Mathematics

Questions tagged [stochastic-analysis]

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

2352 questions
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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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31
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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
29
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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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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
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22
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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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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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20
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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
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19
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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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18
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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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15
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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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15
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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
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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
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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
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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". I am starting to suspect that it isn't easy and they might have just wanted to…
Felix Benning
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13
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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
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