Oh God I loved Wink. He was the best really knew how to run a show not like that swishy Howi Mandel! But when did he get into predicting outcomes!
First some backround- Originally, martingale referred to a class of betting strategies that was popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since as a gambler's wealth and available time jointly approach infinity his probability of eventually flipping heads approaches 1, the martingale betting strategy was seen as a sure thing by those who practiced it. Of course in reality the exponential growth of the bets would eventually bankrupt those foolish enough to use the martingale for a long time.
IT's called DOUBLE DOWN IN STOCKS!!! Man thread closed. Nothing is new- it's all human instincts, you got to go put a fancy name on it I don't understand.
It's really fascinating stuff-
A discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for all n
\mathbf{E} ( \vert X_n \vert )< \infty
\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n,
i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation.
Somewhat more generally, a sequence Y1, Y2, Y3, ... is said to be a martingale with respect to another sequence X1, X2, X3, ... if for all n
\mathbf{E} ( \vert Y_n \vert )< \infty
\mathbf{E} (Y_{n+1}\mid X_1,\ldots,X_n)=Y_n.
Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t
\mathbf{E} ( \vert Y_t \vert )<\infty
\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s, \ \forall\ s \leq t.
This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time s, is equal to the observation at time s (of course, provided that s ≤ t).
In full generality, a stochastic process Y : T Ã Ω → S is a martingale with respect to a filtration Σ∗ and probability measure P if
* Σ∗ is a filtration of the underlying probability space (Ω, Σ, P);
* Y is adapted to the filtration Σ∗, i.e., for each t in the index set T, the random variable Yt is a Σt-measurable function;
* for each t, Yt lies in the Lp space L1(Ω, Σt, P; S), i.e.
\mathbf{E}_{\mathbf{P}} ( | Y_{t} | ) < + \infty;
* for all s and t with s < t and all F ∈ Σs,
\mathbf{E}_{\mathbf{P}} \left([Y_t-Y_s]\chi_F\right)=0,
where χF denotes the indicator function of the event F. In [1], this last condition is denoted as
Y_s = \mathbf{E}_{\mathbf{P}} ( Y_t | \Sigma_s ),
which is a general form of conditional expectation.
It is important to note that the property of being a martingale involves both the filtration and the probability measure (with respect to which the expectations are taken). It is possible that Y could be a martingale with respect to one measure but not another one; the Girsanov theorem offers a way to find a measure with respect to which an Itō process is a martingale. That's easy! Well yes we should incorporate all of this into helping us invest better.
~stoney
First some backround- Originally, martingale referred to a class of betting strategies that was popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since as a gambler's wealth and available time jointly approach infinity his probability of eventually flipping heads approaches 1, the martingale betting strategy was seen as a sure thing by those who practiced it. Of course in reality the exponential growth of the bets would eventually bankrupt those foolish enough to use the martingale for a long time.
IT's called DOUBLE DOWN IN STOCKS!!! Man thread closed. Nothing is new- it's all human instincts, you got to go put a fancy name on it I don't understand.
It's really fascinating stuff-
A discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for all n
\mathbf{E} ( \vert X_n \vert )< \infty
\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n,
i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation.
Somewhat more generally, a sequence Y1, Y2, Y3, ... is said to be a martingale with respect to another sequence X1, X2, X3, ... if for all n
\mathbf{E} ( \vert Y_n \vert )< \infty
\mathbf{E} (Y_{n+1}\mid X_1,\ldots,X_n)=Y_n.
Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t
\mathbf{E} ( \vert Y_t \vert )<\infty
\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s, \ \forall\ s \leq t.
This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time s, is equal to the observation at time s (of course, provided that s ≤ t).
In full generality, a stochastic process Y : T Ã Ω → S is a martingale with respect to a filtration Σ∗ and probability measure P if
* Σ∗ is a filtration of the underlying probability space (Ω, Σ, P);
* Y is adapted to the filtration Σ∗, i.e., for each t in the index set T, the random variable Yt is a Σt-measurable function;
* for each t, Yt lies in the Lp space L1(Ω, Σt, P; S), i.e.
\mathbf{E}_{\mathbf{P}} ( | Y_{t} | ) < + \infty;
* for all s and t with s < t and all F ∈ Σs,
\mathbf{E}_{\mathbf{P}} \left([Y_t-Y_s]\chi_F\right)=0,
where χF denotes the indicator function of the event F. In [1], this last condition is denoted as
Y_s = \mathbf{E}_{\mathbf{P}} ( Y_t | \Sigma_s ),
which is a general form of conditional expectation.
It is important to note that the property of being a martingale involves both the filtration and the probability measure (with respect to which the expectations are taken). It is possible that Y could be a martingale with respect to one measure but not another one; the Girsanov theorem offers a way to find a measure with respect to which an Itō process is a martingale. That's easy! Well yes we should incorporate all of this into helping us invest better.
~stoney
You're one smart guy Stoney!