Applying Itô’s Lemma to a Product of a Diffusion and Decay Process

Summary

The example derives the stochastic differential for a state variable defined as one half the product of a geometric Brownian motion and a deterministic coefficient that decays exponentially. It applies the multivariable form of Itô’s lemma, listing the first and second partial derivatives with respect to both variables. Because the product function is linear in each variable separately, its pure second derivatives vanish; the coefficient’s deterministic finite-variation change also contributes no quadratic-variation term.

Substituting the given dynamics yields a drift proportional to the sum of the diffusion’s growth rate and the coefficient’s decay rate, with a diffusion term inherited from the Brownian component. The exercise illustrates how to combine stochastic and deterministic processes. In general, a product of two variables can have a mixed-variation contribution when both carry Brownian risk; here the coefficient has only a time drift, so that contribution is zero. The result assumes the stated dynamics and does not discuss parameter interpretation or boundary conditions.

Key ideas

• Apply multivariable Itô’s lemma to the function defining the product process.
• The pure second derivatives vanish because the function is linear in each input separately.
• The deterministic coefficient contributes drift but has no quadratic variation.
• The resulting drift combines the diffusion growth rate with the coefficient’s decay rate.
• A mixed quadratic-variation term would matter if both factors had stochastic components.

Tags

# Ito`s Lemma problem


# Ito`s Lemma problem












Can someone help me with calculus for this problem.

I have these 3 equations and with Ito`s Lemma I have to find $dXt$.

\begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}

## Answer by ZRH (score 6, accepted)

https://quant.stackexchange.com/a/45575

write down Ito's lemma for the function X:

$$dX=\frac{\partial X}{\partial Y}dY+\frac{1}{2}\frac{\partial^2 X}{\partial Y^2}(dY)^2+\frac{\partial X}{\partial c}dc+\frac{1}{2}\frac{\partial^2 X}{\partial c^2}(dc)^2+\frac{\partial^2 X}{\partial Y \partial c}dYdc+\frac{\partial^2 X}{\partial c \partial Y}dcdY$$

Using the following:

$\frac{\partial X}{\partial Y}=\frac{1}{2}c$, $\frac{\partial^2 X}{\partial Y^2}=0$

$\frac{\partial X}{\partial c}=\frac{1}{2}Y$, $\frac{\partial^2 X}{\partial c^2}=0$

$\frac{\partial^2 X}{\partial Y \partial c}=\frac{\partial^2 X}{\partial c \partial Y}=0$

Inserting these 4 expressions into the above Ito formula, one gets to:

$$dX=\frac{1}{2}cdY+\frac{1}{2}Ydc=cY(\frac{\mu}{2}-\frac{a\alpha}{2})dt+\frac{\sigma}{2}YcdB$$

where the initial expressions for $dY$ and $dc$ have been substituted back in the last step. The solutions for $Y$ and $c$, are trivial: They are the solution of the SDE for a GBM, and an exponential decay, respectively

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