About stochastic differential equation and probability density

[td21]

I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics.

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?

 

[andrewkirk]

For question (a) you first need to specify exactly which physical process and which SDE you are referring to. The answer may vary between different cases. But in general one should keep in mind that any mathematical model, including Newton's laws, is only ever a simplification of the physical process and hence in some sense an approximation.

To answer question (b) you need to specify a SDE. The equation you give is not an SDE because $t$, assuming it refers to time, is not a stochastic process. To be an SDE it needs to contain one or more components that are differentials of stochastic processes.

But take a classic stochastic differential equation like
$$\frac{dS}S=\mu_t\,dt+\sigma_t\, dB_t$$
where $B$ is a Brownian Motion. Such an equation is exact, given the model, not a first-order approximation (the model is an approximation to the physical process, but that's a different question). If one tries to solve the SDE numerically using small, finite steps $\delta t$ then one will typically be making a first-order approximation, but that issue is not specific to SDEs. The same thing happens if one uses numerical techniques to solve a non-stochastic DE.

Also, your references to pdfs are unclear. A pdf is a property of a random variable, not of a stochastic process, which is a collection of random variables. Unless the stochastic process - call it $X$ - is stationary, the pdf of the random variable $X_t$ that is the value of $X$ at time $t$, will vary with $t$.