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Shown below is a snippet from the paper Arbitrage-free SVI volatility surfaces by Jim Gatheral and Antoine Jacquier (2013) (https://arxiv.org/pdf/1204.0646.pdf) .

The formulae shown below are on page 12, Theorem 4.1.

Is the first line basically saying "The partial derivative of theta with respect to t is always greater than equal to zero"?

In the second line what is the middle condition? Is that "Partial derivative of (theta * phi(theta) with respect to theta"?

Can somebody with math background please explain the notation to me:

  • $\partial_t \theta_t \ge 0$ for all $t \ge 0$;
  • $0 \le \partial_\theta (\theta \varphi(\theta)) \le \frac{1}{\rho^2}(1+\sqrt{1-\rho^2} )\varphi(\theta)$.
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  • $\begingroup$ edit the question with a link to or title of the paper might help, but it would be better if you add a list of what the paper's own definitions are for the variables displayed here, since the other symbols requested depend on the context of those variables' definitions $\endgroup$
    – develarist
    Sep 23, 2020 at 19:02
  • $\begingroup$ Please reopen my question. Thanks. $\endgroup$
    – KTC
    Sep 23, 2020 at 23:42
  • $\begingroup$ Reopened but please improve your presentation further using LaTeX. $\endgroup$
    – Bob Jansen
    Sep 24, 2020 at 7:42
  • $\begingroup$ In what page are these equations in the paper? What is $\theta$ in the second equation? is it a real number? so in the first equation $\theta$ is a function and in the second a number? $\endgroup$
    – user39119
    Sep 24, 2020 at 11:10

1 Answer 1

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From the words that follow (or precede) these equations in the paper it seems that your interpretation is correct. $\partial_t \theta(t)$ is just an abbreviation for $\frac{\partial \theta_t}{\partial t}$. Both should be read as "the partial [derivative] of theta t with respect to t". This usage is common in Stochastic Calculus and the author has decided to use the same notation for ordinary calculus. (Although slightly non-standard it does reduce the amount of writing you have to do, and is especially convenient when you are at the blackboard, speaking and writing at the same time).

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  • $\begingroup$ Thanks for editing the post noob2. Now I know how to use LaTex. And also, thanks for the answer. $\endgroup$
    – KTC
    Sep 24, 2020 at 13:53
  • $\begingroup$ When you mean such notation is common in stochastic calculus - do you mean expressions like $dW_t$? $\endgroup$ Sep 26, 2020 at 19:56
  • $\begingroup$ Yes, never a fraction like $\frac{dy}{dx}$ with two differentials one on top of the other, always everything written on one line with $dW_t,dX_t,dt$ etc. appearing by themselves. $\endgroup$
    – nbbo2
    Sep 26, 2020 at 21:29

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