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Removing Linear Drift from an SDE with an Integrating Factor

Article Quant Q&A · Author: Bohdan_
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Summary

The document introduces the integrating factor method for a stochastic differential equation whose drift is proportional to the state. For the example with constant drift coefficient and additive Brownian noise, it uses the deterministic factor that solves the corresponding drift-canceling ordinary differential equation. Applying Itô's product rule to the factor times the process makes the drift terms cancel, leaving a stochastic integral that can be handled directly; multiplying back recovers the original process.

The answer also describes extending the factor to a time-varying linear drift coefficient by integrating that coefficient over time. The method is useful for linear equations of this form, but the worked reasoning does not establish a general algorithm for arbitrary SDEs. It also briefly motivates the factor through the mean's deterministic equation; the actual cancellation in the stochastic equation follows from Itô's product rule. The discussion offers a method and example, not a treatment of existence conditions or broader nonlinear cases.

Key ideas

  • Choose a deterministic factor by solving an ordinary differential equation that cancels the linear drift.
  • Apply Itô's product rule to the factor multiplied by the process.
  • For additive Brownian noise, the resulting equation can be expressed as a stochastic integral.
  • A time-varying linear drift can be handled by integrating its coefficient over time.
  • The method described applies to linear drift structures, not arbitrary SDEs.

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# Solving SDE using integration factor and Ito's lemma


# Solving SDE using integration factor and Ito's lemma












I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential Equation using integrating factor. And in this book Stochastic Differential Equations in some exercises hints recommend to use integrator factor, but how to get it.

Is there some mechanism(algorithm) which allows us to find this integrator factor?

## Answer by siou0107 (score 0)

https://quant.stackexchange.com/a/68824

I believe that Wikipedia gives a fair definition ; see it as a "helper function".

In your example, $e^{-\mu t}$ helps you getting rid of the drift in your SDE, and you end up solving it by mere stochastic integration ; then just "multiply back" by $e^{\mu t}$ and your initial SDE is solved.

## Answer by Bennnn (score 0)

https://quant.stackexchange.com/a/74076

Essentially you want to get rid of the drift of the process $\mu X_t d t$ to get it in a solvable form

$$E[d X_t] = E[\mu X_t d t]\\ \implies d E[X_t] = \mu E[X_t] d t\\ \equiv d y(t) = \mu \cdot y(t) d t\\ \implies \frac{d y(t)}{d t} - \mu \cdot y(t)=0$$

Hence solving this ODE via the integrating factor method one has

$$I(t) = e^{\int_0^t -\mu d t} = e^{-\mu t}$$

So getting the SDE

$$d (e^{-\mu t} X_t) = X_t \cdot d(e^{-\mu t}) + e^{- \mu t} \cdot d X_t + (d e^{- \mu t})(d X_t)\\ = X_t \cdot (- \mu e^{- \mu t}dt) + e^{- \mu t}(\mu X_t d t + \sigma d W_t) + 0\\ = - \mu e^{- \mu t}X_t d t + \mu e^{-\mu t}X_t d t + \sigma e^{- \mu t }d W_t\\ = \sigma e^{-\mu t} d W_t$$

Hence we have removed the drift and get $$e^{-\mu t} X_t = X_0 + \sigma \int_0^t e^{-\mu t} d W_t$$

From there it should be obvious. This method should work with any SDE with drift term $\mu(t)X_t d t$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.