Itô’s Lemma and the Transformed Diffusion SDE

Summary

The document presents the Itô differential for a twice-differentiable function of a one-dimensional diffusion. The transformed process inherits a drift containing the function’s time derivative, the original drift times its first spatial derivative, and a second-derivative correction proportional to variance. Its random component scales the Brownian increment by the original volatility and the function’s spatial derivative.

The question asks whether this resulting equation has a formal name and what can be said about existence, uniqueness, or convergence to a stationary distribution. The displayed identity is an application of Itô’s lemma; the document does not answer those follow-up questions. Such properties depend on the coefficients and on conditions for the transformed process, so the formula alone does not establish them.

Key ideas

• Applying Itô’s lemma gives an SDE for a smooth function of a diffusion process.
• The transformed drift includes a second-derivative correction based on the diffusion variance.
• The transformed noise coefficient is the original volatility multiplied by the function’s spatial derivative.
• Existence, uniqueness, and stationarity require additional assumptions beyond the displayed formula.

Tags

# The solution of SDE after Itô lemma for diffusion process


# The solution of SDE after Itô lemma for diffusion process












Consider the one-dimensional diffusion process $dX_t = \mu_tdt + \sigma_tdB_t$ and function $f : \mathbb{R} \to \mathbb{R}$, which is twice differentiable. Here we have another SDE by using Itô lemma as follows; $$ d f\left(t, X_t\right)=\left(\frac{\partial f}{\partial t}+\mu_t \frac{\partial f}{\partial x}+\frac{\sigma_t^2}{2} \frac{\partial^2 f}{\partial x^2}\right) d t+\sigma_t \frac{\partial f}{\partial x} d B_t $$

I know that Itô lemma is usually used to solve SDE about $X_t$. However, I also would like to know about this new SDE of $f(t, X_t)$.

Does this new SDE have any formal name? Are there any properties about this new SDE, for example the existence and uniqueness of the solution or the convergence to stationary distribution?

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