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Euler method adapted for the Ito solution of stochastic differential equations
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I am trying to replicate the analysis from the paper
Brock, W. A., and S. R. Carpenter. 2006. Variance as a leading indicator of regime shift in ecosystem services. Ecology and Society 11(2): 9. [online] URL: http://www.ecologyandsociety.org/vol11/iss2/art9/
A system of stochastic differential equations is provided
$$dP_i = [M_i + D(P_j - P_i)-cP_i - sf(P_i)]dt + sigma dW_i$$
$$f(P) equiv fracm^qm^q + P^q$$
$$dS_i = left(fracrkP_i-hS_iright)dt $$
where "$dW$ is a Wiener noise process that is N(0,dt) and independent for each i"
Later, it is stated: "To study the dynamics of variance, we simulated time series of P and S using the Euler method adapted for the Ito solution of stochastic differential equations (Horsthemke and Lefever 1984). We do this by iteratively computing using Eq. 2 below. Each time step dt is of length 1/n. Over each small increment we compute"
$$dP_i,t approxeq [M_i,t + D(P_j,t - P_i,t)-cP_i,t - sf(P_i,t)]/n + sigma Z_i,t/sqrtn$$
$$dS_i,t approxeq left(fracrkP_i,t-hS_i,tright)/n $$
where "$Z_i,t$ is an independently drawn random number (normal, mean 0, variance 1)". The reference for the method is
Horsthemke, W., and R. Lefever. 1984. Noise-induced transitions. Springer-Verlag, New York, New York, USA
I wish to obtain the solutions $P_1$, $P_2$, $S_1$, and $S_2$ while "M1
(pollution discharge rate in region 1) increases linearly from 2.0 to 2.1".
I applied the forward Euler method to this problem in an attempt to obtain a numerical solution. However, my solution does not match the published result. In particular, my solution does not exhibit the transitions between multiple steady states over time.
How can I numerically solve this system of stochastic differential equations? What is the "Euler method adapted for the Ito solution of stochastic differential equations"?
differential-equations numerical-methods stochastic-calculus
asked Aug 3 at 19:56
Nat
1133
add a comment |Â
up vote
0
down vote
favorite
I am trying to replicate the analysis from the paper
Brock, W. A., and S. R. Carpenter. 2006. Variance as a leading indicator of regime shift in ecosystem services. Ecology and Society 11(2): 9. [online] URL: http://www.ecologyandsociety.org/vol11/iss2/art9/
A system of stochastic differential equations is provided
$$dP_i = [M_i + D(P_j - P_i)-cP_i - sf(P_i)]dt + sigma dW_i$$
$$f(P) equiv fracm^qm^q + P^q$$
$$dS_i = left(fracrkP_i-hS_iright)dt $$
where "$dW$ is a Wiener noise process that is N(0,dt) and independent for each i"
Later, it is stated: "To study the dynamics of variance, we simulated time series of P and S using the Euler method adapted for the Ito solution of stochastic differential equations (Horsthemke and Lefever 1984). We do this by iteratively computing using Eq. 2 below. Each time step dt is of length 1/n. Over each small increment we compute"
$$dP_i,t approxeq [M_i,t + D(P_j,t - P_i,t)-cP_i,t - sf(P_i,t)]/n + sigma Z_i,t/sqrtn$$
$$dS_i,t approxeq left(fracrkP_i,t-hS_i,tright)/n $$
where "$Z_i,t$ is an independently drawn random number (normal, mean 0, variance 1)". The reference for the method is
Horsthemke, W., and R. Lefever. 1984. Noise-induced transitions. Springer-Verlag, New York, New York, USA
I wish to obtain the solutions $P_1$, $P_2$, $S_1$, and $S_2$ while "M1
(pollution discharge rate in region 1) increases linearly from 2.0 to 2.1".
I applied the forward Euler method to this problem in an attempt to obtain a numerical solution. However, my solution does not match the published result. In particular, my solution does not exhibit the transitions between multiple steady states over time.
How can I numerically solve this system of stochastic differential equations? What is the "Euler method adapted for the Ito solution of stochastic differential equations"?
differential-equations numerical-methods stochastic-calculus
asked Aug 3 at 19:56
Nat
1133
1
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to replicate the analysis from the paper
Brock, W. A., and S. R. Carpenter. 2006. Variance as a leading indicator of regime shift in ecosystem services. Ecology and Society 11(2): 9. [online] URL: http://www.ecologyandsociety.org/vol11/iss2/art9/
A system of stochastic differential equations is provided
$$dP_i = [M_i + D(P_j - P_i)-cP_i - sf(P_i)]dt + sigma dW_i$$
$$f(P) equiv fracm^qm^q + P^q$$
$$dS_i = left(fracrkP_i-hS_iright)dt $$
where "$dW$ is a Wiener noise process that is N(0,dt) and independent for each i"
Later, it is stated: "To study the dynamics of variance, we simulated time series of P and S using the Euler method adapted for the Ito solution of stochastic differential equations (Horsthemke and Lefever 1984). We do this by iteratively computing using Eq. 2 below. Each time step dt is of length 1/n. Over each small increment we compute"
$$dP_i,t approxeq [M_i,t + D(P_j,t - P_i,t)-cP_i,t - sf(P_i,t)]/n + sigma Z_i,t/sqrtn$$
$$dS_i,t approxeq left(fracrkP_i,t-hS_i,tright)/n $$
where "$Z_i,t$ is an independently drawn random number (normal, mean 0, variance 1)". The reference for the method is
Horsthemke, W., and R. Lefever. 1984. Noise-induced transitions. Springer-Verlag, New York, New York, USA
I wish to obtain the solutions $P_1$, $P_2$, $S_1$, and $S_2$ while "M1
(pollution discharge rate in region 1) increases linearly from 2.0 to 2.1".
I applied the forward Euler method to this problem in an attempt to obtain a numerical solution. However, my solution does not match the published result. In particular, my solution does not exhibit the transitions between multiple steady states over time.
How can I numerically solve this system of stochastic differential equations? What is the "Euler method adapted for the Ito solution of stochastic differential equations"?
differential-equations numerical-methods stochastic-calculus
asked Aug 3 at 19:56
Nat
1133
I am trying to replicate the analysis from the paper
Brock, W. A., and S. R. Carpenter. 2006. Variance as a leading indicator of regime shift in ecosystem services. Ecology and Society 11(2): 9. [online] URL: http://www.ecologyandsociety.org/vol11/iss2/art9/
A system of stochastic differential equations is provided
$$dP_i = [M_i + D(P_j - P_i)-cP_i - sf(P_i)]dt + sigma dW_i$$
$$f(P) equiv fracm^qm^q + P^q$$
$$dS_i = left(fracrkP_i-hS_iright)dt $$
where "$dW$ is a Wiener noise process that is N(0,dt) and independent for each i"
Later, it is stated: "To study the dynamics of variance, we simulated time series of P and S using the Euler method adapted for the Ito solution of stochastic differential equations (Horsthemke and Lefever 1984). We do this by iteratively computing using Eq. 2 below. Each time step dt is of length 1/n. Over each small increment we compute"
$$dP_i,t approxeq [M_i,t + D(P_j,t - P_i,t)-cP_i,t - sf(P_i,t)]/n + sigma Z_i,t/sqrtn$$
$$dS_i,t approxeq left(fracrkP_i,t-hS_i,tright)/n $$
where "$Z_i,t$ is an independently drawn random number (normal, mean 0, variance 1)". The reference for the method is
Horsthemke, W., and R. Lefever. 1984. Noise-induced transitions. Springer-Verlag, New York, New York, USA
I wish to obtain the solutions $P_1$, $P_2$, $S_1$, and $S_2$ while "M1
(pollution discharge rate in region 1) increases linearly from 2.0 to 2.1".
I applied the forward Euler method to this problem in an attempt to obtain a numerical solution. However, my solution does not match the published result. In particular, my solution does not exhibit the transitions between multiple steady states over time.
How can I numerically solve this system of stochastic differential equations? What is the "Euler method adapted for the Ito solution of stochastic differential equations"?
differential-equations numerical-methods stochastic-calculus
asked Aug 3 at 19:56
Nat
1133
asked Aug 3 at 19:56
Nat
1133
asked Aug 3 at 19:56
Nat
1133
asked Aug 3 at 19:56
Nat
1133
1133
1
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04
add a comment |Â
1
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04
1
1
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04
add a comment |Â
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1
Have you heard of the Euler-Maruyama method? (Note: I'm no expert in Ito Calculus, but I've used this in some naive simulations.)
– Clarinetist
Aug 3 at 20:04