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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.
The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.
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.
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. [1] It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
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.
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
where is the Radon–Nikodym derivative of with respect to , and therefore is still a martingale. [3] If in a financial market there is just one risk-neutral measure, then there is a unique arbitrage-free price for each asset in the market.
A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite-variation process starting at zero. (Rogers & Williams 1987, p. 358) For example, if X is an Itō process satisfying the stochastic differential equation dX t = σ t dW t + b t dt, then