What is the meaning of Submartingale?

What is the meaning of Submartingale?

submartingale (plural submartingales) (mathematics) A stochastic process for which the conditional expectation of future values given the sequence of all prior values is superior or equal to its current value. If a gambler repeatedly plays a game with positive expectation, his payoff over time is a submartingale.

Is a Submartingale a martingale?

Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is both a submartingale and a supermartingale is a martingale.

What is discrete random walk?

A discrete random walk is the path produced by a sequence of unit steps at discrete times. The unit steps can be positive or negative, and the discrete times are taken to be integers (in some time unit, such as days).

What is symmetric random walk?

A symmetric random walk is a random walk in which p = 1/2. Thus, a symmetric simple random walk is a random walk in which Xi = 1 with probability 1/2, and Xi = − 1 with probability 1/2.

Are martingales useful?

Martingales are critical in models of gambling (and by extension, stochastic control and optimal stopping).

How can you tell if its a martingale?

3.1. In general, if Yt+1-Yt = bt(Xt+1-Xt) where (Xt,ℱt) is a martingale and bt is measurable ℱt, then Yt is also a martingale with respect ℱt.

Why it is called martingale?

He got the name from a thesis by Ville. A martingale is the name for a Y-shaped strap used in a harness — it runs along the horse’s chest and then splits up the middle to join the saddle. “Martingale pants” are from Martigues, and have, according to Rabelais, “a drawbridge on the ass that makes excretion easier.”

How can you tell a martingale?

What is random walk used for?

It is the simplest model to study polymers. In other fields of mathematics, random walk is used to calculate solutions to Laplace’s equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics. In computer science, random walks are used to estimate the size of the Web.

What is Gaussian random walk?

4.1 Gaussian Random Walk. Let’s start with a simple stochastic process that we’ve already met: a Gaussian Random Walk, which is essentially a series of i.i.d. N(0,1) N ( 0 , 1 ) random variables through time: X0=0 X 0 = 0 Xt=Xt−1+ϵt.

What is recurrent random walk?

Definition 2.15 [Recurrence & transience] We say that a random walk is recurrent if it visits its starting position infinitely often with probability one and transient if it visits its starting position finitely often with probability one.

Why do we care about martingales?

In mathematical finance and economics, martingales are crucial for pricing models. For example, if we model a financial asset as a random process, we demand pricing rules (measures) under which the asset is a martingale.

What is an example of a submartingale?

Examples of submartingales and supermartingales Every martingale is also a submartingale and a supermartingale. Consider again the gambler who wins $1 when a coin comes up heads and loses $1 when the coin comes up tails. A convex function of a martingale is a submartingale, by Jensen’s inequality.

How do you prove that a walk is a martingale?

Proof: E [Y t+1 -Y t |ℱ t] = E [b t (X t+1 -X t )|ℱ t] = b t E [X t+1 -X t |ℱ t] = b t ⋅0 = 0. Special case: A random ±1 walk is a martingale. Another random walk: Let X t+1 = X t ±1 with equal probability and let Y t = X t2 – t.

What is a martingale in statistics?

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale.

How do you find the submartingale of a function?

Define a function μ = μX on J by μ ( J) := E X ( J ). According to the previous paragraph, μ is an additive set function on the ring J. It follows from the definition of a submartingale that X is a submartingale if and only if μ is nonnegative on R and, hence, on J (recall that we consider only right-continuous submartingales).