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Coauthor with Simo Särkkä of a textbook on applied stochastic differential equations, developing analytic solutions and computational methods for continuous-time random models.^[ref]

For a Brownian motion with constant diffusion coefficient $D>0$, conditioned to start at $x>0$ and end at $y>0$ after time $h>0$, its Brownian bridge hits zero during the interval with probabilityThe Brownian reflection principle gives the transition density killed at zero as $p_h(x,y)-p_h(x,-y)$. Divide by $p_h(x,y)$ to obtain the conditional survival probability, and use $p_h(x,-y)/p_h(x,y)=e^{-xy/(Dh)}$. A constant drift cancels upon conditioning on both endpoints. The formula does not apply unchanged to an arbitrary state-dependent Itô diffusion.

This supplies the missing absorption test between two positive endpoints in a numerical approximation of a stochastic differential equation; checking the endpoints alone can miss a barrier crossing.

For a maturity-differentiable Heath-Jarrow-Morton model forward field with the integrability needed for stochastic differentiation, the short rate $r_t=f(t,t)$ has the displayed dynamics. Indeed changing observation time uses the forward-rate stochastic differential equation, while changing maturity adds ${\partial }_{T}f(t,t)dt$. Under a risk-neutral measure, $\alpha (t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so $\alpha (t,t)=0$. The remaining drift depends on the diagonal slope of the whole forward curve, which need not be determined by $r_t$; a Markov process for the short rate is an additional modelling restriction. Without sufficient maturity regularity, the diagonal need not admit this classical Itô process formula.

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 $d{X}_t=\tanh X_{t}dt+d{W}_t$, $X_0=x$, has a global strong stochastic solution and pathwise uniqueness because its coefficients are globally Lipschitz continuous. The Itô formula gives the normalized positive martingaleBoundedness of $\tanh$ verifies Novikov's condition. Under $dQ=Z_{T}dP$, the Girsanov theorem makes $X_t-x$ a Brownian motion through time $T$. The factor $\cosh x$ ensures $Z_0=1$.

The transition density under the original measure isThis is a Doob h-transform with $h=\cosh$. Expanding the hyperbolic cosine gives a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$, showing directly that the density integrates to one.

The drift coefficient is the coefficient $b$ of $dt$ in a stochastic differential equation. It describes the local deterministic tendency of the process.

For any step $\Delta >0$, set $a=e^{-\theta \Delta }$. The Ornstein-Uhlenbeck transition distribution gives the exact updateusing a fresh independent standard normal random variable for each interval. The update reproduces the joint probability distribution on the selected time grid. It can start from a fixed value or use Ornstein-Uhlenbeck stationary initialization. On an equally spaced grid it is an autoregressive process of order one after subtracting $\mu$. Its stationary variance follows directly from $v=a^{2}v+v_{\infty }(1-a^2)$.

For the stipulated example with $\Delta =0.1$, $a=e^{-0.1}\approx 0.90484$ and the innovation variance is $1-e^{-0.2}\approx 0.18127$. Given $X_t=6$, the next conditional expected value is about $5.61935$ and the conditional standard deviation is about $0.42576$.

The Euler-Maruyama method instead approximates the same stochastic differential equation byFor $0<\theta \Delta <2$ its stationary variance solves $v=(1-\theta \Delta {)}^{2}v+{\sigma }^2\Delta$, givingAt $\Delta =0.1$ in the example, this is $1/0.95\approx 1.05263$, about $5.26\%$ above the continuous model's stationary variance. If $\theta \Delta \ge 2$, this Euler recurrence has no finite stationary variance despite the stable continuous model. Exact transitions avoid this discretization error at sampled times; neither update alone determines the unresolved motion between samples.

For the displayed stochastic differential equation, the infinitesimal generator is $\mathcal{L}g=(f^2/2)(g^{\prime \prime }-g^{\prime })$, so $\mathcal{L}s=0$ and the Itô formula makes $e^{X_t}$ a positive local martingale. If $f$ is bounded, strictly positive, and globally Lipschitz, the equation has a unique global strong solution of a stochastic differential equation. For $a<x<b$ and $T_y=\inf \{t:X_t=y\}$, the Itô isometry for the bounded stopped exponential gives finite mean exit time: the bracket density $e^{2z}f(z{)}^2$ has a positive minimum on $[a,b]$. The optional stopping theorem then yields the boundary hitting probability from a diffusion scale functionTaking $a\to -\infty$ gives ${\mathbb{P}}_x(T_b<\infty )=e^{x-b}$ for $b>x$. Passing to strict exceedances and then to their limits shows that ${\sup }_{t\ge 0}{X}_t-x$ has the exponential distribution of rate one. The probability distribution of the maximum is independent of the particular $f$; changing the speed of motion preserves this scale function.

The Hasselmann stochastic climate model represents slowly responding climate variables driven by fast weather fluctuations as well as external forcing. Klaus Hasselmann's 1976 paper separates fast and slow variables, retains fluctuations lost by simple averaging, and explains why stabilizing climate feedbacks are needed to prevent indefinitely growing variability. Its general framework is broader than a single-temperature equation.^[ref]

Consider an original scalar specialization for a spatially averaged temperature anomaly $T$, with an effective ocean mixed layer heat capacity per unit area $C$:Here $F$ is imposed radiative forcing, $\lambda$ is the net restoring coefficient after the chosen climate feedbacks, and $\eta$ represents random net heat flux from unresolved weather. The anomaly energy per unit area is $CT$, so its tendency equals the net input. The weather term represents exchange with omitted degrees of freedom; it does not assert that an isolated ocean creates energy. $F$ and $\eta$ have units $\mathrm{W}{\mathrm{m}}^{-2}$, and $\lambda$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

First set $\eta =0$, start from $T(0)=0$, and switch to constant $F_0>0$. The integrating factor $e^{\lambda t/C}$ givesFor $C=2.0\times 10^8\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $\lambda =2.0\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and $F_0=4.0\mathrm{W}{\mathrm{m}}^{-2}$, the relaxation time is $10^8\mathrm{s}$, about $3.17$ years using 365.25 days per year. At $t=\tau$, $T=2(1-e^{-1})\mathrm{K}\simeq 1.264\mathrm{K}$, with eventual warming $2\mathrm{K}$. Doubling $C$ doubles the response time but leaves this equilibrium unchanged. This separates thermal inertia from climate sensitivity.

Now set $F=0$ and idealize the weather forcing as Gaussian white noise withThe Dirac delta function has inverse-time units, so $D$ has units $(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$. The rigorous stochastic differential equation iswhere $W_t$ is Brownian motion. This is an Ornstein-Uhlenbeck process. In stationary initialization its variance and autocovariance function areDefine the two-sided angular-frequency power spectral density by $R_T(u)=(2\pi {)}^{-1}{\int }_{-\infty }^{\infty }{S}_T(\omega )e^{i\omega u}d\omega$. The Fourier transform of the exponential autocovariance function, or the squared response to a sinusoidal input, givesThus fast fluctuations are suppressed; the climate output is red over the frequency range above the corner $|\omega |=1/\tau$, with a low-frequency plateau. This is a colored noise spectrum without a periodic driver. The corner is $f_c=1/(2\pi \tau )$ in cycles per second, not $1/\tau$. The continuous white input is only an approximation on times longer than weather's correlation time.

For the same $C,\lambda$ and $D=10^8(\mathrm{W}{\mathrm{m}}^{-2}{)}^2\mathrm{s}$, $v_{\infty }=0.25{\mathrm{K}}^2$ and the stationary standard deviation is $0.50\mathrm{K}$. From fixed $T(0)=0$, the variance grows as $v_{\infty }(1-e^{-2t/\tau })$, even when the forced expected value is predictable. Integrating $S_T/(2\pi )$ over all angular frequencies returns $D/(C\lambda )$, checking the normalization independently. If $\lambda =0$ the unforced variance instead grows as $2Dt/C^2$; if $\lambda <0$, deviations amplify. Neither case has the stationary spectrum above.

A red spectrum alone cannot establish attribution of recent climate change: internal variability, changing forcing and measurement effects must be distinguished. This scalar model supplies a calculable baseline for variability and persistence, not a universal spectrum for all climate timescales or a full attribution analysis.

A stochastic differential equation with drift and diffusion coefficients affine in its state. In the constant-coefficient additive-noise case $d{X}_t=(a{X}_t+b)dt+\sigma d{W}_t$, multiplying by $e^{-at}$ givesConsequently its conditional distribution is normal, with mean $e^{a\tau }x+b(e^{a\tau }-1)/a$ and variance ${\sigma }^2(e^{2a\tau }-1)/(2a)$ when $a\ne 0$. For $a=0$, these become $x+b\tau$ and ${\sigma }^2\tau$. Additive noise permits negative states; it differs from geometric Brownian motion.

A local solution is a locally defined stochastic process taking values in a specified open domain $U$ and satisfying the stochastic differential equation on every stopped interval before its lifetime. For $dX=b(X)dt+\sigma (X)dB$, the integral equation holds after each announcing stop, with the requisite local drift and noise integrability. A maximal local solution of a stochastic differential equation cannot be extended while remaining in $U$; a finite lifetime can be a boundary hit rather than divergence to infinity.

Matthias Heymann coauthored the geometric minimum action with Eric Vanden-Eijnden. Their algorithm minimizes a geometric form of the Freidlin--Wentzell action to calculate rare transition paths and the quasipotential. Examples include a bistable stochastic differential equation and a genetic-switch Markov jump process, showing how one variational principle applies to different stochastic models.^[ref]

For a scalar stochastic differential equation, the Milstein method adds the quadratic-variation correctionUnder standard smoothness assumptions it has strong order of convergence one and weak order of convergence one.

An Ornstein-Uhlenbeck process combines mean reversion with continuing random fluctuations. In the scalar model, its stochastic differential equation iswhere $W$ is a standard Brownian motion. The drift points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of mean reversion, while $\sigma$ sets the strength of the random forcing. We consider a fixed or normal initial value independent of future Brownian increments, giving a Gaussian process and a Markov process.

For a stipulated example, use $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the expected value relaxes as $2+4e^{-t}$ and the variance grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting variance is one. The sample paths keep fluctuating even after the expected value is close to two. The Ornstein-Uhlenbeck transition distribution describes this evolving uncertainty; Ornstein-Uhlenbeck stationary initialization instead starts with that limiting normal distribution already in place.

A physical motivation is the velocity of a particle undergoing Brownian motion: linear drag removes a persistent initial velocity, while random forcing continually renews it. The original calculation gives a normal distribution for the velocity after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's mass.^[ref] A position coordinate in an overdamped harmonic optical trap gives another interpretation of the same scalar equation, with $\theta =k/\zeta$ for spring constant $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in thermal calibration of an optical trap.

The linear stochastic differential equation has the explicit solution $Y_t=e^{-\lambda t}{Y}_0+\sigma {\int }_0^t{e}^{-\lambda (t-s)}d{W}_s$. Its integrating factor is $e^{\lambda t}$. For $\lambda >0$, an independent initial normal variable of variance ${\sigma }^2/(2\lambda )$ makes its zero-mean Gaussian process stationary.

Let $f(y)=|y|$ away from zero. Its first and second partial derivatives areThus its Laplace operator is $\Delta f(y)=(n-1)/|y|$. The quadratic covariations of the independent Brownian coordinates satisfy $d[B_j,B_k{]}_s={\delta }_{jk}ds$. The stopped Itô formula therefore givesFor $t<\tau$, this is the required integral representation.

To identify the driving Brownian motion, the stopped stochastic term alone is insufficient: its quadratic variation is $t\wedge \tau$. Extend its unit direction after exit. Define a vector predictable process $u(s)$ byPredictability follows from the continuous adapted direction before stopping and the left-continuous adapted indicator $1_{\{s\le \tau \}}$. This $W$ is a zero-starting continuous martingale, andThe Lévy characterization of Brownian motion identifies $W$ as a one-dimensional Brownian motion. Before $\tau$, the preceding representation becomes the stochastic differential equation of the dimension-$n$ Bessel process:The fully stopped equation has both integrals stopped at $\tau$.

For $n=2$, apply the one-dimensional Itô formula to the logarithm. Before $\tau$, $d[X{]}_t=dt$, so the two drift terms cancel:Consequently,Indeed $X_s\ge r$ before stopping, so the Itô isometry bounds the stochastic term's second moment by $t/r^2$, and $\log X_{t\wedge \tau }\in [\log r,\log R]$. This is the martingale assertion under the paper's standing stopping assumption. Without stopping, the same calculation establishes only a local martingale; the stopped qualification is essential for the stated argument.

The integrated stochastic differential equation readsThe Gaussian white noise has zero expected value, soIts Dirac delta covariance givesThus $\boxed{⟨[y(t)-y(0){]}^2⟩=\Gamma t}$, which is also the variance because the displacement has zero expected value. More generally its covariance is $⟨Y(s)Y(t)⟩=\Gamma \min (s,t)$. Equivalently $Y(t)=\sqrt{\Gamma }{W}_t$ for standard Brownian motion, and the diffusion coefficient is $D=\Gamma /2$.

On a filtered probability space carrying a standard Brownian motion $W$ with independent increments relative to the filtration, a strictly positive security price follows geometric Brownian motion with constant drift $\mu$ and volatility $\sigma$ if it solves the stochastic differential equationHere $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and has the normal distribution $N(0,t-s)$ for $s<t$. The solution isIndeed applying the Itô formula to this exponential gives the specified drift and volatility. Consequently the price is positive and continuous, andThus disjoint log-return increments are independent and stationary. The parameter $\mu$ is the instantaneous expected relative return under the physical probability measure, not the mean log-return. Under the pricing risk-neutral measure it becomes the risk-free rate for a non-dividend-paying stock.

Separate $L$ into a diffusion generator and the constant killing term $-1/2$. The appropriate stochastic differential equation and its explicit geometric Brownian motion solution areThe second-order term in the Itô formula cancels the drift of $\log |X|$ when $x\ne 0$; if $x=0$, the solution stays zero.

Fix a terminal time $t$. Apply the Itô formula to $e^{-s/2}u(t-s,X_s)$ for $0\le s\le t$. Its drift isAfter localization this is a local martingale, and it is bounded because $u$ is bounded. It is therefore a true martingale. Taking expectations at its endpoints gives the Feynman-Kac formula

For $t>0$ and $x\ne 0$, the change of variable $y=x{e}^z$ in the normal density produces the killed geometric Brownian heat kernel with respect to Lebesgue measure:For $xy>0$, an equivalent form isIn particular,The apparent missing $1/y$ in this last expression is absorbed into the completed square, together with the killing factor; it is not a transcription error. The total mass is $e^{-t/2}$, as expected for killing at rate $1/2$. At $x=0$ the fundamental kernel is instead the measure $e^{-t/2}{\delta }_0(dy)$; it has no Radon-Nikodym derivative with respect to Lebesgue measure. Thus $u(t,0)=e^{-t/2}g(0)$. At $t=0$ the kernel is ${\delta }_x$, and continuity of bounded $g$ gives the initial condition. This also specifies the fundamental solution at the degenerate point omitted by a density-only formula.

Differentiate the material derivative of the passive scalar component by component. With ${\sigma }_{ij}={\partial }_j{u}_i$,The chain rule along a Lagrangian trajectory $\dot{X}=u(X,t)$ therefore gives the covector evolution of a material scalar gradientThe transpose matters: a gradient is transported as a covector, rather than as a material separation vector. When $u_3=0$, the third row of $\sigma$ vanishes, andLet $k_h=(k_1,k_2{)}^T$ and setFor $c>0$, the matrix exponential gives the exact horizontal solutionThe two projections are onto the eigenspaces with eigenvalues $-c$ and $+c$. If $p_{-}\ne 0$, write $p_{-}=k_0(\cos \varphi ,\sin \varphi {)}^T$ with $k_0=|p_{-}|>0$. ThenThe stated growing asymptotic requires this nonzero projection. An initial gradient entirely in the other eigenspace decays, and when $c=0$ the horizontal gradient is constant.

For constant vertical shear, write $h=({\sigma }_{13},{\sigma }_{23}{)}^T$. Integrating the third component using the exact horizontal solution yieldsConsequently, if $D=h\cdot (\cos \varphi ,\sin \varphi )$ is nonzero, $k_3\sim -k_{0}D{e}^{ct}/c$, andIf $D=0$, exponential growth of $k_3$ is absent, but the ratio still tends to zero on the growing horizontal branch. Thus exponential vertical growth is generic, not unconditional. The scalar-gradient inclination in a steady planar strain is the tangent of the angle between the scalar's normal and the horizontal plane. Equivalently, it is the cotangent of the inclination of an isoconcentration surface to that plane. Large $\alpha$ corresponds to nearly horizontal scalar surfaces; small $\alpha$ corresponds to nearly vertical ones.

For the probability calculation, take $\Gamma ,\Lambda >0$. The independent random variables $a,b$ with normal distributions of variance ${\Gamma }^2$ make $c$ distributed according to the Rayleigh distribution, withThe advertised Gaussian-over-Rayleigh inclination distribution also needs $D$ to be independent of $(a,b)$; its Gaussian marginal alone does not imply this. Under that intended assumption, for $A>0$,The cumulative distribution function is zero for $A\le 0$. Differentiating givesand the probability density function is zero for negative $A$. Its integral is one because the cumulative distribution function tends to one at infinity. Independence is a real hypothesis: $D=(\Lambda /\Gamma )a$ has exactly the stated Gaussian marginal but makes $\alpha \le \Lambda /\Gamma$ almost surely, contradicting the unbounded support of this density. A sufficient physical model is isotropic Gaussian vertical shear, independent of $a,b$, whose projection in any horizontal direction has variance ${\Lambda }^2$.

If the strain varies slowly, the horizontal gradient must align with the current growing direction, and the vertical-to-horizontal ratio must adjust before that strain changes. Both adjustment times are of order $c^{-1}$. Thus the local condition is $c{\tau }_{\sigma }\gg 1$, and the typical condition for the specified statistics isThis is an adiabatic approximation, not a uniform statement about arbitrarily weak instantaneous strains: samples with $c{\tau }_{\sigma }\lesssim 1$ retain memory of earlier strain. The instantaneous eigenvector must likewise turn slowly compared with $c$.

For rapidly fluctuating vertical shear, insert the large-time horizontal solution into its stochastic differential equation:The linear combination $W=\cos \varphi W^{(1)}+\sin \varphi W^{(2)}$ is a Brownian motion: it has independent Gaussian increments and increment variance $({\cos }^2\varphi +{\sin }^2\varphi )dt=dt$. For the signed inclination $\beta =k_3{e}^{-ct}/k_0$, the Itô product rule introduces no extra cross term because the exponential factor is deterministic. It follows thatThis is an Ornstein-Uhlenbeck process, explaining why white-noise vertical shear gives an Ornstein-Uhlenbeck inclination. Its Fokker-Planck equation isIn a stationary state the Fokker-Planck probability current $j=-cBp-(g^2/2)p^{\prime }$ is constant. A nonzero constant current cannot give an integrable, nonnegative density at both ends of the real line, so $j=0$. Integrating $p^{\prime }/p=-2cB/g^2$ and normalizing with the Gaussian integral givesThe stationary variance is $g^2/(2c)$. More explicitly the explicit Ornstein-Uhlenbeck solution has mean $\beta (0)e^{-ct}$ and variance $g^2(1-e^{-2ct})/(2c)$, confirming convergence to this density. Here $c>0$ and $g\ne 0$ are needed for the stated smooth stationary density; at $c=0$ there is no confinement, while $g=0$ gives a point mass at zero.