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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.
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.
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
is a Q local martingale on the filtered probability space ... is the process Z which solves the stochastic differential equation = +. The ...
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 ...
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.
Then is a stopping time with respect to the martingale (), so () is also a martingale, referred to as a stopped martingale. In particular, ( Y n ) n ∈ N {\displaystyle (Y_{n})_{n\in \mathbf {N} }} is a supermartingale which is bounded below, so by the martingale convergence theorem it converges pointwise almost surely to a random variable Y ...
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