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Why can we assume that interest rate is stationary (identically distributed), Gaussian (has multivariate normal distribution), Markovian (the future is determined only by the present), and continous in probability so it produce the Ornstein-Uhlenbeck process.

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Mostly these assumptions are being made for mathematical simplicity and tractability.

Non-Markovian processes are very difficult to work with. Gaussian processes are easy and convenient. Continuous time is a powerful assumption, although empirical data is usually in discrete time.

Second order stationarity is appropriate because interest rates vary over time both in terms of level and volatility, but it seems reasonable that the mean interest rate around which they vary is constant over time and not unreasonable to neglect the volatility changes as a first approximation. (For stock prices OTOH we usually assume a non-stationary process with an upward trend).

OU is the simplest interest rate model, it is by no means the best or only.

HJM interest rate models are non-Markovian.

GARCH models are widely used in finance to model changes in the second moment over time.

"Jump models" can be used to introduce discontinuities into financial processes.

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There are two ways to look at this question.

You can analyse the historical data to check whether it is in conformity with the stated assumptions around mean reversion and stationarity. This, by the nature of the real market prices/rates which are discrete, will be done in terms of the discrete analog of the OU process.

Alternatively you can derive the pricing formula, and then check how well does it fit the prices. It won’t be perfect as we already know but then you can compare its performance to alternative models and take a view whether any additional complexity is justified based on the desired application.

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