For questions about stochastic analysis or stochastic calculus, for example the Itô integral.
Questions tagged [stochastic-analysis]
2341 questions
48
votes
4 answers
What is "white noise" and how is it related to the Brownian motion?
In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering equations of the form $$\frac{{\rm d}X_t}{{\rm…
0xbadf00d
- 14,208
30
votes
2 answers
Could someone explain rough path theory? More specifically, what is the higher ordered "area process" and what information is it giving us?
http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{X})$ where $X$ is a continuous process and…
user223391
28
votes
1 answer
Why predictable processes?
So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator and admitted integrands to be progressively…
JohnSmith
- 1,544
23
votes
1 answer
Solution to General Linear SDE
In order to find a solution for the general linear SDE
\begin{align}
dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t,
\end{align}
I assume that $a(t), b(t), g(t)$ and $h(t)$ are given deterministic Borel functions on…
iJup
- 2,049
- 1
- 16
- 40
22
votes
1 answer
Initial Distribution of Stochastic Differential Equations
consider the SDE
\begin{align}
\begin{cases}
X_t= \mu (t,x_t)dt + \sigma(t,X_t) d W_t \quad \forall t\in [0,T] \ (\text{or } t\geq 0),\\
X_0 \sim \xi.
\end{cases}
\end{align}
Suppose that, somehow, I could show (weak or strong) existence of a…
Ecthelion
- 460
21
votes
1 answer
Infinitesimal Generator of Ito Diffusion Process
Suppose one has the an Ito process of the form:
$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$
The following is an excerpt from wikipedia
My question is on how to derive this operator? It looks very similar to what you get when using Ito's Lemma. So I start…
Brenton
- 3,796
20
votes
1 answer
When is a stochastic integral a martingale?
In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded progressively measurable process (i.e. bounded on…
Chill2Macht
- 22,055
- 10
- 67
- 178
19
votes
1 answer
Why do we unavoidably (or not) use Riemann integral to define Itô integral?
https://en.wikipedia.org/wiki/Itô_calculus
Define
$$\int_0^tH_tdB_t\equiv \lim_{n\rightarrow\infty}\sum_{i=1}^nH_{t_i}(B_{t_i}-B_{t_{i-1}})$$
But I'm wondering why not defining this using Lebesgue Integral?
It looks more consistent, meaning we can…
ZHU
- 1,352
18
votes
2 answers
Relative entropy for martingale measures
I need some help understanding a note given in a lot of papers I've read.
Let $(\Omega,\mathcal{F},P)$ be a complete probability Space, $\mathbb{F} = (\mathcal{F}_t)_{t\in[0,T]}$ a given filtration with usual conditions, $S$ be a locally bounded…
Gono
- 5,788
15
votes
1 answer
Stochastic Leibniz Rule
I have come up with the following Leibniz stochastic rule and I want to check that:
The result is correct;
The proof is right.
Statement: let $f(\cdot,t):s \rightarrow f(s,t)$, $s \in \mathbb{R}^+$, be some function parameterised by a real number…
Morris Fletcher
- 651
15
votes
1 answer
Other versions of a weak Ito formula?
I am aware that this question is rather unspecific, but I was curious what versions of the Ito formula for Sobolev functions exist (and what methods are used to prove them).The only result I am aware of is the following, but I would be interested in…
Flo
- 311
14
votes
1 answer
Application of the Burkholder Davis Gundy inequality
The proof of the Feynman-Kac formula uses a lemma which I need to prove, but I can not figure it out.
The lemma is the folllowing:
Let $X$ be a weak solution of
$$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$
With $b$ and $\sigma$ continuous and satisfying…
user202723
- 351
13
votes
2 answers
Girsanov: Change of drift, that depends on the process
Known:
If I am looking at an SDE like:
$dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$.
I know that I can change the drift by using Girsanov to
$dX_t = (b(t,\omega)+c(t,\omega)) dt + d\bar{W}_t$ with $\bar{W}_t$ a…
mimi
- 913
13
votes
2 answers
Itô's formula: Differential form
I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book:
It is sometimes useful to use the following shorthand version of [Itô's formula]: $$…
martino
- 133
12
votes
1 answer
Rigorous Book on Stochastic Calculus
I have already taken a couse in Stochastic Calculus.
Due to time constraints on many ocassions we had to skip some formalities among the proofs.
I'm trying now to fill the gaps left, and I have been searching for a book to do so. My problem is that…
Jarana
- 735