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 […]

I would like to prove that the sum of Gaussian processes is also Gaussian, to be precise, $M_t=W_t+W_{t^2}$, where $W_t$ is standard Wiener process. That is kind of obvious, but I am looking for some more rigorous, as short as possible proof, other than just saying that it is the sum of two Gaussian processes. […]

In Ross’s Stochastic processes: A stochastic process $\{X(t), t \geq 0\}$ is said to be a Brownian motion process if $X(0) = 0$, $\{X(t), t \geq 0\}$ has stationary independent increments, and for every t > 0, $X(t)$ is normally distributed with mean 0 and variance $c^2t$. Brownian motion could also be defined as a […]

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 < + […]

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t – \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in \mathbb{R}$. I now want to prove that for every $a>0$ the random variable $$ \sup_{t \geq 0} \left(W_t […]

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 $X_0=0$. […]

Let $N_t$ be a Poisson process with rate $\lambda$. $T_k$ the inter arrival times of $N_t$. $\{X_k\}$ a collection of i.i.d. random variables with mean $\mu$. $X_k$ is independent of $N_t$. Calculate the expectation of $$ S_t= \sum_{k=1}^{N_t} X_k e^{t-T_k}. $$ Given $N_t$, the inter arrival times are uniformly distributed on $[0,t]$. Hence, $T_k \sim […]

No idea on how to start this question. Any help would be much appreciated. A ﬂea lives on a polyhedron with N vertices, labelled $1, . . . , N$. It hops from vertex to vertex in the following manner: if one day it is on vertex $i > 1$, the next day it hops […]

For a $F_t$ adapted process $X$, please prove that $\int_0^t X(s)ds$ is $F_t$ measurable. For simple processes $X$, the conclusion is obvious. I think we should use monotone class theorem to prove the case of general processes. However, I don’t know the details. Thank you.

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