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Solving a Linear SDE with Proportional Drift and Additive Noise

Article Quant Q&A · Author: Kiann
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Summary

The document derives a solution for a stochastic differential equation with drift proportional to the state and additive Brownian noise. Multiplying by an exponential integrating factor removes the drift term, after which integration expresses the terminal value as a deterministic growth component plus a weighted stochastic integral. This is the standard route to finding the process distribution and its moments; the stochastic integral is Gaussian, so its variance can be evaluated from the squared integrand.

The answer compares the equation to a zero long-run-mean Ornstein–Uhlenbeck process, with a sign adjustment for the drift parameter. Its scope is narrower than the original question: it treats additive noise, not the multiplicative-noise geometric form, and does not provide explicit mean and variance formulas. It also does not develop the requested LIBOR market model drift adjustment. Those limits matter when applying the derivation to a different SDE.

Key ideas

  • An exponential integrating factor removes the proportional drift from a linear SDE with additive noise.
  • The terminal state is a deterministic exponential term plus a weighted Brownian integral.
  • The stochastic integral is Gaussian, and its variance follows from integrating the squared weight.
  • The derivation addresses additive noise and does not solve the multiplicative-noise case.

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# Stochastic solution (mean, variance) to lognormal drift and normal volatility


# Stochastic solution (mean, variance) to lognormal drift and normal volatility












I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model

I have this equation : df = u * F * dt + sigma * dW(F)

I am having trouble getting the expected mean, variance, and it's final stochastic differential equation of the form : F(T) = F(0) * exp(...)

For example, given the geometric form df = u * F * dt + sigma * F * dW(F)

I know the

expected mean =

expected variance =

For example arithmetric form, given df = u * dt + sigma * dW(F)

expected mean = u * t

expected variance = sigma * T

But I am getting stumped on the equation which is a mixture of the two.

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/45776

Just use the integrating factor method.

$df=\mu f dt +\sigma dW_t$

Multiply by the integrating factor:

$e^{-\mu t} df =e^{-\mu t} \mu f dt +\sigma e^{-\mu t} dW_t$

$d\left( e^{-\mu t} f \right)=\sigma e^{-\mu t} dW_t$

Now just integrate from 0 to T:

$e^{-\mu T}f_T-f_0=\sigma \int_0^T{e^{-\mu t} dW_t}$

And you can then just isolate $f_T$ on the left hand side.

$f_T=f_0 e^{\mu T}+ \sigma \int_0^T{e^{\mu (T-t)} dW_t}$

And you see it’s like Ornstein Uhlenbeck with long term mean equal to zero, and just look up the mean and variance of that process, but you will have to replace $\mu$ with minus $\mu$ if you would like to make the comparison clearer.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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