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I am confusing about the term $\sum_{ij}^{n} \frac{\partial^2 f }{\partial x_i \partial x_j} $ in the $L_0f(x(t),t)$ operator, Apparent in the Itô–Taylor expansion expression. Is the following formula is correct :

$\sum_{ij}^{n} \frac{\partial^2 f }{\partial x_i \partial x_j} = \sum_{i}^{n} \left[\sum_{j}^{n} \frac{\partial}{\partial x_i} \left( \frac{\partial f }{\partial x_j} \right) \right] = \left[ \frac{\partial}{\partial x_1} \left( \frac{\partial f }{\partial x_1} \right) + \cdots + \frac{\partial}{\partial x_1} \left( \frac{\partial f }{\partial x_n} \right) \right] + \left[ \frac{\partial}{\partial x_2} \left( \frac{\partial f }{\partial x_1} \right) + \cdots + \frac{\partial}{\partial x_2} \left( \frac{\partial f }{\partial x_n} \right) \right]+ \cdots + \left[ \frac{\partial}{\partial x_n} \left( \frac{\partial f }{\partial x_1} \right) + \cdots + \frac{\partial}{\partial x_n} \left( \frac{\partial f }{\partial x_n} \right) \right]$

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    $\sum_{ij}^{n}$ is a short-hand that your particular source is using, not a universally accepted notation. Without knowing that source, I can't say for sure what they mean by it, but $$\sum_{i=1}^n\sum_{j=1}^n$$ does seem the most likely. On the other hand, the partial derivatives are standard notation, and yes, that is what it means. – 2017-02-18
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    @ Paul Sinclair, the source : page 33 of https://users.aalto.fi/~ssarkka/course_s2012/pdf/handout4.pdf – 2017-02-18
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    That is only an overview of one lecture in a series. The notation is not defined in it. But I note that the $n$ superscript, which was the part I was least sure of since there were two index variables but only one limit, was your own invention, or at least is not used in that handout. The notation $$\sum_{i,j}$$ is very common for the double summation when the ranges of the indices are understood from the context. Dropping the comma is not a major variant. So yes, i can practically guarantee that this is the right interpretation. – 2017-02-18

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