For questions about stochastic analysis or stochastic calculus, for example the Itô integral.

# Questions tagged [stochastic-analysis]

1969 questions

**29**

votes

**2**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

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**24**

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

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**24**

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

**17**

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

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**16**

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

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**16**

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

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**13**

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

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**12**

votes

**1**answer

### Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?:
Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that the process has stationary increments,
meaning that…

user42734

- 121
- 3

**12**

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

- 1,839
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**12**

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

- 123
- 4

**11**

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

- 803
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**11**

votes

**0**answers

### Convergence of a Stochastic Process - Am I missing something obvious?

In the paper On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (Jaakkola et al. 1994) the authors claim that a statement is "easy to show". Now I am starting to suspect that it isn't easy and they might have just wanted to…

Felix B.

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**11**

votes

**1**answer

### Voronoi cell volume inside the ball

I have the following problem:
Let us denote a ball with center $C$ and radius $R$ in $\mathbb{R}^d$ as $B(C, R)$. Given a unit ball $B(0, 1)$ and vector $u$ has a uniform distribution inside the ball: $u \sim U(B(0, 1))$. Then we sample $M$ points…

Stanislav Morozov

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**11**

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

- 19,382
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- 120

**10**

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

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