What is the martingale representation theorem?
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
Why is the martingale representation theorem important?
The Martingale Representation Theorem says that indeed you can get a very large class of random processes in this way (starting with dB(t), integrating it in a time-varying manner and adding an external predictable input).
Is stochastic integral martingale?
It is a local martingale, by definition, with quadratic variation given by Qt=∫t0H2udu. Now, QT is upper bounded by an almost surely finite random variable times T. So that the expectation of the quadratic variation is finite, because M is a martingale, and hence the stochastic integral is a martingale.
Is the stochastic integral continuous?
Continuous Local Martingales First, the continuous local martingale property is always preserved by stochastic integration. is a continuous local martingale. Next, the quadratic variation of a continuous local martingale X provides us with a necessary and sufficient condition for X-integrability.
Is Ito integral a martingale?
We give one and a half of the two parts of the proof of this theorem. If b = 0 for all t (and all, or almost all ω ∈ Ω), then F(T) is an Ito integral and hence a martingale. If b(t) is a continuous function of t, then we may find a t∗ and ǫ > 0 and δ > 0 so that, say, b(t) > δ > 0 when |t − t∗| < ǫ.
Is Ito process continuous?
This process is adapted, continuous, equal to zero in zero, and its trajectories are almost surely increasing.
Is stochastic integral linear?
probability – Pathwise stochastic integral as a linear operator on continuous functions – MathOverflow.
Is Ito integral martingale?
If b = 0 for all t (and all, or almost all ω ∈ Ω), then F(T) is an Ito integral and hence a martingale. If b(t) is a continuous function of t, then we may find a t∗ and ǫ > 0 and δ > 0 so that, say, b(t) > δ > 0 when |t − t∗| < ǫ.