Parameter estimation for Stochastic differential equation











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I have a process $X(t)$ defined on some finite time horizon $[0,T]$ and I know that my process satisfies the following SDE:



$dX(t)=mu dt + sigma dB_t$.



where $B$ is a standard Brownian motion. In particular I'm assuming that both the drift and volatility are constant over time.



Question:



Given data points $x_{t_1}, ldots x_{t_n}$ that are realizations of the random variable $X(t)$ at times $t_1 ldots t_n$, how do I estimate the drift and volatility parameters $mu, sigma$ ? I'm interested in a method that is relatively easy to implement.



I would also like to know if there already exist libraries in say Python that might help for this task.



What would be a method for estimating $mu$ and $sigma$ if they change over time? (I.e. they are are time dependent)






reference-request stochastic-processes parameter-estimation







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    I have a process $X(t)$ defined on some finite time horizon $[0,T]$ and I know that my process satisfies the following SDE:



    $dX(t)=mu dt + sigma dB_t$.



    where $B$ is a standard Brownian motion. In particular I'm assuming that both the drift and volatility are constant over time.



    Question:



    Given data points $x_{t_1}, ldots x_{t_n}$ that are realizations of the random variable $X(t)$ at times $t_1 ldots t_n$, how do I estimate the drift and volatility parameters $mu, sigma$ ? I'm interested in a method that is relatively easy to implement.



    I would also like to know if there already exist libraries in say Python that might help for this task.



    What would be a method for estimating $mu$ and $sigma$ if they change over time? (I.e. they are are time dependent)






    reference-request stochastic-processes parameter-estimation







    share|cite|improve this question

















    This question has an open bounty worth +100
    reputation from sigmatau ending in 6 days.


    This question has not received enough attention.


















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I have a process $X(t)$ defined on some finite time horizon $[0,T]$ and I know that my process satisfies the following SDE:



      $dX(t)=mu dt + sigma dB_t$.



      where $B$ is a standard Brownian motion. In particular I'm assuming that both the drift and volatility are constant over time.



      Question:



      Given data points $x_{t_1}, ldots x_{t_n}$ that are realizations of the random variable $X(t)$ at times $t_1 ldots t_n$, how do I estimate the drift and volatility parameters $mu, sigma$ ? I'm interested in a method that is relatively easy to implement.



      I would also like to know if there already exist libraries in say Python that might help for this task.



      What would be a method for estimating $mu$ and $sigma$ if they change over time? (I.e. they are are time dependent)






      reference-request stochastic-processes parameter-estimation







      share|cite|improve this question















      I have a process $X(t)$ defined on some finite time horizon $[0,T]$ and I know that my process satisfies the following SDE:



      $dX(t)=mu dt + sigma dB_t$.



      where $B$ is a standard Brownian motion. In particular I'm assuming that both the drift and volatility are constant over time.



      Question:



      Given data points $x_{t_1}, ldots x_{t_n}$ that are realizations of the random variable $X(t)$ at times $t_1 ldots t_n$, how do I estimate the drift and volatility parameters $mu, sigma$ ? I'm interested in a method that is relatively easy to implement.



      I would also like to know if there already exist libraries in say Python that might help for this task.



      What would be a method for estimating $mu$ and $sigma$ if they change over time? (I.e. they are are time dependent)






      reference-request stochastic-processes parameter-estimation




      reference-request stochastic-processes parameter-estimation






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      share|cite|improve this question













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      edited 23 hours ago









      Mefitico

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      829117










      asked Nov 5 at 14:15









      sigmatau

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      1,8891923






      This question has an open bounty worth +100
      reputation from sigmatau ending in 6 days.


      This question has not received enough attention.








      This question has an open bounty worth +100
      reputation from sigmatau ending in 6 days.


      This question has not received enough attention.
























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          I'm not so familiar with this formulation of the problem, but a common practice in engineering is the method based on Allan Variance. You basically assume the high frequency noise to be the volatility while the low frequency is the drift and identify btoh from the slopes on the chart..



          As for the case when they are not constant it would be necessary to make some assumptions on how they vary over time. Either by giving them specific time functions with parameters to be estimated or possibly assuming that they are themselves performing soem kind of random (possibly Brownian) motion, in which case some technique like an Extended Kalman Filter or a Particle Filter could be envisaged.






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            I'm not so familiar with this formulation of the problem, but a common practice in engineering is the method based on Allan Variance. You basically assume the high frequency noise to be the volatility while the low frequency is the drift and identify btoh from the slopes on the chart..



            As for the case when they are not constant it would be necessary to make some assumptions on how they vary over time. Either by giving them specific time functions with parameters to be estimated or possibly assuming that they are themselves performing soem kind of random (possibly Brownian) motion, in which case some technique like an Extended Kalman Filter or a Particle Filter could be envisaged.






            share|cite|improve this answer



























              up vote
              0
              down vote













              I'm not so familiar with this formulation of the problem, but a common practice in engineering is the method based on Allan Variance. You basically assume the high frequency noise to be the volatility while the low frequency is the drift and identify btoh from the slopes on the chart..



              As for the case when they are not constant it would be necessary to make some assumptions on how they vary over time. Either by giving them specific time functions with parameters to be estimated or possibly assuming that they are themselves performing soem kind of random (possibly Brownian) motion, in which case some technique like an Extended Kalman Filter or a Particle Filter could be envisaged.






              share|cite|improve this answer

























                up vote
                0
                down vote










                up vote
                0
                down vote









                I'm not so familiar with this formulation of the problem, but a common practice in engineering is the method based on Allan Variance. You basically assume the high frequency noise to be the volatility while the low frequency is the drift and identify btoh from the slopes on the chart..



                As for the case when they are not constant it would be necessary to make some assumptions on how they vary over time. Either by giving them specific time functions with parameters to be estimated or possibly assuming that they are themselves performing soem kind of random (possibly Brownian) motion, in which case some technique like an Extended Kalman Filter or a Particle Filter could be envisaged.






                share|cite|improve this answer














                I'm not so familiar with this formulation of the problem, but a common practice in engineering is the method based on Allan Variance. You basically assume the high frequency noise to be the volatility while the low frequency is the drift and identify btoh from the slopes on the chart..



                As for the case when they are not constant it would be necessary to make some assumptions on how they vary over time. Either by giving them specific time functions with parameters to be estimated or possibly assuming that they are themselves performing soem kind of random (possibly Brownian) motion, in which case some technique like an Extended Kalman Filter or a Particle Filter could be envisaged.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 23 hours ago

























                answered 23 hours ago









                Mefitico

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