[Math] Why are stochastic processes with decreasing expected value called supermartingales

expectationmartingalesstochastic-processesterminology

I am curious to know why a process which has decreasing expected value is called a supermartingale.

From a beginners perspective it would seem reasonable to have the following picture:

                ________ (increasing) above ==> super
            ___/
       ____/
      /
E[X]: ------------------ (constant) martingale
      ___
         \_____________
                       \ (decreasing) below ==> sub

Is there a reason why the names where choosen the way they are?

Edit: here is an additional reference:

Snell: Your book established martingales as one of the small number of important types of > stochastic processes. How do you get interested in martingales?

Doob: [… ] The martingale definition led at once
to the idea of sub and super martingales, and it was clear that these
were the appropriate names but, as I remarked in my 1984 book
((Classical Potential Theory and Its Probabilistic Counterpart,
Springer-Verlag 1984), the name supermartingale was spoiled for me by
the fact that every evening the exploits of "Superman" were played on
the radio by one of my children. If I had been doing my work at the
university rather than at home I am sure I would not have used the
ridiculous names semi- and lower semimartingales for sub- and
supermartingales in my 1953 book. Perhaps I should have noted that one
reason for the success of that book is the prestigious sounding title,
a translation of a name in a German Khintchine paper.

Best Answer

As far as I know, this naming has its origin in the connection between Brownian motion and harmonic functions. Let $B$ be a $d$-dimensional Brownian motion. Let $f:\mathbb{R}^d\to\mathbb{R}$ be twice continuously differentiable. By Ito's formula, we have $$ f(B_t) = f(0) + \sum_{i=1}^d\int_0^t \frac{\partial f}{\partial x_i}(B_s) dB^i_s + \frac{1}{2}\sum_{i=1}^d \int_0^t \frac{\partial^2 f}{\partial x_i^2}(B_s) ds. $$ Here, the sum of the first two terms is a local martingale, but let's just think of it as a martingale, call it $M$. Also introduce the Laplace operator as $$ \Delta f = \sum_{i=1}^d \frac{\partial^2 f}{\partial x_i^2}. $$ We then have $$ f(B_t) = M_t + \frac{1}{2} \int_0^t \Delta f(B_s) ds. $$ Now, a function $f$ is called harmonic when $\Delta f = 0$, subharmonic when $\Delta f\ge0$ and superharmonic when $\Delta f\le 0$.

From the above, we then see that $f(B_t)$ is a martingale when $f$ is harmonic, a supermartingale when $f$ is superharmonic and a submartingale when $f$ is subharmonic. This shows the benefit of the definitions of supermartingale and submartingale: It allows for a notational correspondence to superharmonic and subharmonic functions. Of course, why functions are called "subharmonic" and "superharmonic" in the way that they are is then a different question...

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