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Just as a continuous-time martingale satisfies E[X t | {X τ : τ ≤ s}] − X s = 0 ∀s ≤ t, a harmonic function f satisfies the partial differential equation Δf = 0 where Δ is the Laplacian operator. Given a Brownian motion process W t and a harmonic function f, the resulting process f(W t) is also a martingale.
By construction, this implies that if is a martingale, then = will be an MDS—hence the name. The MDS is an extremely useful construct in modern probability theory because it implies much milder restrictions on the memory of the sequence than independence , yet most limit theorems that hold for an independent sequence will also hold for an MDS.
The following result allows to express martingales as Itô integrals: if M is a square-integrable martingale on a time interval [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted square integrable process on [0, T] such that = + almost surely, and for all t ∈ [0, T] (Rogers & Williams 2000 ...
Itô's lemma can be used to derive the Black–Scholes equation for an option. [2] Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation dS = S(σdB + μ dt). Then, if the value of an option at time t is f(t, S t), Itô's lemma gives
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.
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the ...
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...