Solving a Linear Mean-Reverting SDE with an Integrating Factor

Summary

The document presents a linear stochastic differential equation with mean-reverting drift and constant diffusion, then shows how to remove the state-dependent drift using an integrating factor. Applying Itô’s lemma to the product of the process and an exponential time factor cancels the drift term, leaving a stochastic integral whose integrand depends only on time.

Integrating this transformed equation gives an explicit representation of the process as its initial value, scaled by exponential decay, plus a Brownian stochastic integral. Because that integral has a deterministic integrand, it is normally distributed, which provides a direct route to the process distribution and its moments. The answer addresses the key algebraic step behind the stated mean and variance, but does not work through the final integral or derive those moments in detail. Its result assumes the stated constant-coefficient model and Brownian driver.

Key ideas

• An exponential integrating factor can eliminate the linear drift in this mean-reverting SDE.
• Itô’s lemma shows that the transformed process has only a time-dependent diffusion term.
• The integrated solution is the decayed initial state plus a Brownian stochastic integral.
• A Brownian integral with deterministic integrand is normally distributed.
• The brief answer does not fully derive the final mean and variance expressions.

Tags

# prove the normality, with given moments, of this process:


# prove the normality, with given moments, of this process:












I have this process:

$dx_t = -\frac{k}{2}x_tdt + \frac{\beta}{2}dz_t$

and must prove it's normally distributed with first two moments:

$\mu = e^{-\frac{1}{2}kt}x_0$

$\sigma^2 = \frac{\beta^2}{4k}(1-e^{-kt})$

I tried to multiply $x_t$ by $e^{kt}$ and apply Ito's Lemma to this 'product process' in order to eventually recover back $x_t$ by taking exponentials.

The normality is straightfoward; the variance is ok but the mean isn't since I'm left with an integral whose integrand includes $x_t$ and I'm stuck.

I don't know whether I made some mistakes or adopted the wrong approach since the beginning.

## Answer by Neeraj (score 1, accepted)

https://quant.stackexchange.com/a/24702

You have following SDE $$dx_t=-\frac{k}{2}x_t dt+ \frac{\beta}{2}dz_t \tag{1}$$ Consider, $f=e^{\frac{k}{2}t}x_t$. Using Ito: $$df= \frac{k}{2}e^{\frac{k}{2}t}x_t\, dt+ e^{\frac{k}{2}t} \,dx_t \tag{2} $$ So, we have \begin{align} d\left(e^{\frac{k}{2}t}x_t\right)&= \frac{k}{2}e^{\frac{k}{2}t}x_t\, dt + e^{\frac{k}{2}t} \left(-\frac{k}{2}x_t dt+ \frac{\beta}{2}dz_t \right)\\ &= \frac{\beta}{2}e^{\frac{k}{2}t}dz_t \end{align} Note: RHS does not involve $x_t$. Now integrate both side to get your answer.

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