Articles of stochastic calculus

Integration of Gaussian process

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.

Characterization of hitting time's law. (Proof check)

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 […]

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

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 < + […]

About stationary and wide-sense stationary processes

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 $E[X(t)] = 0$, […]

$p$-variation of a continuous local martingale

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 $X_0=0$. […]

A book/text in Stochastic Differential Equations

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 […]

Continuous local martingales with same crochet have the same law?

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.

Ito Integral surjective?

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 […]

What is the difference between “filtration for a Brownian motion” and “filtration generated by a Brownian motion”?

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 […]

Slight generalisation of the distribution of Brownian integral

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 […]