Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)?

share|cite|improve this question
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.