Itô formula and diffusion models

Write dX=b(t,X)dt+sigma(t,X)dB as shorthand for an ordinary drift integral plus an Itô integral, with an initial condition. A solution is an adapted process satisfying that integral identity, not a differentiable function whose velocity includes a random derivative of Brownian motion. Standard sufficient existence and uniqueness assumptions include global Lipschitz bounds in the state variable and linear growth, with an appropriately integrable initial value. These are useful sufficient conditions, not necessary conditions for every solvable model.

Local Lipschitz conditions can give a solution up to an explosion time, so a separate nonexplosion argument may be needed. Coefficients that grow too rapidly or are discontinuous require specialized results rather than automatic application of the easiest theorem. A strong solution uses a specified Brownian motion and filtration; a weak solution allows the probability space and driver to be part of the construction. Here we work with well-behaved one-dimensional examples and explicit solutions, while keeping those notions separate.

For a function f(t,x) with one continuous time derivative and two continuous state derivatives, Itô's formula adds one-half sigma squared times the second state derivative to the ordinary drift chain rule. The first derivative multiplies both drift and noise. A Taylor expansion suggests the origin: Brownian increments are of square-root time size, so their squares contribute at time order. Higher-order terms vanish under the regularity and localization conditions behind the theorem. The correction is not an optional fitting parameter.

Take X=B and f(x)=x². Ordinary calculus would predict d(B²)=2B dB, but Itô adds dt, giving the integral identity established in the previous lesson. For a linear function the second derivative vanishes, so there is no correction. This provides a fast diagnostic when transforming an SDE: determine whether the transformation is curved, identify the diffusion coefficient, and calculate the correction before simplifying the expression. A wrong sign often becomes visible through an inconsistent expected second moment.

For geometric Brownian motion dX=mu X dt+sigma X dB with positive initial value and constant coefficients, applying Itô to log X gives drift mu−sigma²/2 and noise coefficient sigma. The explicit exponential solution stays positive. Its mean grows at rate mu, while its log median grows at rate mu−sigma²/2. These are different observables. Calling the noise mean zero does not imply that every nonlinear statistic follows the deterministic equation, because curvature converts fluctuation into drift.

A numerical Euler–Maruyama step replaces the drift integral by b at the old state times h and the noise integral by sigma at the old state times √h Z, with independent standard normal Z. The square root is essential. For time-independent globally Lipschitz coefficients with linear growth and square-integrable initial data, the basic method has strong order one-half over a fixed time interval. Additional smoothness and moment conditions can yield weak order one for suitable regular observables. Neither claim licenses every unbounded payoff or unstable coefficient. Simulated paths have discretization error as well as Monte Carlo error, and these should be assessed separately.