Why are stochastic processes with decreasing expected value called supermartingales?

Question

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.

Answer 1

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...

Answer 2

There is a quote on this point from page 808 in J.L. Doob's Classical Potential Theory and Its Probabilistic Counterpart.

Before martingales had been formally christened, Lévy [1, 1935; 2, 1937], Bernstein [1, 1937], and other mathematicians had analyzed some of their properties in special contexts; usually the martingales in question arose as partial sums $n\mapsto \sum_0^n y_j$ of a sequence $y$ of random variables under the condition $\mathbb{E}(y_j\mid y_0,\dots, y_{j-1})=0$ so that the sums arose as generalizations of sums of independent random variables with zero means. Ville [1, 1939] defined a martingale very nearly as a positive martingale is now defined but tied it to a sequence of independent random variables under analysis. His fundamental tool, a fact he proved, was that almost every sample sequence of a positive martingale is bounded (see Theorem 9). Doob [1, 1940] discussed martingales and proved the basic convergence properties under the name "family of random variables with the property $E$." ("$E$" was chosen not as the initial letter of "expectation" but as the first letter in the alphabet following "$D$".) Under the respective names "semimartingale" and "lower semimartingale," submartingales and supermartingales were introduced in [Snell 1, 1952] and [Doob 4, 1953]. This obviously inappropriate nomenclature was chosen under the malign influence of the noise level of radio's SUPERman program, a favorite supper-time program of Doob's son during the writing of [Doob 4, 1953]"

Bernstein [1] = Serge Bernstein, On some transformations of the Chebyshev inequality. (Russian) Dokl. Akad. Nauk. SSSR 17 (1937), 275-277.

Doob [1] = Doob, J. L., Regularity properties of certain families of chance variables. Trans. Amer. Math. Soc. 47, (1940). 455–486.

Doob [4] = Doob, J. L., Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.

Lévy [1] = Paul Lévy, Propriétés asymptotiques des sommes de variables aléatoires enchainées, Bull. Soc. Math. Fr. 59, (1935), 1-32.

Lévy [2] = Paul Lévy, Théorie de l'Addition des Variables Aléatoires. Paris, Gauthier-Villars, 1937.

Snell [1] = Snell, J. L. Applications of martingale system theorems. Trans. Amer. Math. Soc. 73, (1952). 293–312.

Ville [1] = Jean Ville, Etude Critique de la Notion de Collectif. Paris, Gauthier-Villars, 1939.