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Kazamaki's Condition
In mathematics, Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's theorem is to be applied to perform a change of measure. Kazamaki's condition is more general than Novikov's condition In probability theory, Novikov's condition is the sufficient condition for a stochastic process which takes the form of the Radon–Nikodym derivative in Girsanov's theorem to be a martingale. If satisfied together with other conditions, Girsanov .... Statement of Kazamaki's condition Let M = (M_t)_ be a continuous local martingale with respect to a right-continuous filtration (\mathcal_t)_. If (\exp(M_t/2))_ is a uniformly integrable submartingale, then the Doléans-Dade exponential ''Ɛ''(''M'') of M is a uniformly integrable martingale. References * Martingale theory {{probability-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doléans-Dade Exponential
In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation dY_t = Y_\,dX_t,\quad\quad Y_0=1,where Y_ denotes the process of left limits, i.e., Y_=\lim_Y_s. The concept is named after Catherine Doléans-Dade. Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since X measures the cumulative percentage change in Y. Notation and terminology Process Y obtained above is commonly denoted by \mathcal(X). The terminology "stochastic exponential" arises from the similarity of \mathcal(X)=Y to the natural exponential of X: If ''X'' is absolutely continuous with respect to time, then ''Y'' solves, path-by-path, the differential equation dY_t/\mathrmt = Y_tdX_t/dt, whose solution is Y=\exp(X-X_0). General formula and special cases * Without any assumptions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Martingale
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale. Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem). Definition Let (\Omega,F,P) be a probability space; let F_*=\ be a filtration of F; let X: adapted stochastic process on the set S. Then X is called an F_*-local martingale if there exists a sequence of F_*-stopping rule">stopping times \tau_k : \Omega \to [0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale (probability Theory)
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. History 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 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 first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girsanov Theorem
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a measure that is absolutely continuous with respect to ''P'' then every ''P''-semimartingale is a ''Q''-semimartingale. State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (probability)
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Novikov's Condition
In probability theory, Novikov's condition is the sufficient condition for a stochastic process which takes the form of the Radon–Nikodym derivative in Girsanov's theorem to be a martingale. If satisfied together with other conditions, Girsanov's theorem may be applied to a Brownian motion stochastic process to change from the original measure to the new measure defined by the Radon–Nikodym derivative. This condition was suggested and proved by Alexander Novikov. There are other results which may be used to show that the Radon–Nikodym derivative is a martingale, such as the more general criterion Kazamaki's condition, however Novikov's condition is the most well-known result. Assume that (X_t)_ is a real valued adapted process on the probability space \left (\Omega, (\mathcal_t), \mathbb\right) and (W_t)_ is an adapted Brownian motion: If the condition : \mathbb\left ^ \right\infty is fulfilled then the process : \ \mathcal\left( \int_0^t X_s \; dW_s \righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right-continuous Filtration
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Integrable
In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in L_1 which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_\, f\, _0 there corresponds a \delta>0 such that : \int_E , f, \, d\mu 0 such that : \sup_\int_A, f, \, d\mu 0 such that, for every measurable A such that P(A)\leq \delta and every X in \mathcal, \operatorname E(, X, I_A)\leq\varepsilon. or alternatively 2. A class \mathcal of random variables is called uniformly integrable (UI) if there exists K\in X, I_)\le\varepsilon\ \text X \in \mathcal, where I_ is the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
