- #1

td21

Gold Member

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- 8

a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second law. So my question is: Is stochastic differential equation simply a first order approximation to the actual stochastic process? Or it is the reverse, the actual stochastic process obeys stochastic differential equation even with finite time interval?

b) Also, in stochastic differential equation, we define the stochastic random variable up to the first order of $$dt$$ only. Let's say $$dp = \frac{0.05}{t}dt$$ and $$\frac{0.05}{t}$$ is the PDF. In a finite time interval, i say the probability of such events occur for probability $$0.05\ln\Delta t$$.

However, it is incomplete. PDF does not say about dependence in second or later order. In actual stochastic process where the probabilty of such event occurs in this interval, for example, maybe

$$0.05\ln\Delta t + 0.00005(\Delta t)^3$$ (unrealistic probabilty example as may blow up). Therefore, can we say that stochastic differential equation and PDF does not predict well for finite $$\Delta t$$ and we should always go back to random process itself and understand that stochastic differential equation only describe things well for very small integration step?