Is a constant a martingale?
The constant, deterministic sequence Xn = 7 is a martingale: in this case E[Xn+1|Fn]=7= Xn for all n ≥ 0. i=1 An for n ≥ 1. Then the sequence Sn is a martingale.
Is a martingale a Supermartingale?
Similarly, a concave function of a martingale is a supermartingale.
What is a continuous martingale?
A continuous time process (Mt) is a martingale with respect to filtration (Ft) if for any 0 ≤ s ≤ t, we have E[Mt|Fs] = Ms. Every martingale has a right continuous modification that is cadlag, and without further mention, we will always assume we are dealing with this version of the martingale.
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.
Are martingales useful?
Essentially, the martingale property ensures that in a “fair game”, knowledge of the past will be of no use in predicting future winnings. These properties will be of fundamental importance in regard to defining Brownian motion, which will later be used as a model for an asset price path.
Are martingales stationary?
Martingales are nonstationary processes (with one singular exception) and martingale differences/increments are generally nonstationary. If the increments are stationary then there is (i) no ergodicity and (ii) no i.i.d. unless the diffusion coefficient is both time and space translationally invariant.
Is W 3 a martingale?
The second piece on the LHS is an Ito integral and thus a martingale. However the first piece on the LHS in not a martingale and thus W3(t) is not a martingale.
Does a martingale have the Markov property?
Any process with independent increments has the Markov property, eg Brownian motion. Martingale means that we expect the future value to be the current value. Standard Brownian motion has the Markov property and is a martingale.
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.
Is w2 t a martingale?
t − t is a martingale. 2 α2t is a martingale.
What can you do with stochastic calculus?
The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model. The physical process of Brownian motion (in particular, a geometric Brownian motion) is used as a model of asset prices, via the Weiner Process.
What’s the difference between Markov chain and martingale?
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.
Which 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.
Which is a natural generalization of martingales and supermartingales?
Quasimartingales are a natural generalization of martingales, submartingales and supermartingales. They were first introduced by Fisk in order to extend the Doob-Meyer decomposition to a larger class of processes, showing that continuous quasimartingales can be decomposed into martingale and finite variation terms ( Quasi-martingales, 1965).
Is the quasimartingale the sum of two submartingales?
That is, every quasimartingale decomposes as the sum of a submartingale and a supermartingale. Equivalently, every quasimartingale is a difference of two submartingales, or alternatively, of two supermartingales.