Product Rules for Stochastic Differentials and Quadratic Covariation

Summary

The document gives the product rule for a process multiplied by a deterministic, finite-variation function, then extends it to the product of two stochastic processes. For a finite-variation function and a process, the ordinary two-term product rule applies. For two processes, an additional quadratic covariation term appears.

That extra term captures the joint infinitesimal variation that ordinary calculus omits. When one process has finite variation, its quadratic covariation with the other process is zero, so the rule reduces to the usual form. The explanation identifies the two-process formula as an application of Itô's formula to the product function. It is a concise rule statement; applying it in a model still requires checking the processes' variation properties and interpreting the covariation term correctly.

Key ideas

• Multiplying a process by a finite-variation function follows the ordinary product rule.
• The product of two stochastic processes includes a quadratic covariation term.
• Quadratic covariation vanishes when one factor has finite variation.
• The two-process product rule follows from Itô's formula applied to the product function.

Tags

# Simple question about stochastic differential


# Simple question about stochastic differential












What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let $X_t$ be a process and $\alpha(t)$ a real function. What would be $d(\alpha(t)X_t)$?

## Answer by quasi (score 6, accepted)

https://quant.stackexchange.com/a/4734

If $\alpha(t)$ is of finite variation, then the product rule is the same as in ordinary calculus:

$$ d(\alpha(t)X_t) = \alpha(t) dX_t + X_t d\alpha(t). $$

If you had $X_t$ and $Y_t$ as processes, you would get

$$ d(X_t Y_t) = X_t dY_t + Y_t dX_t + d [X,Y]_t. $$

If $Y$ has finite variation, the last quadratic covariation term is zero. The second equation is just applying Ito's Formula to $f(x,y) = xy$.

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