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 ?
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2$\begingroup$ 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. $\endgroup$– nbbo2Oct 12, 2015 at 15:25
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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.
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1$\begingroup$ But why f'' does not get vanish in stochastic calculus ? $\endgroup$– NeerajOct 13, 2015 at 8:08
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1$\begingroup$ 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/… $\endgroup$– Richi WOct 13, 2015 at 8:56