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The Wedge Symbol as the Minimum in Stochastic Calculus

Article Quant Q&A · Author: noisyoscillator
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Summary

The document explains notation in the covariance function of an Itô process with deterministic drift and volatility. In that expression, the integration limit written as the wedge of two times means the smaller of those times. Thus the covariance integral only accumulates volatility up to the time shared by both observations; the later observation does not extend their common Brownian history.

The answer also notes a paired convention: the vee symbol denotes the larger of two values. It links wedge and vee to minimum and maximum, respectively, and observes that this notation is common in financial mathematics, although readers from other mathematical fields may not recognize it immediately. The material is a notation clarification, not a derivation of the covariance formula or a discussion of stochastic calculus beyond this convention. The formula assumes the stated deterministic coefficients, and the answer does not cover cases with random drift or volatility.

Key ideas

  • In this financial mathematics notation, the wedge of two times denotes their minimum.
  • The covariance integral ends at the earlier time because it represents shared Brownian increments.
  • The vee symbol is used for the maximum of two values.
  • The notation is common in finance but may be unfamiliar in other mathematical contexts.

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# Trouble understanding Notation in Stochastic Calculus (wedge symbol ∧)


# Trouble understanding Notation in Stochastic Calculus (wedge symbol ∧)












I am a beginner in Stochastic Calculus. I am having trouble understanding the meaning behind a specific notation which appears in the topic of Ito process which in differential notation can be written as

$dX(t)=\mu (t)dt+\sigma (t)dW(t)$

Now it is mentioned as a fact that if $X(0)$, $\mu (t)$ and $\sigma (t)$ are deterministic functions then $X(t)$ is a Gaussian Process with mean and covariance functions given by

$m(t)=X(0)+\int_{0}^{t}\mu (s)ds$, $c(t_{1},t_{2})=\int_{0}^{t_{1}\land t_{2}}\sigma (s)^{2}ds$

I have trouble understanding the upper limit of the integrand appearing the covariance function i.e. $t_{1}\land t_{2}$. What does that mean logically?

## Answer by Dimitri Vulis (score 2, accepted)

https://quant.stackexchange.com/a/52904

Many financial math (stochastic calculus) books use $\wedge$ to mean minimum: $a \wedge b = \min(a,b)$ and likewise $\vee$ to mean maximum: $a \vee b = \max(a,b)$

For example: Steven E. Shreve. Stochastic Calculus for Finance II. Continuous-Time Models (volume 2) Springer (2004). Section 8.2 Stopping Times:

This notation is actually not immediately recognized even by some math people outside finance. It is sort of consistent with $\wedge$ denoting conjunction or infimum and $\vee$ denoting disjunction or supremum.

Personally, I would have preferred to use dyadic floor and ceiling that the language APL used to use: $a⌊b = \min(a,b)$ and likewise $a ⌈ b = \max(a,b)$. But no one uses that.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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