Stochastic differential equation problem

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SUMMARY

The discussion centers on solving the stochastic differential equation (SDE) represented by the formula \(\frac{dv}{dt} = -\alpha v + \lambda F + \eta\), where \(\alpha\), \(\lambda\), and \(F\) are constants, \(v\) is the variable representing speed, and \(\eta\) is a random value. The user seeks analytical solutions to this equation, which resembles Brownian motion with an applied field, and intends to compare these solutions with numerical methods. A reference to a course on SDEs from Berkeley is provided for further guidance on solving such equations.

PREREQUISITES
  • Understanding of stochastic differential equations (SDEs)
  • Familiarity with Brownian motion concepts
  • Basic knowledge of analytical and numerical methods for solving differential equations
  • Proficiency in mathematical notation and calculus
NEXT STEPS
  • Study the course material on stochastic differential equations from the provided Berkeley link
  • Learn about numerical methods for solving SDEs, such as the Euler-Maruyama method
  • Explore the implications of noise in SDEs and its impact on solutions
  • Investigate the change of variables technique to simplify SDEs
USEFUL FOR

Researchers, mathematicians, and students in applied mathematics or physics who are working with stochastic processes and differential equations will benefit from this discussion.

johnt447
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Sorry if this is in the wrong section but i have a problem, I have no experience with stochastic equations well analytically anyway.

The equation i have is the following;

\frac{dv}{dt} = - \alpha v+ \lambda F+\eta

Where alpha lambda and F are constants, v is a variable (speed in this case) and eta is a random value. I believe this is similar to Brownian motion with an applied field, although i have no idea how to solve this analytically i plan to solve it analytically and compare it to a numerical solution. So any help will be most appreciated!
 
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This will turn into a standard equation of the type dv/dt=kv+ noise after a change of variable. For some general methods for solving SDEs, I hope the following link will be of much help -
http://math.berkeley.edu/~evans/SDE.course.pdf
 

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