Connecting AR(1) Processes to Ornstein–Uhlenbeck Dynamics

Summary

The document asks whether the Ornstein–Uhlenbeck (OU) process is the continuous-time counterpart of a stationary AR(1) process. It observes that rewriting the discrete process in terms of changes produces a mean-reverting form resembling the OU stochastic differential equation, with the discrete coefficient related to the strength of reversion.

The post also asks how to translate an AR(1) coefficient, innovation variance, and sampling interval into OU parameters. The accepted response does not derive the mapping; it points to related discussions elsewhere. As a result, this is a useful statement of the modeling connection and conversion problem, but it supplies no derivation, numerical example, or validation. Readers should consult a full treatment before applying a parameter conversion, especially because the mapping depends on the time interval and on how the discrete innovations relate to continuous-time noise.

Key ideas

• Rewriting an AR(1) process as a change equation reveals a mean-reverting term.
• An OU process is a candidate continuous-time analogue of a stationary AR(1) model.
• Converting parameters requires accounting for the sampling interval and innovation variance.
• The document poses the conversion question but provides no derivation or worked answer.

Tags

# Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?


# Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?












I see the AR(1) process (with $|\alpha| < 1$) can be written in the following way: $$x_{t+1} = \alpha x_t + \epsilon_t$$ $$\Delta x_t = - (1 - \alpha) x_t + \epsilon_t$$ which looks quite like the formula of Ornstein–Uhlenbeck process without a drift term as $$dx_t = -\theta x_t + \sigma dW_t$$ then is OU process the continuous-time correspondence of AR(1) process?

Ignore the question below if it is not. If I have the value of parameters of AR(1) process, i.e. $\alpha$, variance of $\epsilon$, and time difference $\Delta_t$, how do I convert these parameters to the parameter of OU process?

## Answer by mark leeds (score 3, accepted)

https://quant.stackexchange.com/a/50499

This link looks very relevant to your question and probably an answer.

https://math.stackexchange.com/questions/345773/how-the-ornstein-uhlenbeck-process-can-be-considered-as-the-continuous-time-anal

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