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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.
where is the Dirac delta function and [] is the quadratic variation. It is a notion invented by Paul Lévy . The basic idea is that L x ( t ) {\displaystyle L^{x}(t)} is an (appropriately rescaled and time-parametrized) measure of how much time B s {\displaystyle B_{s}} has spent at x {\displaystyle x} up to time t {\displaystyle t} .
In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be ...
Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. [1] [2] The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that ...
The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that ( M t ) 0 ≤ t < ∞ {\displaystyle \left(M_{t}\right)_{0\leq t<\infty }} is a Q-martingale process, whose volatility σ t {\displaystyle \sigma _{t}} is always non-zero.
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.