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?

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    $\begingroup$ The wedge symbol $\wedge$ is used to represent minimum, whilst $\vee$ is used to represent the maximum. So $t_1 \wedge t_2=\min \left(t_1,t_2 \right)$ $\endgroup$ – Magic is in the chain Mar 29 '20 at 14:31
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    $\begingroup$ Thanks a lot for clarifying. $\endgroup$ – noisyoscillator Mar 29 '20 at 15:45
  • $\begingroup$ One gimmick I use to remember it, is that $\wedge$ looks like the letter A, you can read it as "And", and logical And can be implemented as the minimum of two Boolean variables. $\endgroup$ – noob2 Mar 30 '20 at 14:32

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.


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