Euler–Maruyama step for a drift-diffusion process

Read the equation

The helper returns x+μdt+σdW, adding deterministic drift and a scaled stochastic increment. It is one discrete approximation step to the displayed Itô stochastic differential equation. For state-dependent coefficients, callers must evaluate the drift and diffusion consistently at the step’s starting state.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

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Original context

// WienerProcess generates Wiener increment dW ~ N(0, dt)
func WienerProcess(dt float64, rng func() float64) float64 {
	// dW = √dt · Z where Z ~ N(0,1)
	return math.Sqrt(dt) * rng()
}

// ItoProcess computes dX = μ·dt + σ·dW (Ito SDE step)
func ItoProcess(x, mu, sigma, dt, dw float64) float64 {
	return x + mu*dt + sigma*dw
}

// OrnsteinUhlenbeck computes mean-reverting process
// dX = θ(μ - X)dt + σdW
func OrnsteinUhlenbeck(x, theta, mu, sigma, dt, dw float64) float64 {
	drift := theta * (mu - x) * dt
	diffusion := sigma * dw
	return x + drift + diffusion

ItoProcess computes dX = μ·dt + σ·dW (Ito SDE step)

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