Brownian Motion and Itô Calculus

Ordinary differential equations describe dynamics once the initial state and parameters are fixed. Random forcing requires a model for fluctuations that continue to enter as the system evolves. Brownian motion supplies the random increments in an SDE (Øksendal, 2003).

A stochastic process \((X_t)_{t\geq 0}\) is a collection of random variables indexed by time. For a fixed time \(t\), the random variable \(X_t\) describes the possible values of the system at that time. One complete realization \(t\mapsto X_t\) is called a sample path. A stochastic process therefore describes how uncertainty evolves over time and how random values at different times are related.

Brownian Motion

A standard Brownian motion, or Wiener process, \((W_t)_{t\geq 0}\) is a stochastic process with four defining properties:

1. \(W_0=0\).

2. Its sample paths are continuous with probability one.

3. Increments over disjoint time intervals are independent.

4. For \(0\leq s<t\), the increment satisfies

   \[ W_t-W_s\sim\mathcal{N}(0,t-s). \]

Consequently, \(\mathbb{E}[W_t]=0\) and \(\operatorname{Cov}(W_s,W_t)=\min(s,t)\). A Brownian increment over an interval of length \(\Delta t\) therefore has a typical size of \(\sqrt{\Delta t}\), much larger than the \(\Delta t\) scaling of an ordinary time increment. Brownian paths are continuous but nowhere differentiable with probability one, so \(dW_t/dt\) is not an ordinary function.

A random event is a yes-or-no statement about the path, such as whether the Brownian path has crossed a specified level by time \(t\). A sigma-algebra is the collection of events whose answers can be resolved from the available information. We write \(\mathcal{F}_t\) for the events that can be resolved by time \(t\). As time advances, information can be gained but not lost, so \(\mathcal{F}_s\subseteq\mathcal{F}_t\) whenever \(s\leq t\). The growing family \((\mathcal{F}_t)_{t\geq 0}\) is called a filtration.

The filtration keeps track of causality. A stochastic process \(H_t\) is adapted when its value at time \(t\) uses only information available by that time. We assume that future Brownian increments are independent of the events that can already be resolved. Thus a model may use the observed past, but it cannot use a fluctuation that has not yet occurred.

We will not define sigma-algebras rigorously because that requires measure theory. Measure theory is necessary to prove that the stochastic processes and integrals we construct are well defined. Our objective here is to develop a formal working understanding that enables engineers and scientists to use SDEs to model systems with randomness.

The Itô Integral

Consider a partition \(0=t_0<t_1<\cdots<t_n=T\). For a process that is constant on each interval and uses the value \(H_{t_k}\) known at its left endpoint, define

\[ \int_0^T H_t\,dW_t = \sum_{k=0}^{n-1}H_{t_k}\bigl(W_{t_{k+1}}-W_{t_k}\bigr). \]

The Itô integral extends this construction by a mean-square limit: as the time partition is refined, the expected squared approximation error tends to zero. We use integrands whose value over each new time interval is fixed using information available at its start and that satisfy

\[ \mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]<\infty. \]

Integrands that use only information available at the start of each interval are called predictable. For the simple process above, \(H_{t_k}\) is fixed before the next Brownian increment is revealed and is therefore independent of that increment. The construction gives the two identities used most often in modeling and computation:

\[ \mathbb{E}\!\left[\int_0^T H_t\,dW_t\right]=0, \]

and the Itô isometry

\[ \mathbb{E}\!\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \mathbb{E}\!\left[\int_0^T H_t^2\,dt\right]. \]

The second identity shows how the local diffusion amplitude accumulates into the variance of a stochastic trajectory.

Itô Stochastic Differential Equations

A scalar Itô SDE has the form

\[ dX_t=a(X_t,t)\,dt+b(X_t,t)\,dW_t, \]

where \(a\) is the drift and \(b\) is the diffusion. The drift gives the systematic local trend, while the diffusion controls the local variability. By definition, the differential notation denotes the integral equation

\[ X_t =X_0 +\int_0^t a(X_s,s)\,ds +\int_0^t b(X_s,s)\,dW_s. \]

Standard sufficient conditions bound how abruptly \(a\) and \(b\) change with the state and how quickly their magnitudes grow. Global Lipschitz continuity means that changing the state changes either coefficient by at most a fixed multiple of the state change. Linear growth bounds their magnitudes by a constant times \(1+|x|\). When these bounds hold uniformly in time and the coefficients are continuous in time, they ensure a unique adapted solution for a specified initial state and Brownian path.

The distinction between drift and diffusion matters in an inverse problem. Data can carry different information about a drift parameter, which affects the local mean, and a diffusion parameter, which affects the local variance.

Some physical models use the Stratonovich integral, denoted by \(\circ\,dW_t\). It is obtained from symmetric rather than left-endpoint sums and obeys the ordinary chain rule. For a scalar model,

\[ dX_t=a_{\mathrm{S}}(X_t,t)\,dt+b(X_t,t)\circ dW_t, \]

the equivalent Itô drift is

\[ a_{\mathrm{I}}(x,t) =a_{\mathrm{S}}(x,t)+\frac{1}{2}b(x,t)\frac{\partial b}{\partial x}(x,t). \]

The convention is therefore part of the model specification. The remainder of this chapter uses the Itô convention.

Itô’s Formula

An ordinary chain rule misses a contribution from the squared Brownian increments. An increment over \(\Delta t\) has size of order \(\sqrt{\Delta t}\), so its square has size of order \(\Delta t\). As we refine the partition, the sum of these squares approaches the elapsed time instead of vanishing. This accumulated quantity is called quadratic variation, and it is why a second-derivative term survives in the stochastic chain rule.

Let \(X_t\) solve the scalar Itô SDE above, and let \(\phi(x,t)\) have one continuous time derivative and two continuous state derivatives. Itô’s formula gives

\[\begin{split} \begin{aligned} d\phi(X_t,t) ={}&\left( \frac{\partial\phi}{\partial t} +a\frac{\partial\phi}{\partial x} +\frac{1}{2}b^2\frac{\partial^2\phi}{\partial x^2} \right)(X_t,t)\,dt \\ &+\left(b\frac{\partial\phi}{\partial x}\right)(X_t,t)\,dW_t. \end{aligned} \end{split}\]

The second-derivative term follows from the quadratic-variation rule \((dW_t)^2=dt\). As a concrete example, geometric Brownian motion satisfies

\[ dX_t=\mu X_t\,dt+\sigma X_t\,dW_t, \qquad X_0>0. \]

Applying Itô’s formula to \(\phi(x)=\log x\) yields

\[ d\log X_t =\left(\mu-\frac{\sigma^2}{2}\right)dt+\sigma\,dW_t. \]

Thus the logarithm has constant diffusion and a drift correction of \(-\sigma^2/2\). The stochastic-exponential-growth example uses this transformation to obtain an explicit likelihood.

For a multidimensional state \(X_t\in\mathbb{R}^d\) driven by an \(m\)-dimensional Brownian motion, write

\[ dX_t=a(X_t,t)\,dt+B(X_t,t)\,dW_t, \]

where \(a\in\mathbb{R}^d\) and \(B\in\mathbb{R}^{d\times m}\). The quadratic covariation of the state components is

\[ dX_{i,t}\,dX_{j,t} =\bigl(BB^{\mathsf{T}}\bigr)_{ij}(X_t,t)\,dt. \]

For a scalar function \(\phi:\mathbb{R}^d\times[0,T]\to\mathbb{R}\) with the same smoothness as above, the multidimensional Itô formula is

\[\begin{split} \begin{aligned} d\phi(X_t,t) ={}&\left[ \frac{\partial\phi}{\partial t} +\nabla\phi^{\mathsf{T}}a +\frac{1}{2}\operatorname{tr}\!\left( BB^{\mathsf{T}}\nabla^2\phi \right) \right](X_t,t)\,dt \\ &+\left(\nabla\phi^{\mathsf{T}}B\right)(X_t,t)\,dW_t. \end{aligned} \end{split}\]

The covariance factor is \(BB^{\mathsf{T}}\) because independent Brownian components satisfy \(dW_{k,t}\,dW_{\ell,t}=\delta_{k\ell}\,dt\).

Euler–Maruyama Simulation

Numerical inference requires a discrete transition model. For the scalar SDE above and a time grid \(t_{n+1}=t_n+\Delta t\), the Euler–Maruyama method takes the following form (Kloeden and Platen, 1992):

\[ X_{n+1} =X_n+a(X_n,t_n)\Delta t +b(X_n,t_n)\Delta W_n, \]

where the independent increments satisfy

\[ \Delta W_n\sim\mathcal{N}(0,\Delta t). \]

Equivalently, \(\Delta W_n=\sqrt{\Delta t}\,Z_n\) with \(Z_n\sim\mathcal{N}(0,1)\). The method resembles explicit Euler, but its random increment scales as \(\sqrt{\Delta t}\).

Suppose that the drift and diffusion depend on a parameter vector \(\theta\). Conditional on \(X_n=x_n\), one scalar Euler–Maruyama step with \(b_\theta(x_n,t_n)\neq 0\) has the Gaussian distribution

\[ X_{n+1}\mid X_n=x_n,\theta \sim \mathcal{N}\!\left( x_n+a_\theta(x_n,t_n)\Delta t, b_\theta^2(x_n,t_n)\Delta t \right). \]

For an observed state path \(x_{0:N}=(x_0,\ldots,x_N)\), the corresponding approximate conditional likelihood factors as

\[ p_\theta(x_{1:N}\mid x_0) =\prod_{n=0}^{N-1}p_\theta(x_{n+1}\mid x_n). \]

Scientific measurements often observe the state only indirectly. Let \(t_k\) be the observation times, \(x_k=X_{t_k}\) the latent state, \(y_k\) the corresponding measurement, and \(\theta\) the model parameters. Numerical propagation from \(t_{k-1}\) to \(t_k\) defines a transition distribution; when it has a density, denote it by \(p_\theta(x_k\mid x_{k-1})\). Together with an initial-state density \(p_\theta(x_0)\) and an observation density \(p_\theta(y_k\mid x_k)\), these distributions define a state-space model. Filtering and smoothing infer distributions over the latent states, while calibration methods infer unknown drift, diffusion, and observation parameters.

The Brownian-motion example first constructs and simulates the driving process. The stochastic-exponential-growth example applies Itô’s formula to a multiplicative-noise model. A companion notebook on the Ornstein–Uhlenbeck process studies linear mean reversion before the chapter turns to sequential state inference.