1
vote
0answers
33 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 ...
0
votes
1answer
20 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
0
votes
1answer
54 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
2
votes
1answer
35 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
0answers
45 views

Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma ...
0
votes
0answers
43 views

The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
1
vote
0answers
90 views

Subtraction of Probability Measures

I have just read that apparently the following two conditions are equivalent: $$ \int f dP \geq \int f dQ \Longleftrightarrow \int f d(P-Q) \geq 0$$ for $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ and ...
1
vote
0answers
24 views

How to find sigma-algebra over omega3 generated by the log-return ln(S2/S1) and ln(S3/S2)?

I calculated {S2/S1} = {u, u*u/d, d, d*d/u}, and then get ln(S2/S1)= {ln(u),ln(u*u/d), ln(d),ln( d*d/u)}. I am not sure my way of doing this question is right, because i m confuse about how to get ...
5
votes
1answer
195 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
0answers
48 views

Distributions representable as Ito diffusions

This is inspired by the following question. Let $X_t$ be an Ito diffusion on the interval $t\in [0,1]$: $$ \mathrm dX_t = a(X_t)\mathrm dt+ b(X_t)\mathrm dW_t $$ where say $a,b$ are Lipschitz ...
3
votes
0answers
100 views

Measurability of number of upcrossing $U_I(\alpha,\beta; X)$ in continuous time

These definitions come from Karatzas and Shreve, Brownian Motion and Stochastic Calculus. We may take for granted that $U_F(\alpha,\beta; X(\omega))$, the number of upcrossings over $[\alpha,\beta]$ ...
5
votes
1answer
228 views

generating set of predictable sigma algebra

I am solving an exercise in Rogers and Williams and want to ask if my solution is correct. Let me first introduce the notation. The space $b\mathcal{E}$ is the space of processes of the form ...
2
votes
1answer
94 views

Some preliminaries for the canonical construction of a Brownian Motion, help needed.

I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
1
vote
3answers
531 views

Exchange integral and conditional expectation

I know that if we have $E[\int_0^1 |X_t|dt] < \infty$ we may apply Fubini's theorem and compute $E[\int_0^1 X_tdt] = \int_0^1 E[X_t]dt$. Is there a similar version that allows the exchange of ...
2
votes
1answer
119 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) = ...
6
votes
0answers
347 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...