Coupled linear stochastic differential equations

In summary: The generic solution to the Ornstein-Uhlenbeck equation is a function of the two independent variables. However, in this case, we are trying to solve for the y-value of a single variable, so we need to find a different solution.
  • #1
Tyler_D
6
0
Homework Statement
In the following system of coupled stochastic differential equations, solve for ##x## (note that ##dW## is a Wiener process)
Relevant Equations
[tex]
dx = (\omega y - \gamma x)dt \\
dy = (-\omega x - \gamma y)dt + gdW \\
[/tex]
In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?
 
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  • #2
Solve the first equation for ##y ## (will be a function of ##x ## and ##dx/dt##), then from this ##y## calculate ##dy/dt## (which will be a function of ##dx/dt## and ##d^2x/dt^2##) and then replace in the second equation the ##y ## and the ##dy/dt## you found.
 
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  • #3
Delta2 said:
Solve the first equation for ##y ## (will be a function of ##x ## and ##dx/dt##), then from this ##y## calculate ##dy/dt## (which will be a function of ##dx/dt## and ##d^2x/dt^2##) and then replace in the second equation the ##y ## and the ##dy/dt## you found.

There's no ##dy/dt## in the second equation. I could of course divide by ##dt##, but I don't believe we are allowed to as ##dW## is not differentiable due to it being Wiener noise (?)
 
  • #4
To tell you the truth it beats me how you can define a differential ##dW## without the function W being differentiable, but anyway I think if it isn't differentiable then you probably can't solve this system.
Anyway after you calculate ##y'=dy/dt## as I told you before , replace ##dy## in the second equation with ##y'dt## and see what you can get.
 
  • #6
Fred Wright said:
Please see equ. 3.31 here https://users.aalto.fi/~ssarkka/course_s2012/pdf/sde_course_booklet_2012.pdf. Of course you will have to read everything leading up to that equation to make sense of it:-)

Thanks Fred. But this is just a generic solution to the Ornstein-Uhlenbeck equation as far as I can tell? The question is, how do I go from this coupled system of SDEs I have above to an SDE in a single variable that I can solve. Or am I misunderstanding you?

Hope you can say a bit more about what you are thinking :) thanks
 

Related to Coupled linear stochastic differential equations

What are coupled linear stochastic differential equations?

Coupled linear stochastic differential equations are a type of mathematical model used to describe the behavior of a system that is influenced by both random and deterministic factors. They consist of a set of differential equations that are linear in their variables and are interdependent, meaning that the variables in one equation affect the variables in another equation.

What is the difference between coupled and uncoupled linear stochastic differential equations?

The main difference between coupled and uncoupled linear stochastic differential equations is that in the former, the equations are interconnected and influence each other, while in the latter, each equation is independent and not affected by the others. This means that the behavior of a system described by coupled equations is more complex and difficult to predict compared to one described by uncoupled equations.

What are some real-world applications of coupled linear stochastic differential equations?

Coupled linear stochastic differential equations are commonly used in various fields of science and engineering, such as physics, biology, economics, and finance. They can be applied to study the behavior of complex systems, such as chemical reactions, population dynamics, and stock market fluctuations. They are also used in control systems and signal processing to model and predict the behavior of physical systems.

How are coupled linear stochastic differential equations solved?

There is no general analytical solution for coupled linear stochastic differential equations, so numerical methods are used to find approximate solutions. These methods involve discretizing the equations and using computer algorithms to solve them iteratively. Some common numerical methods used for solving coupled equations include the Euler-Maruyama method, the Runge-Kutta method, and the Milstein method.

What are some challenges in working with coupled linear stochastic differential equations?

One major challenge in working with coupled linear stochastic differential equations is the difficulty in obtaining accurate and reliable solutions, as the presence of random factors can make the system unpredictable. This requires careful selection of numerical methods and parameters, as well as rigorous testing and validation of the results. Another challenge is the computational complexity and time required to solve these equations, especially for systems with a large number of variables and parameters.

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