0
votes
1answer
24 views

Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
1
vote
0answers
40 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
2
votes
1answer
36 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
0
votes
1answer
14 views

Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
2
votes
1answer
125 views

Laplace functional of a Poisson random measure with stochastic intensity

This is one of the problems from Cinlar's 2011 book - "Probability and Stochastics" (Chapter VI, page 262, exercise 2.36) : Let $N$ be a Poisson random measure on $R^{+}$, defined by $N(\omega, B) = ...
0
votes
1answer
123 views

properties about stochastic integral

I have a question about stochastic / Lebesgue Stieltjes integrals. I'm following Revuz / Yor. The space $H^2$ is the space of all $L^2$ bounded continuous martingales. If $M\in H^2$ then they call ...