# Difference between stochastic calculus and newton calculus

As I am not a student of hard core mathematics,I just want to know how stochastic calculus is different from newton calculus. What make stochastic calculus different from simple newton calculus ?

• Stochastic calculus deals with random things, newton's calculus deals with things having no randomness in them. A bullet flies a Newtonian trajectory, but a leaf that tumbles to the ground follows a random path. – noob2 Oct 12 '15 at 15:25

Talking about stochastic calculus in the sense of Ito the basic buidling block is a process with iid Gaussian increments called Brownian motion $(B_t)_{t \ge 0}$.
Then a basic observation that can be generalized in numerous ways is that for a bounded function $f$ it holds that $$f(B_T) = f(B_0) + \int_{0}^T f'(B_t) dB_t + 1/2 \int_{0}^T f''(B_t) dt,$$ where the definition of the integral with respect to Brownian motion is fundamental. Furthermore in usual calculus the $f''$ would not be present in the above equation. In stochastic calculus the second order derivative does not vanish. This is what pops up everywhere.
• Beause the quadratic variation of $B_t$ does not vanish but the limit is $t$ - time. That's why you get integration with respect to $t$ above. You could read Oksendal's book .. googeling I found a pdf link too: imcs.dvfu.ru/lib.int/NEW/Math/MV_Probability/… – Richard Oct 13 '15 at 8:56