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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} .
An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with W 0 = 0 and quadratic variation [W t, W t] = t (which means that W t 2 − t is also a martingale).
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.
A sample path of an Itō process together with its surface of local times. In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level.
A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that X n 2 − n is a martingale). Similarly, a concave function of a martingale is a supermartingale.
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.
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.