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The following approach of deriving Ito’s Formula is in the spirit of Evans handout ‘An introduction to stochastic differential equations’. His approach is intuitive at the cost of some omission of detail and precision.

The experiemently measured trajectories of systmes modeled by ordinary differential equation (ODE) do not always give give good prediction. The ODE looks like

$X^. (t) = b(X(t)) (t>0)$ with $X(0) = x_0$,

where $b: R^n\to R^n$ is a given, smooth vector field and the solution is the trajectory $X(\dot):[0,\infty)\to R^n.$

In reality, systems behave with some randomness. Hence, it is intuitive or reasonable to extend the ODE in some way to capture the random effects which disturbing the system.

Not finished post.