Are Markov processes martingales?

Are Markov processes martingales?

The Markov property states that a stochastic process essentially has “no memory”. The Martingale property states that the future expectation of a stochastic process is equal to the current value, given all known information about the prior events.

Is every Markov chain a martingale?

Intuitively, this means that the conditional expectation of the future value of the process, given all its historical values, equals to its current value. Note that this is not the same as conditioning only knowing the current value. No, not all martingales are Markov processes.

Are Markov processes stationary?

1: A stochastic process Y is stationary if the moments are not affected by a time shift, i.e., A theorem that applies only for Markov processes: A Markov process is stationary if and only if i) P1(y, t) does not depend on t; and ii) P1|1(y2,t2 | y1,t1) depends only on the difference t2 − t1.

Is Markov chain stationary?

Ergodic Markov chains have a unique stationary distribution, and absorbing Markov chains have stationary distributions with nonzero elements only in absorbing states.

Is Brownian motion a martingale?

The Brownian motion process is a martingale: for s < t, Es(Xt ) = Es(Xs) + Es(Xt − Xs) = Xs by (iii)’.

What is martingale model?

The Martingale system is a system of investing in which the dollar value of investments continually increases after losses, or the position size increases with the lowering portfolio size. The Martingale system was introduced by French mathematician Paul Pierre Levy in the 18th century.

What is irreducible Markov chain?

A Markov chain in which every state can be reached from every other state is called an irreducible Markov chain. If a Markov chain is not irreducible, but absorbable, the sequences of microscopic states may be trapped into some independent closed states and never escape from such undesirable states.

What is a reversible Markov chain?

A Markov chain whose stationary distribution π and transition probability matrix P satisfy (1) is called reversible. Then, the length of the queue is a Markov chain, and in fact it turns out to be reversible.

Are martingales useful?

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

Is the martingale a Markov process?

In terms of a probability function… The martingale described above is also a Markov process unless the wager at t depends on past outcomes (e.g. If you lost or won the last index, you change the wager at t, this is not a Markov process)

What is an example of a Markov process?

If you lost or won the last index, you change the wager at t, this is not a Markov process) We can describe Markov processes in both discrete and continuous-time indexes, where diffusion is defined as a continuous Markov process. The Random Walk Model is the best example of this in both discrete and continuous time.

Do stock prices have the Markov property?

To say that stock prices have the Markov propertyis to assume much greater stability in the data-generating process than is generally believed to be the case. In particular, if prices were a Markov process, then knowledge of merely the current pricewould be a sufficient statistic for the probability distribution of future prices.

Is diffusion a Markov process or not?

If you lost or won the last index, you change the wager at t, this is not a Markov process) We can describe Markov processes in both discrete and continuous-time indexes, where diffusion is defined as a continuous Markov process.