Martingale (probability Theory) - Submartingales, Supermartingales, and Relationship To Harmonic Functions

Submartingales, Supermartingales, and Relationship To Harmonic Functions

There are two popular generalizations of a martingale that also include cases when the current observation X_n is not necessarily equal to the future conditional expectation E but instead an upper or lower bound on the conditional expectation. These definitions reflect a relationship between martingale theory and potential theory, which is the study of harmonic functions. Just as a continuous-time martingale satisfies E − X_s = 0 ∀s ≤ t, a harmonic function f satisfies the partial stochastic differential equation Δf = 0 where Δ is the Laplacian operator. Given a Brownian motion process W_t and a harmonic function f, the resulting process f(W_t) is also a martingale.

• A discrete-time submartingale is a sequence of integrable random variables satisfying

Likewise, a continuous-time submartingale satisfies

In potential theory, a subharmonic function f ]satisfies Δf ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball are bounded above by the harmonic function for all points inside the ball. Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the prefix "sub-" is consistent because the current observation X_n is less than (or equal to) the conditional expectation E. Consequently, the current observation provides support from below the future conditional expectation, and the process tends to increase in future time.

• Analogously, a discrete-time supermartingale satisfies

Likewise, a continuous-time supermartingale satisfies

In potential theory, a superharmonic function f satisfies Δf ≤ 0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball are bounded below by the harmonic function for all points inside the ball. Similarly, if a supermartingale and a martingale have equivalent expectations for a given time, the history of the supermartingale tends to be bounded below by the history of the martingale. Roughly speaking, the prefix "super-" is consistent because the current observation X_n is greater than (or equal to) the conditional expectation E. Consequently, the current observation provides support from above the future conditional expectation, and the process tends to decrease in future time.