Table of Contents
- 1 What is the intuition of a martingale?
- 2 Why are martingales so important?
- 3 Is a martingale a random walk?
- 4 Is W 3 a martingale?
- 5 Are all Brownian motions martingales?
- 6 Is Brownian motion with drift a martingale?
- 7 Is w2 t a martingale?
- 8 Is a constant a martingale?
- 9 Where does the breakdown of a martingale occur?
- 10 What is the quantitative barrier to a local martingale being true?
What is the intuition of a martingale?
The martingale property says precisely that in any particular round your total earnings are expected to stay the same regardless of the outcomes of the previous rounds. For me the most important intuition was that filtration represent information.
Why are martingales so important?
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.
How do you prove a martingale?
The useful property of martingales is that we can verify the martingale property locally, by proving either that E[Xt+1|ℱt] = Xt or equivalently that E[Xt+1 – Xt|ℱt] = E[Xt+1|ℱt] – Xt = 0. But this local property has strong consequences that apply across long intervals of time, as we will see below.
Is a martingale a random walk?
A Martingale process is similar to a one-dimensional random walk. The only difference is that you can vary the size of your bet. For the standard Martingale, you try to guarantee a win. Every time you lose, you double your stake.
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.
Is martingale a good strategy?
The Martingale Strategy is a strategy of investing or betting introduced by French mathematician Paul Pierre Levy. It is considered a risky method of investing. It is based on the theory of increasing the amount allocated for investments, even if its value is falling, in expectation of a future increase.
Are all Brownian motions martingales?
Martingale properties: The Brownian motion process is a martingale: for s < t, Es(Xt ) = Es(Xs) + Es(Xt − Xs) = Xs by (iii)’. = Ms because Es(X) = 0 and Es(X)2 = t − s.
Is Brownian motion with drift a martingale?
When the drift parameter is 0, geometric Brownian motion is a martingale. If , geometric Brownian motion is a martingale with respect to the underlying Brownian motion .
Is 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.
Is w2 t a martingale?
2 α2t is a martingale. Show that the Itô integral defined above is a martingale with respect to the standard Brownian motion.
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. Then the sequence Sn is a martingale.
What is the difference between submartingale and martingale?
Firstly, a Submartingale has increasing or equal expectation (not decreasing). Secondly, the process d X t = X t d W t is a true martingale (not strictly local), since its solution (by Ito): has E ( X t) = X 0 constant expectation ( e − t 2 E ( e W t) = 1, W t ∼ N ( 0, t) ). The negative drift comes from W t ‘s nonzero quadratic variation.
Where does the breakdown of a martingale occur?
The breakdown occurs in the limit. In fact, this is where the name comes from: locally, a local martingale does look like a martingale. Simulation When you simulate, you should get zero drift. In the limit is where the mass will be lost.
What is the quantitative barrier to a local martingale being true?
A quantitative barrier to a local martingale being a true martingale is integrability. An example is as follows: ∫ 0 t f ( s) d B s where B s is a Brownian motion and f progressively measurable is a strict local martingale if ( E ∫ 0 t | f ( s) | 2 d s) α / 2 = ∞ for some α between 0 and 1.
Is the Ito process a martingale?
All Ito processes desrcibed by a ‘driftless SDE’ are in fact local martingales, but not martingales (which is surprising to many). For example the familiar Geometric Brownian motion is a local martingale but not a true martingale. In fact, starting from Y 0 the expectation