Interpreting Expected Products of Stochastic Differentials

Summary

The document asks what an expression involving the expected product of two stochastic differentials means, given that differentials of random processes are not ordinary standalone quantities. The answer explains a stochastic differential equation as shorthand for an integral representation: a drift integral with respect to time and an Itô integral with respect to Brownian motion. It notes that stochastic integrals can be defined rigorously through limits of partition sums.

The reply interprets the differential product as a covariation increment, related to quadratic variation and often viewed informally as instantaneous covariance. This intuition is useful, but the response is tentative and does not state precise conditions under which the notation is valid. It also gives two differing bracket symbols, one apparently containing a typographical error. A careful treatment should distinguish quadratic covariation from covariance and specify the processes’ assumptions and interpretation of the expectation.

Key ideas

• A stochastic differential equation represents a process through drift and stochastic integrals.
• The Itô integral is defined through a limiting procedure over increasingly fine partitions.
• The product of stochastic differentials is associated with quadratic covariation.
• Covariation can be viewed informally as an instantaneous covariance contribution.
• The answer is informal and leaves notation and mathematical assumptions underspecified.

Tags

# Expectation of Stochastic Differential


# Expectation of Stochastic Differential












First of all, I am a mathematician, so I apologize for my ignorance regarding stochastic calculus. What exactly does an expression like:

$$ \mathbb{E}[dX_tdY_t] $$ here $X_t,Y_t$ are stochastic processes? The differentials $dX_t, dY_t$ are not exactly well defined, so I was wondering what this expression means.

## Answer by byouness (score 6, accepted)

https://quant.stackexchange.com/a/46531

In stochastic calculus, expressions of the type: $$ dX_t = a(t, X_t)dt + b(t, X_t) dW_t $$

are called stochastic differential equations.

What the one above means for example is that $X_t$ has the following expression: $$ X_t = X_0 + \int_0^t a(u, X_u)du + \int_0^t b(u, X_u) dW_u $$

The first integral is a regular one, and the second is called a stochastic integral or Ito integral. You can find a rigorous definition of stochastic integrals in any stochastic calculus notebook. It is defined as the limit of a sum over some subdivision when its mesh goes to zero (similar to how the Riemann integral is defined).

See for example:https://en.wikipedia.org/wiki/It%C3%B4_calculus

As for the symbol's definition, I think $\mathbb{E} \left[dX_tdY_t \right]$ denotes the covariation of $X$ and $Y$, which is sometimes denoted $d\langle X, Y\rangle_t$ or $d[X, T]_t$.

This quantity also has a rigorous mathematical definition (also as a sum, resembling that of a covariance, over a partition when the mesh goes to zero). You can think of it as the instantaneous covariance between $X$ and $Y$.

See for example: https://en.wikipedia.org/wiki/Quadratic_variation

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