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I work with stock price time series where I check for structural breaks in the series. To do that I fit simple models such as AR and ARIMA.
However, I was proposed to express the stock price in terms of not classic time series models but in terms of (system of), possibly stochastic, differential equations. The idea is that the model should be similar to Black&Scholes. However, the BS model is about option price, not stock price, so I cannot use it (plus, if I am not mistaken, the stock price is a Brownian motion that is a random variable, so cannot be predicted). I failed to find publications that show the details.

Can anyone suggest a way or literature regarding how time series can be expressed or forecasted using DE?

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    $\begingroup$ People do use "Black-Scholes model" to refer to the SDE for the stock price that implies the option pricing formulae derived by them. I don't know what you mean by "is a random variable so cannot be predicted"; there's nothing fundamentally different in that regard between that model and an ARIMA model. It's just that one is defined in discrete time and the other in continuous time, but it's only observed discretely either way. What are you trying to do with these models? Check for structural breaks? Forecast? What feature of the data do you hope to better capture? $\endgroup$
    – Chris Haug
    Jul 4, 2021 at 14:24
  • $\begingroup$ I'm trying to find a way of expressing a stock price time series using SDE. I agree that "cannot be predicted" is not the best choice of word. I meant that BS models stock price as a Brownian motion that is an approximation of real stock price. I'm trying to express stock price, not as a random normal variable. $\endgroup$
    – student
    Jul 4, 2021 at 15:41

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