Itô processes
An Itô process is a continuous stochastic process whose change can be decomposed into a drift part and a Brownian-driven diffusion part.
\[\boxed{dX_t=a(X_t,t)\,dt+b(X_t,t)\,dW_t}.\]Why introduce the name “Itô process”?
The previous page introduced drift and diffusion. An Itô process is the broader mathematical object obtained when these local contributions are accumulated through continuous time.
The notation
\[dX_t=a(X_t,t)dt+b(X_t,t)dW_t\]looks like an ordinary differential equation, but it cannot be interpreted using ordinary derivatives because Brownian motion is nowhere differentiable.
What the differential notation really means
The precise meaning is the integral equation
\[\boxed{X_t=X_0+\int_0^t a(X_s,s)\,ds+\int_0^t b(X_s,s)\,dW_s}.\]This equation says that the value at time \(t\) equals the initial value plus all deterministic drift accumulated up to time \(t\), plus all stochastic diffusion accumulated up to time \(t\).
| Term | Meaning |
|---|---|
| \(X_0\) | initial state |
| \(\int_0^t a(X_s,s)ds\) | accumulated drift |
| \(\int_0^t b(X_s,s)dW_s\) | accumulated Brownian fluctuation |
Why the first integral is ordinary
The drift integral
\[\int_0^t a(X_s,s)\,ds\]is an ordinary time integral. It accumulates many small contributions of the form
\[a(X_s,s)\Delta t.\]If the drift is constant, \(a(X_t,t)=\mu\), then
\[\int_0^t\mu\,ds=\mu t.\]Why the second integral is different
The diffusion integral
\[\int_0^t b(X_s,s)\,dW_s\]cannot be defined by treating \(W_t\) as an ordinary differentiable function. Brownian paths have no ordinary derivative.
Instead, the Itô integral is built from sums involving Brownian increments:
\[\sum_k b(X_{t_k},t_k)\,[W(t_{k+1})-W(t_k)].\]As the time partition becomes finer, these sums converge—under suitable conditions—to the Itô stochastic integral.
Why use the left endpoint?
In the Itô construction, the coefficient multiplying each future Brownian increment is evaluated using information available at the beginning of that interval:
\[b(X_{t_k},t_k)\,\Delta W_k.\]This is important because the model should not use knowledge of a future random increment before that increment has occurred.
“Adapted” means no knowledge of the future
Mathematically, the process used inside an Itô integral must be adapted to the available information. Informally, this means its value at time \(t\) may depend on the present and past, but not on future Brownian motion.
This is one of the reasons Itô calculus fits causal stochastic models naturally.
A simple Itô process
Consider
\[\boxed{dX_t=\mu\,dt+\sigma\,dW_t}.\]Because both coefficients are constant, integrating gives
\[\boxed{X_t=X_0+\mu t+\sigma W_t}.\]This is perhaps the simplest non-trivial Itô process.
How the two accumulated parts combine
At each time,
\[\boxed{X_t-X_0=\underbrace{\mu t}_{\text{accumulated drift}}+\underbrace{\sigma W_t}_{\text{accumulated diffusion}}}.\]The stochastic contribution may be positive or negative and can temporarily dominate the drift.
Expectation and variance in the simple case
For
\[X_t=X_0+\mu t+\sigma W_t,\]and using \(E[W_t]=0\),
\[\boxed{E[X_t]=X_0+\mu t}.\]Since \(\operatorname{Var}(W_t)=t\),
\[\boxed{\operatorname{Var}(X_t)=\sigma^2t}.\]So the drift controls the mean while the diffusion controls how rapidly uncertainty spreads in this constant-coefficient model.
An Itô process is not generally Gaussian
The simple constant-coefficient process above is Gaussian. But a general Itô process
\[dX_t=a(X_t,t)dt+b(X_t,t)dW_t\]need not have a normal distribution, because the drift and diffusion coefficients may depend nonlinearly on the evolving random state.
State-dependent coefficients
For example,
\[dX_t=rX_t\left(1-\frac{X_t}{K}\right)dt+\sigma X_t\,dW_t\]has logistic drift and multiplicative diffusion. Both coefficients depend on the current state \(X_t\).
This means that when \(X_t\) changes, both the deterministic tendency and the strength of the random fluctuation can change.
Local view versus accumulated view
| Differential form | Integral form |
|---|---|
| \(dX_t=a\,dt+b\,dW_t\) | \(X_t=X_0+\int a\,ds+\int b\,dW\) |
| describes local change | describes total accumulated change |
| compact modelling notation | precise mathematical meaning |
Connection to Euler–Maruyama
The integral interpretation leads directly to numerical approximation. Over one short interval,
\[X_{n+1}-X_n\approx a(X_n,t_n)\Delta t+b(X_n,t_n)\Delta W_n,\]with
\[\Delta W_n=\sqrt{\Delta t}\,Z_n.\]Hence
\[\boxed{X_{n+1}=X_n+a(X_n,t_n)\Delta t+b(X_n,t_n)\sqrt{\Delta t}\,Z_n}.\]This is the Euler–Maruyama method.
Connection to Itô's formula
Once \(X_t\) is an Itô process, we often want to know how a function of it,
\[Y_t=f(X_t,t),\]changes through time.
Ordinary chain-rule calculus is no longer sufficient because Brownian increments have non-zero quadratic variation. The correct stochastic chain rule is Itô's formula.
Why ordinary calculus needs correction
In ordinary calculus, terms involving \((dx)^2\) vanish faster than \(dt\). For Brownian motion,
\[(dW_t)^2\sim dt.\]So second-order terms involving the diffusion coefficient survive. This is why Itô's formula contains an additional second-derivative term.
Biological interpretation
An Itô process can represent a biological quantity whose evolution combines systematic mechanisms and continuous stochastic fluctuations. For example, deterministic epidemic growth may form the drift while random fluctuations inherited from infection and recovery events may motivate the diffusion term.
Whether this approximation is appropriate depends on the biological scale. When individual discrete events are important, a CTMC may be preferable; when a continuous stochastic approximation is suitable, an Itô process can be useful.
What an Itô process is—and is not
| It is | It is not |
|---|---|
| a continuous stochastic process built from drift and Brownian diffusion | a deterministic differential equation with random-looking decoration |
| defined rigorously through ordinary and Itô integrals | based on an ordinary derivative \(dW_t/dt\) |
| able to have state-dependent drift and diffusion | necessarily Gaussian |
What comes next?
The next step is to understand the stochastic integral itself more carefully and then the stochastic chain rule, Itô's formula, which explains how functions of Itô processes evolve.