Defining stochastic differential equations and simulating a system of three SDEs
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I am trying to work on stochastic differential equations and I have been trying to use Mathematica's built-in function to simulate the system of equations below. When i use the randomfunction to simulate it using the Milstein method. I keep getting an output implying the RandomFunction method is not a random process recognized by the system.
Please look at my codes and help.
This is the system of equations:
dx[t] = (-a*s(x[t] + y[t]) - s*z[t])dt + 0.1*x[t] dw1
dy[t] = (p*x[t] - l*y[t] + s*z[t])dt + 0.1*y[t] dw2
dz[t] = (-p*x[t] -l*y[t] -(s + m)*z[t]) + 0.1*z[t] dw3
where w1, w2 and w3 are standard Wiener processes.
a = 10; l = 24.625; m = 14.925; s = 0.415; p = 5;
proc1 =
ItoProcess[
{[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t] )[DifferentialD]t + 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p* x[t] - l*y[t] + s*z[t]) [DifferentialD]t + 0.1*y[t] [DifferentialD]w2[t], [DifferentialD]z[t] == (-p*x[t] + {{l*y[t], -(s + m)*z[t]}}) [DifferentialD]t + 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1, w2, w3} [Distributed] WienerProcess]
paths = RandomFunction[proc1, {0, 100, 0.01}, 250, Method -> "Milstein"];
stochastic-calculus
edited Feb 11 at 10:36
m_goldberg
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
$endgroup$
add a comment |
$begingroup$
I am trying to work on stochastic differential equations and I have been trying to use Mathematica's built-in function to simulate the system of equations below. When i use the randomfunction to simulate it using the Milstein method. I keep getting an output implying the RandomFunction method is not a random process recognized by the system.
Please look at my codes and help.
This is the system of equations:
dx[t] = (-a*s(x[t] + y[t]) - s*z[t])dt + 0.1*x[t] dw1
dy[t] = (p*x[t] - l*y[t] + s*z[t])dt + 0.1*y[t] dw2
dz[t] = (-p*x[t] -l*y[t] -(s + m)*z[t]) + 0.1*z[t] dw3
where w1, w2 and w3 are standard Wiener processes.
a = 10; l = 24.625; m = 14.925; s = 0.415; p = 5;
proc1 =
ItoProcess[
{[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t] )[DifferentialD]t + 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p* x[t] - l*y[t] + s*z[t]) [DifferentialD]t + 0.1*y[t] [DifferentialD]w2[t], [DifferentialD]z[t] == (-p*x[t] + {{l*y[t], -(s + m)*z[t]}}) [DifferentialD]t + 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1, w2, w3} [Distributed] WienerProcess]
paths = RandomFunction[proc1, {0, 100, 0.01}, 250, Method -> "Milstein"];
stochastic-calculus
edited Feb 11 at 10:36
m_goldberg
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
$endgroup$
$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22
add a comment |
$begingroup$
I am trying to work on stochastic differential equations and I have been trying to use Mathematica's built-in function to simulate the system of equations below. When i use the randomfunction to simulate it using the Milstein method. I keep getting an output implying the RandomFunction method is not a random process recognized by the system.
Please look at my codes and help.
This is the system of equations:
dx[t] = (-a*s(x[t] + y[t]) - s*z[t])dt + 0.1*x[t] dw1
dy[t] = (p*x[t] - l*y[t] + s*z[t])dt + 0.1*y[t] dw2
dz[t] = (-p*x[t] -l*y[t] -(s + m)*z[t]) + 0.1*z[t] dw3
where w1, w2 and w3 are standard Wiener processes.
a = 10; l = 24.625; m = 14.925; s = 0.415; p = 5;
proc1 =
ItoProcess[
{[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t] )[DifferentialD]t + 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p* x[t] - l*y[t] + s*z[t]) [DifferentialD]t + 0.1*y[t] [DifferentialD]w2[t], [DifferentialD]z[t] == (-p*x[t] + {{l*y[t], -(s + m)*z[t]}}) [DifferentialD]t + 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1, w2, w3} [Distributed] WienerProcess]
paths = RandomFunction[proc1, {0, 100, 0.01}, 250, Method -> "Milstein"];
stochastic-calculus
edited Feb 11 at 10:36
m_goldberg
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
$endgroup$
I am trying to work on stochastic differential equations and I have been trying to use Mathematica's built-in function to simulate the system of equations below. When i use the randomfunction to simulate it using the Milstein method. I keep getting an output implying the RandomFunction method is not a random process recognized by the system.
Please look at my codes and help.
This is the system of equations:
dx[t] = (-a*s(x[t] + y[t]) - s*z[t])dt + 0.1*x[t] dw1
dy[t] = (p*x[t] - l*y[t] + s*z[t])dt + 0.1*y[t] dw2
dz[t] = (-p*x[t] -l*y[t] -(s + m)*z[t]) + 0.1*z[t] dw3
where w1, w2 and w3 are standard Wiener processes.
a = 10; l = 24.625; m = 14.925; s = 0.415; p = 5;
proc1 =
ItoProcess[
{[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t] )[DifferentialD]t + 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p* x[t] - l*y[t] + s*z[t]) [DifferentialD]t + 0.1*y[t] [DifferentialD]w2[t], [DifferentialD]z[t] == (-p*x[t] + {{l*y[t], -(s + m)*z[t]}}) [DifferentialD]t + 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1, w2, w3} [Distributed] WienerProcess]
paths = RandomFunction[proc1, {0, 100, 0.01}, 250, Method -> "Milstein"];
stochastic-calculus
stochastic-calculus
edited Feb 11 at 10:36
m_goldberg
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
edited Feb 11 at 10:36
m_goldberg
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
edited Feb 11 at 10:36
m_goldberg
87.4k872198
edited Feb 11 at 10:36
m_goldberg
87.4k872198
edited Feb 11 at 10:36
m_goldberg
87.4k872198
87.4k872198
asked Feb 11 at 6:00
Abiy DAbiy D
183
asked Feb 11 at 6:00
Abiy DAbiy D
183
asked Feb 11 at 6:00
Abiy DAbiy D
183
183
$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22
add a comment |
$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22
$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22
$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22
add a comment |
1 Answer
1
active
oldest
votes
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Two things:
1) You can't use {} to group terms as in your z equation. See, for example, here for more info.
2) You need to define each noise term separately. {w1, w2, w3} is a list of length three but WienerProcess is a scalar, so they don't have the same shape.
The following works:
proc1 = ItoProcess[{
[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t]) [DifferentialD]t
+ 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p*x[t] - l*y[t] + s*z[t]) [DifferentialD]t
+ 0.1*y[t] [DifferentialD]w2[t],
[DifferentialD]z[t] == (-p*x[t] + l*y[t] - (s + m)*z[t]) [DifferentialD]t
+ 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1 [Distributed] WienerProcess, w2 [Distributed] WienerProcess, w3 [Distributed] WienerProcess}];
answered Feb 11 at 8:08
Chris KChris K
6,78921942
$endgroup$
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
add a comment |
Your Answer
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1 Answer
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oldest
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
Two things:
1) You can't use {} to group terms as in your z equation. See, for example, here for more info.
2) You need to define each noise term separately. {w1, w2, w3} is a list of length three but WienerProcess is a scalar, so they don't have the same shape.
The following works:
proc1 = ItoProcess[{
[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t]) [DifferentialD]t
+ 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p*x[t] - l*y[t] + s*z[t]) [DifferentialD]t
+ 0.1*y[t] [DifferentialD]w2[t],
[DifferentialD]z[t] == (-p*x[t] + l*y[t] - (s + m)*z[t]) [DifferentialD]t
+ 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1 [Distributed] WienerProcess, w2 [Distributed] WienerProcess, w3 [Distributed] WienerProcess}];
answered Feb 11 at 8:08
Chris KChris K
6,78921942
$endgroup$
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
add a comment |
$begingroup$
Two things:
1) You can't use {} to group terms as in your z equation. See, for example, here for more info.
2) You need to define each noise term separately. {w1, w2, w3} is a list of length three but WienerProcess is a scalar, so they don't have the same shape.
The following works:
proc1 = ItoProcess[{
[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t]) [DifferentialD]t
+ 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p*x[t] - l*y[t] + s*z[t]) [DifferentialD]t
+ 0.1*y[t] [DifferentialD]w2[t],
[DifferentialD]z[t] == (-p*x[t] + l*y[t] - (s + m)*z[t]) [DifferentialD]t
+ 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1 [Distributed] WienerProcess, w2 [Distributed] WienerProcess, w3 [Distributed] WienerProcess}];
answered Feb 11 at 8:08
Chris KChris K
6,78921942
$endgroup$
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
add a comment |
$begingroup$
Two things:
1) You can't use {} to group terms as in your z equation. See, for example, here for more info.
2) You need to define each noise term separately. {w1, w2, w3} is a list of length three but WienerProcess is a scalar, so they don't have the same shape.
The following works:
proc1 = ItoProcess[{
[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t]) [DifferentialD]t
+ 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p*x[t] - l*y[t] + s*z[t]) [DifferentialD]t
+ 0.1*y[t] [DifferentialD]w2[t],
[DifferentialD]z[t] == (-p*x[t] + l*y[t] - (s + m)*z[t]) [DifferentialD]t
+ 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1 [Distributed] WienerProcess, w2 [Distributed] WienerProcess, w3 [Distributed] WienerProcess}];
answered Feb 11 at 8:08
Chris KChris K
6,78921942
$endgroup$
Two things:
1) You can't use {} to group terms as in your z equation. See, for example, here for more info.
2) You need to define each noise term separately. {w1, w2, w3} is a list of length three but WienerProcess is a scalar, so they don't have the same shape.
The following works:
proc1 = ItoProcess[{
[DifferentialD]x[t] == (-a*s x[t] - a*s y[t] - s*z[t]) [DifferentialD]t
+ 0.1*x[t] [DifferentialD]w1[t],
[DifferentialD]y[t] == (p*x[t] - l*y[t] + s*z[t]) [DifferentialD]t
+ 0.1*y[t] [DifferentialD]w2[t],
[DifferentialD]z[t] == (-p*x[t] + l*y[t] - (s + m)*z[t]) [DifferentialD]t
+ 0.1*z[t] [DifferentialD]w3[t]},
{x[t], y[t], z[t]}, {{x, y, z}, {0.115, -0.115, 0}}, t,
{w1 [Distributed] WienerProcess, w2 [Distributed] WienerProcess, w3 [Distributed] WienerProcess}];
answered Feb 11 at 8:08
Chris KChris K
6,78921942
answered Feb 11 at 8:08
Chris KChris K
6,78921942
answered Feb 11 at 8:08
Chris KChris K
6,78921942
answered Feb 11 at 8:08
Chris KChris K
6,78921942
6,78921942
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
add a comment |
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
$begingroup$
sure the above works well and i can see where my problem is. Thank you!
$endgroup$
– Abiy D
Feb 11 at 13:50
add a comment |
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$begingroup$
Welcome to Mathematica.SE, Abiy! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour and check the faqs! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign!
$endgroup$
– Chris K
Feb 11 at 8:22