Simulating an Ornstein–Uhlenbeck Process from Its Conditional Distribution

Summary

The document asks how to simulate paths for a mean-reverting diffusion without stepping through its stochastic differential equation directly. It gives the process solution over an interval and its conditional mean and variance, which together specify the Gaussian distribution of the next value given the current value. The initial condition is zero, and the parameters control the rate of mean reversion and the scale of random shocks.

The answer does not provide a simulation algorithm or Matlab implementation; it points to a separate introduction to numerical SDE simulation. Thus, the useful starting point is the stated conditional distribution, while the practical steps for generating repeated path values must be obtained elsewhere. The document also does not discuss time-grid choices, parameter estimation, or validation of simulated paths.

Key ideas

• The process value conditional on its current state is described by a Gaussian distribution.
• The conditional mean decays exponentially toward zero at a rate set by the mean-reversion parameter.
• The conditional variance increases with the interval length and depends on the shock scale and mean-reversion rate.
• The answer refers readers to an external numerical SDE resource but supplies no implementation details.

Tags

# How to simulate a path through its solution and conditional expectation / variance


# How to simulate a path through its solution and conditional expectation / variance












Hi I want to simulate in Matlab the following stochastic integral:

$ x(t) = x(s) e^{-a(t-s)} + \sigma \int_s^t e^{-a(t-u)} dW_1(u)$

with

$E[x(t) \vert F_s] = x(s) e^{-a(t-s)}$

$Var[x(t) \vert F_s] = \frac{\sigma^2}{2a} [1-e^{-2a(t-s)}]$

The dynamics is given by :

$dx(t) = -a x(t) dt + \sigma dW_1(t), x(0) = 0 $

I want to implement this in Matlab without using the dynamics but the stochastic integral and the distribution property. I want to model paths for x(t).

## Answer by phdstudent (score 0, accepted)

https://quant.stackexchange.com/a/39289

This reference: An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations should give you a hint on where to start.

Plenty of similar examples along with matlab codes.

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