dWt

dWt is most commonly encountered as the differential dW_t, denoting the infinitesimal increment of a standard Wiener process (Brownian motion) at time t. The Wiener process W_t is a continuous-time stochastic process with W_0 = 0, independent and normally distributed increments, and Var(W_t − W_s) = t − s for t > s. The differential dW_t represents an infinitesimal random change over an infinitesimal time interval dt.

In stochastic calculus, dW_t has expectation zero and variance dt, and satisfies the informal rule dW_t^2 = dt. It is used to build stochastic integrals and functions of stochastic processes through Ito calculus. For an adapted process X_t, integrals with respect to W_t are written ∫ b(X_t,t) dW_t, and stochastic differential equations take the form dX_t = a(X_t,t) dt + b(X_t,t) dW_t. The term dW_t is not a traditional derivative; it is a differential in the sense of stochastic processes, and algebra with dW_t follows Ito rules rather than ordinary calculus.

A common application is in financial mathematics, where asset price dynamics are modeled by dS_t = μ S_t dt + σ S_t dW_t. Here the diffusion term σ S_t dW_t captures random fluctuations, while the drift term μ S_t dt accounts for systematic growth. The dW_t notation can also appear in various textbooks and software when describing Brownian motion or stochastic differential equations.