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- Doléans-Dade_exponential abstract "In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the stochastic differential equation dYt = Yt dXt with initial condition Y0 = 1. The concept is named after Catherine Doléans-Dade. It is sometimes denoted by Ɛ(X).In the case where X is differentiable, then Y is given by the differential equation dY/dt = Y dX/dt to which the solution is Y = exp(X − X0).Alternatively, if Xt = σBt + μt for a Brownian motion B, then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale X, applying Itō's lemma with ƒ (Y) = log(Y) givesExponentiating gives the solutionThis differs from what might be expected by comparison with the case where X is differentiable due to the existence of the quadratic variation term [X] in the solution.The Doléans-Dade exponential is useful in the case when X is a local martingale. Then, Ɛ(X) will also be a local martingale whereas the normal exponential exp(X) is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale X to ensure that its stochastic exponential Ɛ(X) is actually a martingale are given by Kazamaki's condition, Novikov's condition and Beneš' condition.It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale X iswhere the product extents over the (countable many) jumps of X up to time t.".
- Doléans-Dade_exponential wikiPageID "16986027".
- Doléans-Dade_exponential wikiPageRevisionID "569330757".
- Doléans-Dade_exponential subject Category:Martingale_theory.
- Doléans-Dade_exponential subject Category:Stochastic_differential_equations.
- Doléans-Dade_exponential comment "In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the stochastic differential equation dYt = Yt dXt with initial condition Y0 = 1. The concept is named after Catherine Doléans-Dade.".
- Doléans-Dade_exponential label "Doléans-Dade exponential".
- Doléans-Dade_exponential label "Stochastisches Exponential".
- Doléans-Dade_exponential sameAs Dol%C3%A9ans-Dade_exponential.
- Doléans-Dade_exponential sameAs Stochastisches_Exponential.
- Doléans-Dade_exponential sameAs Q5289708.
- Doléans-Dade_exponential sameAs Q5289708.
- Doléans-Dade_exponential wasDerivedFrom Doléans-Dade_exponential?oldid=569330757.