Is martingale a fair game?
Martingales The concept of martingale has its origin in gambling. It describes a fair game of chance. Favorable and unfavorable games are described by submartingales and supermartingales. Because option pricing is based on a “fair game” assumption, martingales play an important, crucial role in it.
What is positive martingale?
The sequence is called a positive martingale if the inequality is replaced by an equality. This definition makes sense as the conditional expectation of an arbitrary positive real-valued r.v. was defined in the first chapter (Lemma 1-2-9).
Is martingale a supermartingale?
Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is both a submartingale and a supermartingale is a martingale. If p is less than 1/2, the gambler loses money on average, and the gambler’s fortune over time is a supermartingale.
Do martingales converge?
The martingale convergence theorems, first formulated by Joseph Doob, are among the most important results in the theory of martingales. The first martingale convergence theorem states that if the expected absolute value is bounded in the time, then the martingale process converges with probability 1.
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.
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.
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.
How do you show a martingale is uniformly integrable?
Definition 2.7. (Mn,Fn)n≥1 is an uniformly integrable martingale, if (Mn,Fn)n≥1 is a martingale, and the family Mn, n ≥ 1 is uniformly integrable. (2.24) Mn = IE[M∞|Fn].
How do I show a process in martingale?
Note that to check a process is a martingale, it suffices to check property (iii) (which is usually called “the martingale property”) since if it holds, then the condi- tional expectation makes sense, so (ii) holds, and since the conditional expectation is measurable with respect to the σ-field being conditioned on, it …
Does martingale work in forex?
The martingale strategy works much better in forex trading than gambling because it lowers your average entry price.
Is w2 t a martingale?
t − t is a martingale. 2 α2t is a martingale.
What is exponential martingale?
Exponential Martingale. Page 1. Math 280C, Spring 2005. Exponential Martingales. In what follows, (Ω,F,P) is the canonical sample space of the Brownian motion (Bt)t≥0 with B0 = 0; other notation is that used in class.
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