Mixing
How to improve the convergence of a stochastic differential equation?
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I have a stochastic differential equation, i.e,
$$ drho_t= hat{A} rho_s dt + hat{B} rho_s nu dt + hat{C}rho_somega_{1t} dt + hat{D}rho_s omega_{2t}dt quad , quad t>s $$
Here A, B, C and D are operators. $nu$ is a white noise. $omega_1$ and $omega_2$ are color noises with specific correlation matrices. As we know that there are several methods to improve the convergence of a SDE, for example,
1) Decrease the number of noises
2) Use higher order numerical algorithms (Range Kutta etc.)
3) Use small time steps and large number of trajectories
Now, apart from above mentioned techniques, is there any other way to decrease the fluctuations of a SDE if we directly simulate it numerically with Euler or Range Kutta algorithms?
stochastic-processes numerical-methods stochastic-calculus stochastic-analysis numerical-calculus
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
$endgroup$
add a comment |
$begingroup$
I have a stochastic differential equation, i.e,
$$ drho_t= hat{A} rho_s dt + hat{B} rho_s nu dt + hat{C}rho_somega_{1t} dt + hat{D}rho_s omega_{2t}dt quad , quad t>s $$
Here A, B, C and D are operators. $nu$ is a white noise. $omega_1$ and $omega_2$ are color noises with specific correlation matrices. As we know that there are several methods to improve the convergence of a SDE, for example,
1) Decrease the number of noises
2) Use higher order numerical algorithms (Range Kutta etc.)
3) Use small time steps and large number of trajectories
Now, apart from above mentioned techniques, is there any other way to decrease the fluctuations of a SDE if we directly simulate it numerically with Euler or Range Kutta algorithms?
stochastic-processes numerical-methods stochastic-calculus stochastic-analysis numerical-calculus
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
$endgroup$
$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05
add a comment |
$begingroup$
I have a stochastic differential equation, i.e,
$$ drho_t= hat{A} rho_s dt + hat{B} rho_s nu dt + hat{C}rho_somega_{1t} dt + hat{D}rho_s omega_{2t}dt quad , quad t>s $$
Here A, B, C and D are operators. $nu$ is a white noise. $omega_1$ and $omega_2$ are color noises with specific correlation matrices. As we know that there are several methods to improve the convergence of a SDE, for example,
1) Decrease the number of noises
2) Use higher order numerical algorithms (Range Kutta etc.)
3) Use small time steps and large number of trajectories
Now, apart from above mentioned techniques, is there any other way to decrease the fluctuations of a SDE if we directly simulate it numerically with Euler or Range Kutta algorithms?
stochastic-processes numerical-methods stochastic-calculus stochastic-analysis numerical-calculus
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
$endgroup$
I have a stochastic differential equation, i.e,
$$ drho_t= hat{A} rho_s dt + hat{B} rho_s nu dt + hat{C}rho_somega_{1t} dt + hat{D}rho_s omega_{2t}dt quad , quad t>s $$
Here A, B, C and D are operators. $nu$ is a white noise. $omega_1$ and $omega_2$ are color noises with specific correlation matrices. As we know that there are several methods to improve the convergence of a SDE, for example,
1) Decrease the number of noises
2) Use higher order numerical algorithms (Range Kutta etc.)
3) Use small time steps and large number of trajectories
Now, apart from above mentioned techniques, is there any other way to decrease the fluctuations of a SDE if we directly simulate it numerically with Euler or Range Kutta algorithms?
stochastic-processes numerical-methods stochastic-calculus stochastic-analysis numerical-calculus
stochastic-processes numerical-methods stochastic-calculus stochastic-analysis numerical-calculus
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
edited Jan 7 at 9:19
Skeptical Khan
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
asked Jan 7 at 6:51
Skeptical KhanSkeptical Khan
193
193
$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05
add a comment |
$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05
$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05
add a comment |
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$begingroup$
Why do you have 3 noise terms? You can combine them to one noise term. Ito or Stratanovich? Why not use the exact solution formula of the geometric Brownian motion?
$endgroup$
– LutzL
Jan 7 at 8:42
$begingroup$
yes i know, the above equation is just an example. I am not talking about exact solution. I am talking about if we do it numerically. The number of noises will play the role in convergence. In my actual equation, i can not decrease my noises, there are white noises and color noises which can't be combined
$endgroup$
– Skeptical Khan
Jan 7 at 9:05