Solving a Linear Stochastic Differential Equation with Itô’s Lemma

Summary

The document derives an explicit solution for a process with a level-proportional drift and constant diffusion, starting from a specified value at time S. It applies an integrating-factor method: multiply the process by an exponential factor, use Itô’s lemma to simplify the dynamics, and integrate from the starting time to the target time. The resulting expression combines the initial value, grown or decayed according to the drift, with a stochastic integral weighted by the same exponential factor.

The answer identifies the process as an Ornstein–Uhlenbeck process when the drift coefficient is negative, in which case it mean-reverts toward zero. The derivation provides a method for obtaining the process path solution, but does not discuss the distribution or moments of the terminal value, nor applications to financial modeling. Its result assumes constant coefficients and standard Brownian motion; further interpretation depends on the signs and values of the parameters.

Key ideas

• An exponential integrating factor converts the linear stochastic differential equation into a form that can be integrated directly.
• Itô’s lemma is used to derive the transformed process dynamics.
• The terminal value consists of a drift-adjusted initial value and a weighted stochastic integral.
• With a negative drift coefficient, the process is an Ornstein–Uhlenbeck process mean-reverting toward zero.

Tags

# The solution to arithmetic brownian motion


# The solution to arithmetic brownian motion












I would like to obtain an explicit solution to $X$ when it satisfies

$$dX_t = \mu X_t dt + \sigma dW_t, X_S = x$$

Here, $S > 0$, and we want an explicit solution for $X_T$, $T > S$.

I am not sure how to approach the problem. Seems more difficult than regular brownian motion!

## Answer by NSZ (score 7)

https://quant.stackexchange.com/a/32724

To solve the SDE you should use the so called variation of constant method. Define a process $Y_t=e^{-\mu t}X_t$, so that using Itô we obtain: $$dY_t=-\mu Y_t dt+ e^{-\mu t}X_t=e^{-\mu t} \sigma dW_t $$ Therefore by integrating we have: $$Y_T=Y_S+\int_S^T e^{-\mu t} \sigma dW_t=e^{-\mu S} X_S +\sigma \int_S^T e^{-\mu t} dW_t$$ $$\Rightarrow \quad X_T=e^{\mu (T-S)} X_S +\sigma \int_S^T e^{\mu (T-t)} dW_t$$ This is a simple Ornstein Uhlenbeck process with mean reversion towards 0 if the coefficient $\mu$ is negative. I think you can easily find it in your Stochastic Calculus reference book.

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.