Applying Itô’s Formula to a Function of One Stochastic Variable

Summary

The document addresses how to apply Itô’s formula to a process defined as a function of another stochastic process, even when the function has no explicit time argument. The key step is to write the quantity as a function of the underlying state variable alone, with time dependence entering through that state variable.

For a function of one variable, the differential includes a first derivative term multiplied by the underlying process differential and a second derivative term multiplied by half the process’s quadratic variation. This explains why the two-variable version of Itô’s formula is not required in this setup. The answer gives the general rule but leaves the substitution and calculation for the specific reciprocal function unfinished; the underlying dynamics are also not supplied in the excerpt.

Key ideas

• A process may depend on time indirectly through its stochastic state variable.
• Represent the quantity as a one-variable function of the underlying process.
• The differential includes both a first derivative term and a quadratic variation correction.
• The specific calculation requires the dynamics of the underlying process, which are not shown.

Tags

# Compute dZ(t) : Ito's formula/lemma


# Compute dZ(t) : Ito's formula/lemma












We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables

But here Z(t) = 1/(2+x(t)), which just has one variable?

So, I am not sure how to proceed. Any tips will be appreciated!

## Answer by ForumWhiner (score 1)

https://quant.stackexchange.com/a/51859

It looks like $Z(t)$ is dependent on $t$ through $X_t$ and not directly on $t$, loosely speaking. Assume that $Z_t = f(X_t)$ and use Ito's formula with just one variable.

$$dZ_t = \frac{df}{dX} dX_t + \frac{1}{2} \frac{d^2f}{dX^2} d[X_t,X_t] $$

Can you proceed and finish it?

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