Link Between Autoregressive and Ornstein-Uhlenbeck Processes

Autoregressive time series models and continuous time series models such as the ubiquitous Ornstein-Uhlenbeck (OU) process generally arise in different contexts. While the AR process is a staple of traditional time series modeling, the OU process is a widely used stochastic process in various fields, including finance, physics, and biology.
Often the connection between the two processes goes unremarked upon. In fact, the discrete-time AR model converges to the continuous-time OU process under certain conditions.
The discrete - time AR model can be written as :
X
t+1
=
ϕX
t
+
ϵ
t
where
X
t
is the time series at time t, ϕ is the autoregressive parameter and
ϵ
t
is the white noise error term with mean zero and variance
2
σ
.
The continuous - time OU process is defined as :
dX
t
=θ(μ-
X
t
)dt+
σdW
t
where
X
t
is the process at time
t
,
θ
is the rate of mean reversion, μ is the long term mean, σ is the volatility, t is an infinitesimally small time interval and

W
t
s a Wiener process (Brownian motion), representing the stochastic component of the process.
To demonstrate the correspondence between the two models, we need to show that under certain conditions, the AR model converges to the OU process as the time interval approaches zero.
To derive the connection, we start by taking the limit as the time interval Δt approaches zero, making the AR model a continuous - time process. The noise term
ϵ
t
will converge to 
W
t
, the Wiener process, under specific conditions.
The convergence is ensured if the following conditions hold :
◼
  • The AR parameter ϕ should be negative and the absolute value of ϕ should be less than 1 to ensure stability.
    • ◼
  • The variance of the noise term
    ϵ
    t
    ​ should be related to
    σ and Δt.
    • ◼
  • The mean of
    ϵ
    t
    ​ should be zero, which is typical in many AR models .
    • By taking the limit as Δt approaches zero, the discrete - time AR model converges to the continuous - time OU process :
    lim
    Δt->0
    X
    t+1
    -
    X
    t
    Δt
    =θ(μ-
    X
    t
    )+σ
    dW
    t
    dt
    We then equate the terms from the AR model and the OU process to establish the connection between ϕ and θ, and relate the variance of ​
    ϵ
    t
    to σ and Δt.
    Step1:RewritethediscreteARmodelusing
    ΔX
    t
    :
    ΔX
    t
    =
    X
    t+1
    -
    X
    t
    =-
    X
    t
    +
    ϵ
    t
    +
    ϕX
    t
    =(ϕ-1)
    X
    t
    +
    ϵ
    t
    Step 2 : Divide by Δt :
    ΔX
    t
    Δt
    =
    (ϕ-1)
    X
    t
    +
    ϵ
    t
    Δt
    Step 3 : Take the limit as Δt approaches zero:
    lim
    Δt0
    ΔX
    t
    Δt
    =
    lim
    Δt0
    (ϕ-1)
    X
    t
    Δt
    +
    lim
    Δt0
    ϵ
    t
    Δt
    Step 4 : Recognize that
    lim
    Δt->0
    ϵ
    t
    Δt
    ​ is equivalent to σ
    W
    t
    in the continuous - time OU process .
    Step 5 : Take the first limit:
    lim
    Δt0
    (ϕ-1)
    X
    t
    Δt
    =(ϕ-1)
    lim
    Δt0
    X
    t
    Δt
    Step 6 : Recognize that
    lim
    Δt0
    ΔX
    t
    Δt
    is equivalent to
    
    X
    t
    t
    in the continuous-time OU process.
    Step 7: Rewrite the limit:
    lim
    Δt0
    ΔX
    t
    Δt
    =(ϕ-1)
    
    X
    t
    t
    +σ
    dW
    t
    dt
    Step 8 : Equate the continuous - time OU process and the AR model:
    
    X
    t
    t
    =
    θ(μ-
    X
    t
    )
    (ϕ-1)
    This shows that as Δt approaches zero, the discrete AR model converges to the continuous - time OU process. The correspondence between the two is established, and we find that ϕ and θ are related by:
    θ=
    2
    σ
    ϕ-1
    Another, simpler approach is to discretize the OU process:
    X
    Δt+t
    -
    X
    t
    =Δtθ(μ-
    X
    t
    )+σ
    Δt
    ϵ
    t
    By comparing the two, we can relate the parameters :
    ϕ = 1 - θ tμ = 0 (for simplicity)​
    σ
    AR
    = σ
    Δt

    Illustration

    Let' s simulate paths for both processes using the following parameters :
    In[]:=
    (*Defineparameters*)​​phi=0.95;​​sigmaAR=0.05;​​dt=0.01;​​theta=(1-phi)/dt;​​sigmaOU=sigmaAR/Sqrt[dt];​​mu=0;​​T=100;​​steps=Round[T/dt];​​​​(*SimulateAR(1)Process*)​​arPath=NestList[phi#+sigmaARRandomReal[NormalDistribution[0,1]]&,0,steps];​​​​(*DiscretizeandSimulateOUProcess*)​​ouPath=NestList[#+theta(mu-#)dt+sigmaOUSqrt[dt]RandomReal[NormalDistribution[0,1]]&,0,steps];​​​​(*Plotthepaths*)​​ListLinePlot[{arPath,ouPath},PlotLegends->{"AR(1)","OU"},AxesLabel->{"Time","Value"}]​​
    Out[]=
    AR(1)
    OU

    Cumulative Paths

    The equivalence between the discrete AR (1) process and the continuous Ornstein - Uhlenbeck (OU) process is more of a theoretical one, particularly when considering infinitesimally small time steps in the OU process. However, when we discretize the OU process, as we' ve done, the two processes can be made to resemble each other over short intervals . Given the parameters we' re using:
    ϕ=0.95​​
    σ
    AR
    =0.05​​Δt=0.01​​θ=
    1-ϕ
    Δt
    ​​
    σ
    OU
    =
    σ
    AR
    Δt
    Here' s what' s happening :
    ◼
  • The AR (1) process is defined by its autoregressive parameter ϕ and the standard deviation of the noise
    σ
    AR
    .
    • ◼
  • The discretized OU process' s parameters are derived to match the AR (1) process over a one - step interval . Specifically, θ is chosen to give a similar autoregressive behavior to ϕ, and
    σ
    OU
    is adjusted to match the noise level.
    • Over short intervals (one - step changes), the two processes should be fairly similar. However, over longer intervals, the cumulative effects of their different dynamics will become evident, especially since the AR (1) lacks the mean - reverting property of the OU process.
    ​
    For the parameters we’ ve chosen :
    ◼
  • The AR (1) process has a strong persistence due to the high ϕ value, meaning it will remember its previous values well.
    • ◼
  • The OU process will have strong mean - reversion due to the high θ value derived from ϕ and Δt.
    • The individual paths of the processes will have similar fluctuations around their current values. However, the cumulative paths will differ because of the reasons mentioned above. The AR (1) process will tend to drift, while the OU process will revert to its mean (which is 0 in this case). To get a closer match in the cumulative paths over a longer duration, you would have to adjust the parameters, especially the mean of the OU process μ and its mean - reversion strength θ.
    In[]:=
    ListLinePlot[{Accumulate@arPath,Accumulate@ouPath},PlotLegends->{"AR(1)","OU"},AxesLabel->{"Time","Value"}]
    Out[]=
    AR(1)
    OU

    References

    ◼
  • Brockwell, P . J ., & Davis, R . A . (1991) . Time Series : Theory and Methods . Springer Science & Business Media. (Chapter 3.5 covers AR processes and their properties).
    • ◼
  • Box, G . E ., Jenkins, G . M ., Reinsel, G . C ., & Ljung, G . M . (2015). Time Series Analysis : Forecasting and Control. Wiley. (Chapter 5 introduces ARIMA models).
    • ◼
  • Uhlenbeck, G . E ., & Ornstein, L . S . (1930). On the Theory of the Brownian Motion . Physical Review, 36 (5), 823 - 841. (Original paper on the Ornstein - Uhlenbeck process).

    • CITE THIS NOTEBOOK

    Momentum and mean reversion in discrete and continuous time processes​
    by Jonathan Kinlay​
    Wolfram Community, STAFF PICKS, August 2, 2023
    ​https://community.wolfram.com/groups/-/m/t/2981627