Solving a Linear SDE with an Integrating Factor

Summary

The document works through the stochastic differential equation dX = r dt + aX dW with initial value x. The questioner proposes an exponential integrating factor and applies Itô’s lemma, but makes an error when combining the differentials of the factor and X. The accepted response lays out the separate terms, including the quadratic covariation term, which cancels the drift contribution proportional to X.

The resulting product satisfies d(XF) = rF dt, so integrating and then multiplying by the inverse factor gives X as a stochastic integral expression in terms of F and the initial condition. This illustrates why Itô’s product rule includes a cross term and how an integrating factor can simplify a linear SDE. The note gives a derivation but no worked numerical example or discussion of parameter restrictions.

Key ideas

• The SDE has a constant drift term and a diffusion term proportional to the state.
• Itô’s product rule includes a cross-variation term when differentiating the product of the process and its integrating factor.
• The cross term cancels the state-dependent drift terms in the product differential.
• Integrating the simplified equation yields a solution expressed using the inverse factor and an integral over the factor.

Tags

# Help on solving a stochastic differential equation


# Help on solving a stochastic differential equation












I am trying to solve the following SDE $$dX(t)=rdt+aX(t)dW(t),\ t>0$$ $$X(0)=x$$ where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor $$F(t)=exp^{-aW(t)+(1/2)a^{2}t}$$ I have calculated dF using Ito's Lemma

$$dF_{t}=(1/2)a^{2}{F}_{t}dt-a{F}_{t}dW+(1/2)a^{2}{F}_{t}dW^{2}=a^{2}F_{t}dt-aF_{t}dW_{t}$$ and then I proceeded in finding $$d(X_{t}F_{t})=X_{t}dF_{t}+F_{t}dX_{t}+dX_{t}dF_{t}=rF_{t}dt+(a-1)X_{t}F_{t}dW_{t}$$ I have 2 questions:

- Am I correct until now?

- How do I proceed in finally solving the SDE and finding X?

## Answer by ir7 (score 4, accepted)

https://quant.stackexchange.com/a/63439

With your SDE for $F$, I get:

$$ dXdF = -a^2XFdt $$

$$FdX = rFdt + aXFdW $$

$$XdF = a^2XF dt -aXF dW$$

So, adding up:

$$ d(XF) = rF dt, $$

giving

$$ X_t = F^{-1}_t X_0 + rF^{-1}_t \int_0^t F_u du $$

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