dSt

dSt is a notation you may encounter as a shorthand for the differential of a stochastic process S_t with respect to time t. In standard stochastic calculus, the differential is written as dS_t and is a central object in stochastic differential equations (SDEs). A common form is dS_t = μ(S_t,t) dt + σ(S_t,t) dW_t, where μ is the drift term, σ is the diffusion term, and W_t is a standard Brownian motion. The variable dW_t represents an increment of a Wiener process with mean zero and variance dt.

Interpretation and use: The term μ(S_t,t) dt describes a deterministic trend over a small interval dt, while σ(S_t,t) dW_t captures random fluctuations. Because dW_t is stochastic, dS_t is not a ordinary derivative; it is an infinitesimal change defined within Itô calculus. Stochastic differential equations model the evolution of S_t over time and are interpreted through stochastic integrals, with the solution typically written as S_T = S_0 + ∫_0^T μ(S_t,t) dt + ∫_0^T σ(S_t,t) dW_t.

Examples and applications: The geometric Brownian motion model, widely used in finance to describe stock prices, uses dS_t = μ S_t dt + σ S_t dW_t. Beyond finance, dS_t forms the backbone of models in physics, biology, and engineering where systems are affected by both deterministic dynamics and random noise.

Notation notes: While dS_t is standard, some texts or fonts may render it as dSt when subscripts are not easily formatted. The essential idea remains the same: dS_t represents the infinitesimal stochastic change of the process S_t over time. See also stochastic differential equations, Itô calculus, Brownian motion, and the Black-Scholes model.