Let $\textbf{G}(t)$ be a zero-mean tight Gaussian process and $f(t)$ be a deterministic function. What theorem can be used to prove that $\int_0^\tau \textbf{G}(t)df(t)$ is a zero-mean Gaussian variable? Thanks.

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob’s optional sampling theorem implies that $$ \mathbb E \left\{ \exp(\lambda \ B_{T_a\ \wedge\ n} -\frac{\lambda^2}{2}({T_a\ \wedge\ n})) \ |\ \mathcal […]

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where $B=(B_t)_{t\geq 0}$ is a standard brownian motion in $\mathbb R$ starting from zero. To show that $T_x < + […]

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the processes $X(t)$ and $X’ (t)$ are orthogonal. $\bullet$If $X(t)$ is a twice mean square differentiable, stationary and Gaussian stochastical process, such that $E[X(t)] = 0$, […]

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| X_{(k+1)2^{-n}} -X_{k2^{-n}} \big|^p, $$ where $p>1$, $X$ is a continuous local martingale in some filtered probability space, with $X_0=0$. […]

Somebody know a book/text about Stochastic Differential Equations? I’m in the last period of the undergraduate course and I have interest in this field, but my university don’t have a specialist in this area. So, I want a book that can introduce me in this field without many difficulty and that permite me study still […]

Consider $M= \left(M\right )_{t \geq0}, \ N=\left(N\right) _{t \geq0} \in \mathcal M_{c,loc} $ starting both from zero, such that, a.e.$ \langle M \rangle_t =\langle N \rangle_t, \ \forall t\geq 0$. Have $M$ and $N$ the same law? If not give a contre exemple.

Let $\Phi\in\mathcal{L}\left(M\right)$ if and only if $\Phi$ is a real predictable process and for every $\left\Vert \Phi\right\Vert_{2,t,M}:=\mathbb{E}\left[\int_{0}^{t}\Phi_{s}^2 d\langle M\rangle_{s}\right]^{\frac{1}{2}}<\infty$. And let $\mathcal{M}_{2}$ be the space of square integrable right continuous martingales. It is well known by Ito Isometry that the mapping $$I_{M}:(\mathcal{L}\left(M\right),\left\Vert .\right\Vert_{2})\rightarrow\left(\mathcal{M}_{2},\left|\left\Vert \right|\right\Vert\right)$$ is injective, where $I_{M}\left(\Phi\right)$ is the Ito Integral of $\Phi$.I would […]

I’m reading Shreve’s book “Stochastic Calculus for Finance: Vol II”. In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: “The assumption that the filtration in Theorem 5.3.1 is the one generated by the Brownian motion is more restrictive than the assumption of Girsanov’s Theorem, Theorem 5.2.3, in which the filtration can be […]

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] = E\left[\exp \left(- \frac{1}{2} u^2 \int_t^T \sigma_s^2 \, ds \right) \mid \mathcal{F} \right] $$ for $\mathcal{F}$ the $\sigma$-algebra generated by $(\sigma_s)_{ s \leq […]

Intereting Posts

Surprise exam paradox?
Is a norm a continuous function?
Set theoretic construction of the natural numbers
Transforming Diophantine quadratic equation to Pell's equation
A simple permutation question – discrete math
Proof of the Pizza Theorem
Books to study for Math GRE, self-study, have some time.
$|G|>2$ implies $G$ has non trivial automorphism
Hard planar graph problem
Trigonometric Uncertainty Propagation
A quick question about categoricity in model theory
Characterization of the trace function
How can I show that it's a Banach space?
The Intuition behind l'Hopitals Rule
A question about the arctangent addition formula.