Good day.
For $k\geq0$ a continuous function on $\mathbb{R}_+$ and $\{W_t\}$ a standard Brownian motion, could you help me find the Taylor's expansion of the following exponential: $e^{-\int_0^t k(s) dW_s}.$
For the case where $e^{-k W_t}$ where $k>0$ is a constant, I was able to recompute...
Many thanks for the reply. What I meant of "law" is the probability density of the given integral. For the case $\mu(s) = -\nu s$ where $\nu$ is a positive constant and $h(s)=1,$ the law was already known (Corollary 1.2, p95) from Mar Yor's book given here Exponential Functionals of Brownian...
Good day!
I have a question regarding the law of the ff:
$$
\int_0^t h(s) e^{2\beta(\mu(s) + W_s)}
$$
where $\beta >0;$ $h,\mu$ are continuous functions on $\mathbb{R}_+$ with $h\geq 0;$
and $W=\{W_s,s\geq 0\}$ is a standard Brownian motion.
Thanks for any help.:D
I see. Below is taken from page 72 of the book. It is part of the definition of a strong solution of the semilinear SDE. This is the setting:
Let $K$ and $H$ be real separable Hilbert spaces, and $W_t$ be a $K$-valued $Q$-Wiener process on a complete filtered probability space...
Good day! I came across this symbol $dt \otimes dP$-a.e. in the book of Mandrekar (page 72) Stochastic Differential Equations in Infinite Dimensions: With Applications ... - Leszek Gawarecki, Vidyadhar Mandrekar - Google Books.
What does this symbol mean? I understand that in real analysis...
Thanks for your time and reply. Its really of great help.
Though I want to ask if you know of some books, that discusses the above topic.
Thanks a lot.
Good day!
I was trying to make sense of the notation $P(X \in dx),$ where $X$ is a continuous random variable. Some also write this one as $P(X \in [x, x+dx])$ to represent the probability that the random variable $X$ takes on values in the interval $[x, x+dx].$
I have seen similar notation a...
Thanks for the suggestion. Please give your comment on the proof I produced. Thanks.
Recall that $\sigma\left(A\right)$ is the smallest $\sigma$-field that contains the set (or family of subsets) $A.$ Hence, for the first question regarding unions, it suffices to show that
$$
\bigcup_{s < t}...
Hi again, I hope you can clarify me on this:
Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $\{\mathcal{F}_t\}_{0\leq t<\infty}$ be a filtration on it. Define
$\mathcal{F}_{\infty} = \sigma\left(\bigcup_{t} \mathcal{F}_t \right)$ where $t \in [0,\infty).$
My question...
Hi! I am reading the book of Karatzas and Shreve (Brownian Motion and Stochastic Calculus - Ioannis Karatzas, Steven E. Shreve - Google Books). On page 4 we have the ff definitions:
$$
\mathcal{F}_{t-} := \sigma \left( \bigcup_{s<t} \mathcal{F}_s \right) \quad \text{and}\quad
\mathcal{F}_{t+} :=...
Good day!
I am reading the paper of Marc Yor (www.jstor.org/stable/1427477). equation (1.a) seems unfamiliar to me since the $ds$ comes first before the exponential part;
$$
\int_0^t ds \exp(aB_s + bs).
$$
Can you please help me clarify if there is a difference with the above notation as...