= Itô process
{c}
{title2=$X_t=X_0+\int_0^t b_sds+\int_0^t\sigma_s dW_s$}
An Itô process has the representation
$$
X_t=X_0+\int_0^t b_sds+\int_0^t\sigma_s\,dW_s,
$$
where $W$ is a <Brownian motion>, $X_0$ is $\mathcal F_0$-measurable, and $b,\sigma$ are <progressively measurable processes> satisfying $\int_0^t(|b_s|+\sigma_s^2)ds<\infty$ <almost surely> on every finite horizon. A <predictable> version of $\sigma$ may equivalently be used for the <Itô integral>. The process is a <continuous semimartingale>; the coefficients may depend on its whole past.
An <Itô diffusion> usually specifies the coefficients as functions of time and the current state. A continuously differentiable deterministic function is an Itô process with zero noise coefficient. For $X_t=\mu t+\sigma W_t$, its <quadratic variation> is $\sigma^2t$, and the <Itô formula> determines how a smooth function of $X$ evolves.
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