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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
edited 23 hours ago
Mefitico
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asked Nov 5 at 14:15
sigmatau
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This question has an open bounty worth +100
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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
edited 23 hours ago
Mefitico
829117
asked Nov 5 at 14:15
sigmatau
1,8891923
This question has an open bounty worth +100
reputation from sigmatau ending in 6 days.
This question has not received enough attention.
add a comment |
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
edited 23 hours ago
Mefitico
829117
asked Nov 5 at 14:15
sigmatau
1,8891923
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
edited 23 hours ago
Mefitico
829117
asked Nov 5 at 14:15
sigmatau
1,8891923
edited 23 hours ago
Mefitico
829117
asked Nov 5 at 14:15
sigmatau
1,8891923
edited 23 hours ago
Mefitico
829117
edited 23 hours ago
Mefitico
829117
edited 23 hours ago
Mefitico
829117
829117
asked Nov 5 at 14:15
sigmatau
1,8891923
asked Nov 5 at 14:15
sigmatau
1,8891923
asked Nov 5 at 14:15
sigmatau
1,8891923
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.
add a comment |
add a comment |
1 Answer
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0
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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.
answered 23 hours ago
Mefitico
829117
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered 23 hours ago
Mefitico
829117
add a comment |
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.
answered 23 hours ago
Mefitico
829117
add a comment |
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.
answered 23 hours ago
Mefitico
829117
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.
answered 23 hours ago
Mefitico
829117
edited 23 hours ago
answered 23 hours ago
Mefitico
829117
answered 23 hours ago
Mefitico
829117
answered 23 hours ago
Mefitico
829117
829117
add a comment |
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