Overview
Euler-Maruyama extends explicit Euler integration by adding a Brownian-motion increment for stochastic dynamics.
ExtendedSource checked
Euler-Maruyama Method
Approximates stochastic differential equations with an Euler step plus random noise.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- Stochastic Differential Equation Solver
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
Core Concepts
stochastic differential equationbrownian incrementtime discretizationsample path
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(steps)
Average
O(paths * steps)
Worst
O(paths * steps)
Space
O(paths)
When to Use
- Use when solving equations needs stochastic differential equation.
- Use when the input matches the assumptions behind stochastic Differential Equation Solver.
When Not to Use
- Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Approximates stochastic differential equations with an Euler step plus random noise
- Uses stochastic differential equation and brownian increment to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
Euler-Maruyama step
Xnext = X + a(X,t)·dt + b(X,t)·√dt·Normal(0,1)
X_next = X + a(X,t)*dt + b(X,t)*sqrt(dt)*Normal(0,1)Advances one stochastic sample path by drift and diffusion terms.
Pseudocode
- 1.Prepare the inputs and state needed for Euler-Maruyama Method.
- 2.Apply the stochastic differential equation step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- Euler-Maruyama Method input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Euler-Maruyama Method applies stochastic differential equation to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the stochastic differential equation step separate enough to test independently.
Common Pitfalls
- Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
- Forgetting to check edge cases can make Euler-Maruyama Method look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Euler-Maruyama Method.
numerical-methods-mechanismNotes
Euler-Maruyama Method should be used with the assumptions and tradeoffs of stochastic Differential Equation Solver in mind.
Sources (1)
What each reference supports for this entry.
Numerical Solution of Stochastic Differential Equations
BookReference text for Euler-Maruyama and stochastic differential equation discretization.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageVariant ofEuler Integration
Euler-Maruyama adapts Euler-style updates to stochastic differential equations.
Alternative toMonte Carlo Simulation
Both model randomness but at different levels of simulation detail.
Related Algorithms
Tags
Numerical MethodsOde SolverStochastic Differential EquationSimulation