral against a differentiable noise function; Brownian motion requires a stochastic integral.

A stochastic differential equation specifies a stochastic process through a combination of ordinary drift and random increments. In the scalar Itô integral convention, the notationmeans the stochastic integral equation $X_t=X_0+{\int }_0^{t}b(s,X_s)ds+{\int }_0^t\sigma (s,X_s)d{W}_s$, with the required measurability and local integrability. The driver $W$ is a Brownian motion; $b$ is the drift coefficient and $\sigma$ the diffusion amplitude.

For example, the Ornstein-Uhlenbeck process satisfies $d{X}_t=-\lambda X_{t}dt+\sigma d{W}_t$: linear restoring drift competes with persistent noise. A strong solution of a stochastic differential equation uses a prescribed driver, whereas a weak solution of a stochastic differential equation may construct the probability space and driver. Pathwise uniqueness, uniqueness in law and the lifetime of a maximal local solution of a stochastic differential equation are separate questions. If the equation uses a Stratonovich integral, its convention changes the drift conversion and must be stated.

A stochastic volatility model gives an asset a separate random volatility or variance process. For example, a stock can have pricing dynamics $d{S}_t/S_t=rdt+\sqrt{v_t}d{W}_t$, with random $v_t$, instead of the constant variance of the Black-Scholes model. The Heston model uses a CIR model for $v$; an Ornstein-Uhlenbeck process can instead model a signed volatility whose square is the variance.

The extra state lets current uncertainty affect future uncertainty, and correlated asset and variance shocks change the distribution of future returns and option values. This differs from a local volatility model, where the coefficient is a deterministic function of time and current spot alone.

With two independent sources of uncertainty and only a stock and bank account, the variance risk is generally not fully hedgeable. A physical variance model then does not uniquely determine its pricing dynamics: a volatility-risk premium or additional traded instruments must be specified. Heston's original model makes the joint asset/variance system tractable through an affine characteristic-function calculation.^[ref]

Underdamped Langevin dynamics retains both position $q$ and momentum $p$. For mass $m>0$, temperature $T>0$, friction rate $\gamma >0$ and smooth potential energy $U$, one common normalization isHere $W$ is vector Brownian motion. The noise amplitude is tied to friction by the fluctuation-dissipation relation for a Langevin particle. If the density is normalizable and the dynamics is nonexplosive, its invariant distribution is the Boltzmann distributionThe Hamiltonian flow preserves the total energy and phase-space volume; the momentum Ornstein-Uhlenbeck process preserves a centered normal distribution with coordinate variance $m{k}_{B}T$. These two facts verify the invariant density. In dimensionless units with $k_{B}T=1$ and momentum variance $\eta$, it is $\exp [-U(q)-|p{|}^2/(2\eta )]$, with the corresponding choice $m=\eta$.

Retaining inertia resolves the short-time ballistic-to-diffusive Brownian displacement. The overdamped Langevin dynamics approximation removes momentum relaxation when only much longer times are relevant.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.

For $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$ with ${\sigma }_0=0$, the explicit stochastic integral is ${\sigma }_t=\sqrt{\kappa }{\int }_0^t{e}^{-(t-u)/\tau }d{W}_u$. The Itô isometry on the overlap of the two integration intervals gives the displayed covariance. The second exponential records the zero initial condition; removing it gives the stationary Ornstein-Uhlenbeck process covariance.