Search results
Results from the WOW.Com Content Network
The quadratic variation exists for all continuous finite variation processes, and is zero. This statement can be generalized to non-continuous processes. Any càdlàg finite variation process X {\displaystyle X} has quadratic variation equal to the sum of the squares of the jumps of X {\displaystyle X} .
First, the Doob–Meyer decomposition theorem is used to show that a decomposition M 2 = N + M exists, where N is a martingale and M is a right-continuous, increasing and predictable process starting at zero. This uniquely defines M , which is referred to as the predictable quadratic variation of M.
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
Stochastic exponential of a local martingale is again a local martingale. All the formulae and properties above apply also to stochastic exponential of a complex -valued X {\displaystyle X} . This has application in the theory of conformal martingales and in the calculation of characteristic functions.
The alternative (and preferred) terminology quadratic pure-jump semimartingale for a purely discontinuous semimartingale (Protter 2004, p. 71) is motivated by the fact that the quadratic variation of a purely discontinuous semimartingale is a pure jump process. Every finite-variation semimartingale is a quadratic pure-jump semimartingale.
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, [1] also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random ...
The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge. As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening.
Visualisation of the Girsanov theorem. The left side shows a Wiener process with negative drift under a canonical measure P; on the right side each path of the process is colored according to its likelihood under the martingale measure Q. The density transformation from P to Q is given by the Girsanov theorem.