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.
1,020 questions
0 votes
2 answers
184 views
Is $\mathcal C(\mathbb R_+, \mathbb R)$ measurable in the cylindrical $\sigma$-algebra $B(\mathbb R)^{\otimes \mathbb R_+}$?
Let $$ \mathcal A(\mathbb R_+, \mathbb R) = \{ f : \mathbb R_+ \to \mathbb R \text{ a map}\}. $$ For each $t \ge 0$, denote by $$ \operatorname{ev}_t : \mathcal A(\mathbb R_+, \mathbb R) \to \mathbb R,...
- 141
1 vote
0 answers
44 views
Can Atlas-model minimize the distance between two stochastic processes?
Consider a filtered probability space in which there exist two independent Brownian motions $W$ and $B$. For every progressively measurable process $u=(u_t)_{t\ge 0}$ taking values in $[0,1]$, define ...
- 189
1 vote
0 answers
54 views
Lifting of non-reversible Markov chains for convergence acceleration
Background and motivation Let $(\Omega, \pi)$ be a finite state space with a stationary distribution $\pi$. Consider an ergodic Markov chain on $\Omega$ with transition matrix $P$ that is irreducible ...
- 313
1 vote
0 answers
69 views
Approximation by Staircase Processes and $L^2$ Convergence of Malliavin Derivatives
Let $(x_s)_{s\in[0,T]}$ be a stochastic process such that, for every $s\in[0,T]$, $$ x_s \in \mathbb{D}^{1,2}(\mathbb{R}), $$ and assume $$ \mathbb{E}\left[\int_0^T |x_s|^2\,ds\right] < \infty, \...
- 666
1 vote
0 answers
72 views
Generalized Malliavin Chain Rule
Let $ E $ be the Banach space of continuous functions on $[0, T]$, equipped with the supremum norm. Let $ \sigma : E \to \mathbb{R} $ be a function of class $ C^1 $ (Fréchet differentiable). Let $ X = ...
- 666
1 vote
0 answers
58 views
Is this also known as BDG inequality?
Let $W=(W_t)$ be a real-valued Brownian motion on some filtered probaiblity space. Let $H=(H_t)$ be a progressively measurable process taking values in some Hilbert space $\mathcal H$, endowed with ...
- 111
2 votes
1 answer
101 views
Pointwise estimate in Ito isometry
Let $W$ be a classical Wiener process on $[0,1]$ and let $$ \mathcal{I}\colon a\mapsto \int_0^1a(t) dW(t) $$ be the stochastic integral with respect to $W$. Ito isometry states that $\mathcal{I}$ is ...
- 607
4 votes
0 answers
122 views
Localized Yamada–Watanabe
I wonder if a Localized Yamada–Watanabe theorem up to a stopping time exists. Here is more details: Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space ...
- 666
6 votes
1 answer
480 views
What curve maximizes the Levy area?
The Levy area of a $C^1$ curve $f:[0,\infty)\to \mathbb R^2$ is defined to be $$L_f(t):=\int_0^t (f_1(s)f_2'(s)-f_2(s)f_1'(s))ds. $$ It is called Levy area because by Green's theorem, it is twice the ...
- 2,221
2 votes
1 answer
261 views
A generalization of freezing lemma
Let $\mathscr G⊆\mathscr F$ be a sub-$\sigma$-algebra, $X\in \mathbb{L}^0(\mathscr G,\mathbb R^d)$, $\varphi : \mathbb R^d×\Omega→\mathbb R$ be bounded and $B(\mathbb R^d)×F$ -measurable. Assume for ...
- 93
3 votes
0 answers
46 views
General Conditions for NFLVR
Background: An $\mathbb{R}^d$-valued process $Y_{\cdot}=(Y_t)_{t \ge 0}$ is called a $\sigma$-martingale if there exists some $\mathbb{R}^d$-valued martingale $M_{\cdot}$ and a $\mathcal{F}_{\cdot}^M :...
0 votes
0 answers
87 views
Piecewise linear approximation: time-independent renormalization constant
We are interested in the piecewise linear approximation of the $\Phi^4_d,d=2,3$ model, interpreted in the mild sense: $(\partial-\Delta)\phi=\phi^3+\xi.$ for this kind of approximation, how to define ...
- 607
1 vote
0 answers
110 views
Lattice and piecewise approximation of SPDE
This question concerns lattice & piecewise linear approximation (version of Wong-zakai theorem). Regularity structures allowed to tackle these topics for several SPDE, for example: https://arxiv....
- 607
9 votes
1 answer
171 views
Which functions preserve semimartingality?
Let $n,m\in \mathbb{N}_+$, $f:\mathbb{R}^n\to \mathbb{R}^m$ be a continuous function, and $X_{\cdot}=(X_t)_{t\ge 0} := M_{\cdot}+A_{\cdot}$ be a semimartingale. Is there a characterization of when $...
- 4,942
1 vote
1 answer
97 views
Kolmogorov continuity theorem for Lipschitz continuity
Kolmogorov's continuity theorem states that if for a $\mathbb{R}$--valued process $(X_t)_{t\geq 0}$ we have $$\mathbb{E}\left[\vert X_t-X_s\vert^\alpha \right] \leq K \vert t-s\vert^{1+\beta},$$ then ...
- 248
3 votes
0 answers
68 views
Convergence question on quadratic variation
Let $(X_t)_{t \in [0,2\pi]}$ be a continuous Ito martingale. Let $ t_1, \ldots, t_N, t_{N+1} $ be a discretization grid on $[0,2\pi]$. Let s $\in [0,2\pi]$. I would like to show the following ...
- 161
1 vote
0 answers
188 views
Probability distribution function on a sphere in d-dimensions
I was reading this post Scalar product of random unit vectors and I cannot understand how the density is obtained. We start by stating that $X$ and $X'$ are uniformly distributed on the sphere — which ...
- 139
2 votes
0 answers
188 views
Integration by parts for Functional Ito Calculus
I briefly recall the definition of Functionnal Ito Calculus, developped by Bruno Dupire in https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551. The paths we consider are càdlàg (right ...
- 161
1 vote
1 answer
125 views
Kolmogorov continuity theorem when the index set is an arbitrary subset of $\mathbb R^d$
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} $ I have used the following version of Kolmogorov Continuity Theorem in this thread. Lemma Let $(E, \mathrm{d})$ be a Polish space and $S$ ...
- 1,163
0 votes
0 answers
72 views
Substitute deterministic with random data
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\coloneq}{:=} \newcommand{\colon}{:} \newcommand{\rD}{D} \newcommand{\...
- 1,163
0 votes
0 answers
102 views
How much can bounded volatility bias a martingale's moving-average exit?
Let $(W_t)_{t\ge 0}$ be a one-dimensional standard Brownian motion on its natural filtration $(\mathcal{F}_t)$. For fixed constants $$0 < \underline{\sigma} < \overline{\sigma} < \infty,$$ ...
2 votes
0 answers
315 views
A kind of generalization of Doob Dynkin lemma for Stochastic processes
My question is how to prove the second part of the theorem in the picture above. The first part can be easliy proved by using the Doob-Dynkin lemma and the monotone class theorem，but the second part ...
- 93
2 votes
2 answers
564 views
What does stochastic quantization have to do with quantization?
Let me define the way I understand stochastic quantization. Stochastic quantization to me is a way of constructing infinite dimensional measures with a given "density". Suppose we are given ...
- 2,221
4 votes
0 answers
96 views
Use of left-continuous with right-limits processes in singular stochastic control
I am trying to wrap my head around some features of singular stochastic control, and one of the things that has been bothering me is that authors sometimes take the singular control to be left-...
- 151
1 vote
1 answer
89 views
stochastic integral on the canonical space
Let $C[0,T]$ be the space of all continuous functions on $[0,T]$. Let $X$ be the coordinate mapping process, that is $$X_t(\pi)=\pi_t,\;\forall \pi\in C[0,T].$$ Finally, let $\mathbb{F}$ be the ...
- 111
3 votes
0 answers
200 views
Simple way to simulate SDE solutions?
To simulate an Ito diffusion, $$dX_t = f(t,X_t)dt + g(t,X_t)dW_t,$$ you can discretize this in time by using an equidistant partition $(t_0^N,t_1^N,\dots,t_N^N)$ of the time interval $[0,T]$. The ...
- 708
1 vote
0 answers
84 views
Generalization of Fernique theorem for SDE
Let $(\Omega,\mathcal{F},\mathbb{P})$ a probability space and $(W_t)_{t \ge 0}$ a standard Brownian motion, with $\mathbb{F}$ is natural filtration. Let $(X_t)_{t \ge 0}$ a continuous adapted process ...
- 143
0 votes
0 answers
128 views
Nelson estimate for $:\phi^2:$ in where $\phi$ is the $3d$ Gaussian free field
If $\phi$ is the Gaussian free field on the $2$-torus then the Nelson estimate says that $e^{-\langle :\phi^n:,z\rangle}\in L^p$ for all $p\geq 1$, smooth $z$, and where $:\phi^n:$ is the Wick power. ...
- 2,221
2 votes
1 answer
369 views
Bound on Wick power of the Gaussian free field
Let $\phi$ be the Gaussian free field and $\phi^{:k:}$ its Wick power. Is it true that $\|\phi^{:k:}\|_{C^{-\alpha}}\lesssim \|\phi\|_{C^{-\alpha}}^k?$
- 2,221
2 votes
0 answers
118 views
Understanding a lemma in some notes by Da Prato on the derivative of Wick powers
I am reading http://eprints.biblio.unitn.it/1189/1/UTM711.pdf Lemma 5.1 which is about Wick powers of the Gaussian free field. They claim that for almost every $\phi,\psi$ we have for $g(\varepsilon):=...
- 2,221
2 votes
0 answers
65 views
A class of non-local linear Cauchy problem
Define an operator, \begin{align*} \mathcal{B}^Q f(t,x,v) \triangleq & \frac{\partial f(t,x,v)}{\partial t} + \left(r - q - \lambda^* \bar{\mu}^* - \frac{v}{2}\right) \frac{\partial f(t,x,v)}{\...
- 31
4 votes
1 answer
158 views
Reference for density of dynamic $\Phi_2^4$ SPDE wrt SHE
I am interested in the SPDE $$\partial_t u=\Delta u-u^3+\xi$$ on the $2$ torus $\Lambda$, started at the invariant measure $\mu$ on $C^{-\alpha}$. It is well known that $\mu$ has a density with ...
- 2,221
0 votes
0 answers
46 views
BSDE not having a local maxima at hitting time almost surely
Let us have an FBSDE: \begin{align*} dX_t &= b(t,X_t)dt + \sigma(t,X_t)dB_t, \quad X_0 = x \\ dY_t &= -f(t,X_t)dt + Z_tdB_t, \quad Y_T = g(X_T) \end{align*} where all the coefficients satisfy ...
- 131
0 votes
1 answer
119 views
Why do 1D diffusion exit times scale with the square of interval width?
Migrated from the MSE. Let $W_t^y$ denote a one-dimensional Brownian motion with initial position $y\in(a,b)$. Furthermore, let $\tau_{a,b}^y=\inf\{t\geq 0:W_t^y\notin(a,b)\}$ denote the exit time of ...
2 votes
1 answer
139 views
Conditional hitting time distribution for OU process between two fixed barriers
For an the OU process $\mathrm dX_t=-\omega X_t\,\mathrm dt+\omega\mathrm dW_t$ with $X_0=x_0$ and $L<x_0<U$ let $$ \tau=\inf\{t>0:X_t\notin(L,U)\} $$ denote the hitting time of either ...
2 votes
2 answers
301 views
KPZ fixed point
It was conjectured, later proved that KPZ converges to a universal limit called the KPZ fixed point. Are there analogue of this for other classes of SPDE like the stochastic quantization equation?
- 607
2 votes
0 answers
176 views
Continuity of Wick powers of the GFF
Let $h$ be the Gaussian Free Field on the $2$-torus $\mathbb{T}^2$. It is well known that one can construct the Wick power $:h^k:$ as a deterministic functional of $h$ through Wick renormalization. Is ...
- 2,221
9 votes
0 answers
356 views
The "easy to show" direction of Hairer's $\Phi_3^4$ is Orthogonal to GFF
I was reading the note https://hairer.org/Phi4.pdf where Hairer shows that the $\Phi_3^4$ measure is orthogonal to the GFF. He defines the following set $$A_\psi:=\{\Phi:\lim_{n\to\infty} e^{-3n/4}\...
- 2,221
0 votes
0 answers
54 views
Conditions on SDE coefficients for well-posedness of Fokker-Planck equation
Consider the following $n$-dimensional Ito-SDE: \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} What are the necessary regularity conditions on $\mu$ and $\sigma$ to ensure that the ...
- 121
2 votes
1 answer
384 views
Self-adjointness of generator and semigroup of an SDE
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
- 1,163
1 vote
0 answers
74 views
$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 321
1 vote
0 answers
132 views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation: $$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 321
0 votes
1 answer
93 views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the $$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$ Then, ...
- 4,942
2 votes
0 answers
74 views
Existence and moment estimation for a linear stochastic differential equation (SDE) with random coefficients
Let $W$ be one-demensional Brownian motion, and suppose $X$ satisfies the following SDE $$ \mathrm{d}X_s=(A_sX_s+B_s)\mathrm{d}s+(C_sX_s+D_s)\mathrm{d}W_s, \quad X_0=x_0\in\mathbb{R}^n, $$ where $A, C\...
- 135
2 votes
0 answers
141 views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE $$dX_t = V(X_t) \, dW_t,$$ where we identify $V(X_t) \in \mathbb R^n$ with ...
- 9,438
3 votes
0 answers
345 views
Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)
There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions. In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
- 3,745
0 votes
0 answers
77 views
Bound on the radon-nikodym derivative between two stochastic processes at a time point
I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution: \begin{align*} dX_t &= b(t, X_t)...
2 votes
0 answers
74 views
Approximate the adjoint generator of the discretization of an SDE
Let $d\in\mathbb N$; $\sigma\in\mathbb R^{d\times d}$; $p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$ $(X_t)_{t\ge0}$ denote ...
- 249
4 votes
1 answer
133 views
Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
1 vote
0 answers
87 views
An application to product formula of multiple integral
It is well-known that from Nualart's book, two multiple integrals can be expanded into a sum of multiple integrals, i.e., $$I_n(f)I_m(g)=\sum_{i=0}^{m\wedge n}i!C_m^iC_n^iI_{m+n-2i}(f\otimes_ig),$$ ...
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