Euler Discretization for a Mean-Reverting Diffusion with a Zero Floor

Summary

The answer shows how to step forward a mean-reverting stochastic process using Euler discretization. At each time step, the next state is formed from the previous state, a drift term that pulls it toward a long-run level, and a random shock scaled by the current state raised to a power. The shock is a normal draw with mean zero and variance equal to the time increment.

After updating, the method floors the state at zero. This gives a simple simulation procedure for a process that should remain nonnegative. The explanation is limited to the discretization rule: it does not discuss parameter estimation, convergence, stability, or the bias introduced by truncating negative updates. The requested time step is one, but the answer states the method using a general increment.

Key ideas

• Euler stepping advances the state with a mean-reverting drift and a state-dependent random shock.
• The Brownian increment is represented by an independent normal draw with variance equal to the time step.
• The update is floored at zero to prevent negative simulated states.
• The answer supplies a discretization rule but does not assess numerical accuracy or truncation bias.

Tags

# How to use Euler discretization for this interest rate model?


# How to use Euler discretization for this interest rate model?












How can I perform Euler discretization on this model where $\delta t=1$ and $\delta x_t = x_t-x_{t-1}$

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/21473

I would proceed as follows: \begin{align*} x_t &= x_{t-\delta t} + \alpha (\beta - x_{t-\delta t}) \delta t + \sigma x_{t-\delta t}^\gamma (w_t - w_{t-\delta t}),\\ x_t &= \max (x_t, \ 0), \end{align*} where $w_t - w_{t-\delta t}$ is a normal random variable with mean $0$ and variance $\delta t$, which can be obtained by an independent draw for each time step.

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