Are Stock Market graphs differentiable?

Question

I read investing in Stock Market was about predicting the future of a graph.

Suppose we know the value of the graph at all points before $a$ and also at $a$. Then one could use a very small number $h$ to calculate the approximate left hand derivative at $a$ by using the formula:

$$f'(a)=\frac{f(a-h)-f(a)}{-h}$$

Similarly, the successive left hand derivatives could be calculated by the formulas:

$$f''(a)=\frac{f(a-2h)-2f(a-h)+f(a)}{h^2}$$

I believe the general formula is:

$$f^n(a)=\frac{\sum_{r=0}^n (-1)^r\binom{n}{r}f(a-(n-r)h)}{(-h)^n}$$

Then one assumes all Left Hand Derivatives=Right Hand Derivatives to plot the approximate future of the graph by using the formula:

$$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+.....$$

Answer 1

Things are often modeled by Levy processes, which generally aren't even continuous much less differentiable. Even Brownian motion, the simplest case, is not differentiable (although continuous).

See http://faculty.baruch.cuny.edu/lwu/papers/handbooklevy.pdf for more on Levy processes in finance.

Answer 2

No because of the random aspect (Brownian motion) of it I believe its not continuous throughout and 'normal' calculus won't be enough. Double check what I'm saying but I think Ito calculus is needed to differentiate functions with a stochastic nature to them.