Itô diffusion

An Itô diffusion solves a stochastic differential equation $d{X}_t=b(X_t)dt+\sigma (X_t)d{B}_t$ driven by Brownian motion.

Couple $d{X}_t^{\epsilon }=b(X_t^{\epsilon })dt+\epsilon d{W}_t$ with ${\dot{x}}_t=b(x_t)$ and the same initial value. If $b$ has Lipschitz constant $L$, the Gronwall inequality gives ${\sup }_{s\le t}|X_s^{\epsilon }-x_s|\le \epsilon e^{Lt}{\sup }_{s\le t}|W_s|$. Applying the Gaussian maximal bound for Brownian motion coordinatewise in dimension $d$ bounds the deviation probability by $2d\exp [-{\delta }^2{e}^{-2Lt}/(2dt{\epsilon }^2)]$ for $t>0$. Matching initial values is essential.

The Itô formula identifies the operator of an SDE with drift $b$ and noise vector fields ${\sigma }_k$ as the displayed second-order operator. With bounded measurable coefficients, every solution satisfies the true martingale identity for $C_b^2$ test functions, hence is an L-diffusion in the martingale-problem sense.

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For deterministic $X_0=x$ and bounded continuous drift and volatility, the initial right derivative of the mean is $b(x)$ and that of the variance is $\sigma (x{)}^2$. The stochastic integral has mean zero and second moment ${\int }_0^t\mathbb{E}\sigma (X_s{)}^{2}ds$; the bounded drift contributes only higher-order terms to the variance.

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The neutral one-dimensional Wright–Fisher diffusion records a proportion in $[0,1]$, with absorbing endpoints, zero drift and generator $Lf(x)=x(1-x)f^{\prime \prime }(x)/2$. Its conditional variance rate is largest near one-half and vanishes at fixation. Mutation or selection variants add drift and must be specified separately.

An urn with $j$ red balls among $n$ is sampled independently $n$ times with replacement, and the next generation's red count is the number of red draws. Its proportion is a bounded discrete-time martingale. Accelerating generation time by $n$ gives a Wright–Fisher diffusion limit: the one-step variance of the proportion is $x(1-x)/n$, and the higher moments make Taylor remainders negligible.

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 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 time spent by a Itô diffusion in a measurable set $I$ up to time $T$ is $A_T(I)={\int }_0^T{1}_I(X_t)dt$. Under stationarity its expected value is $T\mu (I)$, where $\mu$ is the Invariant distribution of an Itô diffusion. Occupation times form the denominator of a local drift estimator.

A power diffusion has a power of its positive state as noise coefficient. Here the normalized zero-drift family is considered on $U=(0,\infty )$ with $X_0>0$. Its coefficients are locally Lipschitz continuous on that domain, so a maximal local solution of a stochastic differential equation is pathwise unique before its lifetime. The Lamperti transform reduces its noise coefficient to a constant; the case $\alpha =1$ is geometric Brownian motion. This normalized family is part of the constant-elasticity-of-variance models, but is not the whole model with arbitrary drift and scale.

For the positive-domain power diffusion with $0<\alpha \le 1$, assuming the lifetime is the limit of the hitting times of $1/n$, it is finite almost surely exactly when $\alpha <1$. For $\alpha <1$, the Itô formula givesThe negative drift bounds this positive process above by $X_0^{1-\alpha }+(1-\alpha )B$, whose first hit of zero is finite by recurrence of one-dimensional Brownian motion. The lifetime must precede that hit. At $\alpha =1$, $X_t=X_0{e}^{B_t-t/2}$ is positive and finite on every compact time interval, so the lifetime is infinite. Approaching zero only as $t\to \infty$ is not a finite boundary lifetime.

On an interval where $\sigma >0$ is continuously differentiable, define $F(x)={\int }^x\sigma (y{)}^{-1}dy$. The Itô formula transforms $dX=b(X)dt+\sigma (X)dB$ intoThe transformed Itô diffusion has constant noise coefficient. For a power diffusion $\sigma (x)=x^{\alpha }$, $F(x)=x^{1-\alpha }/(1-\alpha )$ when $\alpha \ne 1$, and $F(x)=\log x$ when $\alpha =1$. This simplifies comparison with Brownian motion and exposes the boundary drift.

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

For continuous observations of an Itô diffusion $d{X}_t=b(X_t)dt+\sigma (X_t)d{W}_t$, a local drift estimator at $x$ ison $\{A_T>0\}$. Its error is the average of $b(X_t)-b(x)$ over that window, plus $A_T^{-1}{\int }_0^T{1}_{\{|X_t-x|\le h\}}\sigma (X_t)d{W}_t$. A local Hölder continuity bound with exponent $\alpha$ bounds the first term by $L{h}^{\alpha }$.

The commonly used noise scale $(Th{)}^{-1/2}$ requires occupation control. For a stationary model with continuous positive density $\pi (x)$ of its invariant distribution and locally bounded $\sigma$, the Itô isometry makes the numerator's second moment $O(Th)$. If additionally $A_T/(2Th\pi (x))\to 1$ in probability in the chosen long-time, shrinking-window regime, division gives stochastic error $O_{\mathbb{P}}((Th{)}^{-1/2})$. Positivity of the invariant density alone does not establish that concentration of the random denominator.

Geometric Brownian motion has the explicit strong solution

The differential operator $L=\frac{1}{2}{x}^2{\partial }_{xx}+\frac{1}{2}x{\partial }_x-\frac{1}{2}$ has solution operator $P_{t}g(x)=e^{-t/2}\mathbb{E}[g(x{e}^{B_t})]$. For $x\ne 0$ its integral kernel with respect to Lebesgue measure is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t}{)}^{-1}\exp [-{\log }^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the measure $e^{-t/2}{\delta }_0$, not a density. The Itô formula applied to $e^{-s/2}u(t-s,X_s)$ proves this Feynman-Kac formula.