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Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. Sometimes, we define quantity such as $\Delta f(W_t)$ (e.g. in this lecture, where one defines a Martingale) . My question is that: a) What does the notation mean? b) strictly speaking, with the notation as it is, doesn't the notation imply the use of chain rule (i.e. that we also need to take the derivative of a Brownian motion with respect to $t$, which isn't defined)?

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it means $(\Delta f)|_{W_t}$ where the meaning of the vertical bat is 'at the point'. So you first consider $f$ as a function of, say $x$ and taking sum of its second pure derivatives and substitute $W_t$ as an argument of the resulting function, i.e. if $$f(x,y) = \sin x +\cos y$$ then $$\Delta f(W^1_t,W^2_t) = -\sin W^1_t-\cos W^2_t$$ –  Ilya Jan 25 '12 at 17:50

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Since $\Delta f$ is a real valued function defined on $\mathbb R^d$ and $W_t$ is an $\mathbb R^d$-valued random variable, $\Delta f(W_t)$ is a real valued random variable.

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