Questions tagged [stochastic-calculus]
Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.
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Intuition and/or visualisation of Itô integral/Itô's lemma
Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum.
The Itô integral has due to the unbounded total variation ...
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What phenomena are better modelled by SDE instead of ODE?
Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
23
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Bochner integral of stochastic process = path by path Lebesgue integral?
After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
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Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
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Big Picture: What is the connection of Malliavin calculus with differential geometry?
I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
18
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Fictitious density of paths of diffusion processes outside the Cameron--Martin space
Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class $C^...
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Do convex and decreasing functions preserve the semimartingale property?
Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
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Brownian motion and spheres
Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ W\left(\frac{k}...
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Why do stochastic integrals depend on the choice of partitioning points?
When we integrate a function, we must make some choice about how we approximate it before we take the limit.
In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...
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Wiener process related counterexample
The Wiener process is defined by the three properties:
1. $W(0) = 0$,
2. $W(t)$ is almost surely continuous, and
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s &...
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surprisingly difficult filtration problem
I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
13
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Geometric characterization of martingales
Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...
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Discrete version of Ito's lemma
Could anyone give me some references where I could find
(a) discrete version(s) of Ito's lemma
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic ...
12
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Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?
There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...
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Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
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Karhunen–Loève approximation of Brownian motion and diffusions
The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:
$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$
where $Z_n \sim \mathcal{...
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'Nonclassical' abstract Wiener space
Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...
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Can every discrete martingale be embedded in a continuous martingale?
Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale $(\tilde{X}...
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Why do we mainly integrate with respect to martingales?
Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
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Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
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Convergence and non-convergence of left-point and mid-point Riemann sums
In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case. Taking the ...
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Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
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Is there any reason to use paracontrolled calculus over regularity structures?
Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
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When is a continuous path stochastic process be representable as diffusion or Ito process?
When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
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Show that this process is not a martingale
I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising ...
9
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1
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Feynman-Kac for jump-diffusion
I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes.
I know that, given a pure diffusion process like
$$dS_t=\mu_tdt+\sigma_tdW_t,$$ if $u(t,s)$ ...
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On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)
Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
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Quadratic variation and predictable quadratic variation for martingales
Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...
9
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Strong Markov property for Poisson point process
The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...
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Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: Take ...
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How to prove Feller property without using heat kernel estimates
I have a question about Markov processes.
Let $\mathbb{M}=(X_t,P_x)$ be a Markov process on a locally compact separable metric measure space $(E,\mu)$.
$\mathbb{M}$ is called Feller process if its ...
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What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
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Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion
Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where $b,\...
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Martingale representation theorem for Levy processes
Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
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A curious martingale
Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely?
Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
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Brownian Motion Winding Number
Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...
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A non-degenerate martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions.
$W_t$ is
a standard Brownian motion.
Let $Y_t$ be a martingale given by
$$...
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Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
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Interpretation of second order term in Fokker-Planck equation
Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by
$$
\begin{align*}
\nabla^2\cdot G(x)
&=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...
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Brownian motion in $n$ dimensions
Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
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Good papers on stochastic differential equations with applications in finance
I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) ...
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How is the Gronwall lemma used in this paper?
Let $(X_t, t \ge 0)$ be a $\mathbb R^d$-valued stochastic process. Let $\lambda>0$. Assume we have $\mathbb E [|X_0|^2] < \infty$ and
$$
\mathbb E [|X_t|^2] - \mathbb E [|X_0|^2] \le -2 \lambda \...
7
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Reference for Feynman-Kac
I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...
7
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comparing diffusions
Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility ...
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Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
7
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1
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Scalar product of random unit vectors
Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
$X,X'$ independent with uniform distribution on the sphere $...
7
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1
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Itô's article "A measure-theoretic approach to Malliavin calculus"
Apart from citations all over the internet, the following paper appears to be off-the-grid.
K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
7
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1
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a $L^1$ convergence for backward martingale
I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...
7
votes
1
answer
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Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...