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Solving a Linear SDE with Additive Brownian Noise

Article Quant Q&A · Author: the_src_dude
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Summary

The discussion solves the linear stochastic differential equation in which the state has drift proportional to its level and receives additive Brownian noise. The answer uses the integrating factor exp(−μt): applying the product rule cancels the drift terms and leaves a stochastic integral with a time-varying integrand. Rearranging yields the process at a later time in terms of its value at an earlier time and that integral.

The answer also gives the conditional distribution: it is normal, with mean equal to the earlier state grown by the drift and variance equal to the diffusion coefficient squared times the integrated squared kernel. This contrasts with the geometric Brownian motion equation mentioned in the question, which has multiplicative noise. The response provides a derivation sketch and distributional result but does not discuss numerical simulation or parameter estimation; its applicability is to this linear equation with constant coefficients.

Key ideas

  • An integrating factor removes the proportional drift from the linear SDE.
  • The resulting solution combines drift growth from the earlier state with a stochastic integral.
  • Conditional on the earlier state, the process is normally distributed.
  • Its conditional variance is determined by integrating the squared stochastic kernel.
  • Additive Brownian noise leads to a different equation from geometric Brownian motion’s multiplicative noise.

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# Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$


# Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$












I want to solve the following SDE:

$$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$

Integrating, I get:

$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$ $$ X_{t} = x_{0} + \mu \int_{0}^{t} X_{s} ds + \sigma [W_{t} - W_{0}] $$ $$ X_{t} = x_{0} + \mu \int_{0}^{t} X_{s} ds + \sigma W_{t} $$

This is where I get stuck -- How do I proceed?

I know that the solution to:

$$ dX_{t} = \mu X_{t} dt + \sigma X{t} dW_{t} \quad X_{0} = x_{0}$$

Is given by:

$$ X_{t} = X_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t} + \sigma W_{t}$$

This is given by the fundamental matrix solution. (Though honestly I am not sure how this is derived, so I can't specialize it to my case).

Sorry for such a basic question. I am trying to learn this from resources on the web. So if anyone could recommend a good book dealing with how to actually solve SDEs, I would really appreciate it.

## Answer by Gordon (score 6, accepted)

https://quant.stackexchange.com/a/49156

It appears that you need to read some books such as Stochastic Differential Equations. For such type of equations, you need to use something called integrating factor such as the function $e^{-\mu t}$ here. Note that \begin{align*} d\big(e^{-\mu t} X_t \big) &= X_t d\big(e^{-\mu t}\big) + e^{-\mu t} dX_t\\ &=-\mu e^{-\mu t} X_t dt + e^{-\mu t} (\mu X_t dt + \sigma dW_t)\\ &=\sigma e^{-\mu t} dW_t. \end{align*} Then, for $t > s \ge 0$, \begin{align*} e^{-\mu t} X_t = e^{-\mu s} X_s+ \int_s^t \sigma e^{-\mu v} dW_v. \end{align*} Consequently, \begin{align*} X_t &= X_s e^{\mu (t-s)} + \sigma\int_s^t e^{\mu (t-v)} dW_v\\ &\sim N\Big(X_s e^{\mu (t-s)}, \, \sigma^2\int_s^t e^{2\mu (t-v)} dv \Big). \end{align*}

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