二维随机微分方程(SDE) [英] Stochastic Differential Equations (SDE) in 2 dimensions

问题描述

我是第一次研究随机微分方程.我正在寻找在二维上模拟和求解随机微分方程的方法.

I am working on stochastic differential equations for the first time. I am looking to simulate and solve a stochastic differential equations in two dimensions.

模型如下:

dp = F(t，p)dt + G(t，p)dW(t)

dp=F(t,p)dt+G(t,p)dW(t)

其中:

• p是2比1的向量:p =(theta(t); phi(t))
F是列向量:F =(sinθ+ Psi * cos(phi); Psi * cot(theta)* sin(phi))
• G是2×2矩阵:G =(D 0; 0 D/sinθ)
• Psi是一个参数，D是扩散常数

• p is a 2-by-1 vector: p=(theta(t); phi(t))
• F is a column vector: F=(sin(theta)+Psi* cos(phi); Psi* cot(theta)*sin(phi))
• G is a 2-by-2 matrix: G=(D 0;0 D/sin(theta))
• Psi is a parameter and D is the diffusion constant

我写的代码如下:

function MDL=gyro_2dim(Psi,D)
% want to solve for 2-by-1 vector:
%p=[theta;phi];
%drift function
F=@(t,theta,phi)  [sinth(theta)+Psi.*cos(phi)-D.*cot(theta);Psi.*cot(theta).*sin(phi)];
%diffusion function
G=@(t,theta,phi) [D 0;0 D./sin(theta)];
MDL=sde(F,G)
end

然后我使用以下脚本调用该函数:

Then I call the function with the following script:

params.t0   = 0;               % start time of simulation
params.tend = 20;              % end time
params.dt =0.1;                % time increment
D=0.1;
nPeriods=10; % # of simulated observations
Psi=1;
MDL=gyro_2dim(Psi,D);
[S,T,Z]=simulate(MDL, nPeriods,'DeltaTime',params.dt);
plot(T,S)

运行代码时，我收到以下错误消息:

When I run the code, I receive this error message:

漂移率在初始条件下或模型不一致时无效 尺寸.

Drift rate invalid at initial conditions or inconsistent model dimensions.

您知道如何解决此错误吗?

Any idea how to fix this error?

推荐答案

摘自 sde :

用户定义的漂移率函数，用F表示. DriftRate是一个函数，当使用两个输入进行调用时，该函数将返回NVARS by-1漂移率向量:
-实值标量观察时间t.
-一个NVARS by-1状态向量Xt.

User-defined drift-rate function, denoted by F. DriftRate is a function that returns an NVARS-by-1 drift-rate vector when called with two inputs:
- A real-valued scalar observation time t.
- An NVARS-by-1 state vector Xt.

为Diffusion功能提供了类似的规范.但是，您将状态向量的元素作为标量传递，因此具有三个而不是两个输入.您可以尝试将模型创建功能更改为:

A similar specification is provided for the Diffusion function. However, you're passing in the elements of your state vector as scalars and thus have three, rather than two, inputs. You can try changing your model creation function to:

function MDL=gyro_2dim(Psi,D)
% State vector: p = [theta;phi];
F = @(t,p)[sin(p(1))+Psi.*cos(p(2))-D.*cot(p(1));
           Psi.*cot(p(1)).*sin(p(2))];            % Drift
G = @(t,p)[D 0;
           0 D./sin(p(1))];                       % Diffusion
MDL = sde(F,G);
MDL.StartTime = 0;   % Set initial time
MDL.StartState = ... % Set initial conditions

我也将sinth(theta)更改为sin(p(1))，因为没有sinth功能.我无法测试此功能，因为我没有财务"工具箱(很少这样做).

I also changed sinth(theta) to sin(p(1)) as there is no sinth function. I can't test this as I don't have the Financial toolbox (few do).

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