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.
376 questions with no upvoted or accepted answers
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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 ...
- 167
2
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0
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93
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$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE
$$\partial_t u=\Delta u-u^3+\xi,$$
where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
- 1,904
2
votes
0
answers
82
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Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
- 135
2
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0
answers
42
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Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
2
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0
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89
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Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
- 61
2
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0
answers
61
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Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
2
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0
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136
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Towards the KPZ: Wiener-Ito integral, Kolmogorov type criterion
Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$
The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\...
- 573
2
votes
0
answers
80
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Stability of Hölder constants of frozen Itô stochastic integrals
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
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Are speed, scale function and killing measures of Itô diffusion absolutely continuous respect to Lebesgue measure and do have smooth derivative?
In Borodin and Salminen's Handbook of Brownian motion (MR1912205, Zbl 1012.60003), pages 16–17, they mention the fact that if the three basic characteristics (speed measure, scale function and killing ...
- 41
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Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
2
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95
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Brownian bridge as a limit of SDEs
Let $B$ be a Brownian motion and with respect to some probability measure $\mathbf{P}$ and filtration $(\mathcal{F})_{t \geq 0}$ and let $S_\epsilon = \{B_1 \in (-\epsilon,\epsilon)\}$.
For every $t \...
- 141
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Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?
Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
- 1,904
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81
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Assumptions for uniform measure of SDE on manifolds
Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
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How to estimate the difference between two Ito diffusions?
Suppose $𝑏:\mathbb R^d \to \mathbb R^d, \sigma:\mathbb R^d \to \mathbb R^{d\times d}$ are measurable functions and satisfy
\begin{equation*} 2\langle 𝑥−𝑦,𝑏(𝑥)−𝑏(𝑦)\rangle +\|\sigma(𝑥)−\sigma(�...
- 622
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103
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Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
2
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75
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Autocovariance of harmonic oscillator in fluid (Langevin Equation)
I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
- 71
2
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62
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Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
2
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203
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Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 167
2
votes
0
answers
68
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On the measurability of stochastic integrals
Let $S(t)$ be a $C_0$-contraction semi-group, $W$ is a cylindrical Wiener process in a separate Hilbert space $U$. Assume the following conditions:
$$
\|F(t,u_1)-F(t,u_2)\|_{H}< C\|u_1-u_2\|_{H},~~...
- 31
2
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answers
111
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Bounding from below the distance between SDE started from different initial conditions
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...
- 6,155
2
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0
answers
121
views
Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
- 573
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0
answers
62
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On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 505
2
votes
0
answers
301
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Ito lemma for SDEs on a Lie group
I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group.
In doing so i'm trying to define a formula ...
- 293
2
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0
answers
356
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KL Divergence between the solution to two SDEs
What is the KL divergence between the laws of solutions to SDEs? That is, let
\begin{align*}
dX^1&=b_1(X^1,t) \, dt+\sigma(X^1,t) \, dB\\
dX^2&=b_2(X^2,t) \, dt+\sigma(X^2,t) \, dB
\end{align*}...
- 21
2
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0
answers
191
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Multiple Wiener integral as Witt polynomial of Brownian motion
I know that if i have a Brownian motion $W_t$ the multiple Wiener integral
$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$
can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is ...
- 293
2
votes
0
answers
65
views
Lipschitzness of conditional law of a stochastic filtering problem wrt the Wasserstein distance
Let $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be a pair of stochastic processes taking values in $\mathbb{R}^n$ and in $\mathbb{R}^m$; defined on a filtered probability spaces $(\Omega,\mathcal{F},(\...
- 169
2
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0
answers
186
views
Time derivative of relative entropy in this setting
I was reading the following article : https://arxiv.org/pdf/2005.13097.pdf and a question came up.
In page 30 in the proof of Lemma 16, when taking the time derivative of the KL divergence, there is ...
- 41
2
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0
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94
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Lyapunov function utility in stochastic optimal control
The article Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching by (X.Mu, Q.Zhang) studies necessary and sufficient conditions on near-optimal controls.
In both ...
- 141
2
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116
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Is a Riccati BSDE explicitly solvable?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
- 335
2
votes
0
answers
53
views
Continuity of translation operator in fractional white noise analysis
Fix $H\in(\frac{1}{2},1)$, and let $\Omega:=C_0([0,T],\mathbb R^d)$ be the space of $\mathbb R^d$-valued continuous functions. There is a probability measure $P^H$ on $(\Omega,\mathcal B(\Omega))$, ...
- 515
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votes
0
answers
171
views
Convergence of conditioned stochastic integral
Let $B_t$ be a standard Brownian motion, $f: [0, T] \to \mathbb R$ a bounded Borel measurable function, and $X_t$ a process independent of $B_t$ with sample paths that almost surely start at $0$, and ...
- 6,155
2
votes
1
answer
389
views
A mean field SDE with hitting time
Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.
$$\int_{(0,\infty)}\mu_0(dx)=1,$$
consider the mean field SDE :
$$dX_t = \mathbf{1}_{\...
- 1,334
2
votes
0
answers
108
views
Existence of solutions to some Mckean-Vlasov SDE
Let $\mathcal P(\mathbb R)$ be the space of probability measures and $(W_t)_{t\ge 0}$ be a standard Brownian motion.
For given functions $b, \sigma, \beta: \mathbb R_+\times \mathbb R\times \mathbb R\...
user128095
2
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answers
61
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Convolution of Wiener measure and measure on $W_0^{1,2}$
Let $F$ by a process adapted to the filtration of a standard Brownian motion. Suppose that the Doleans Dade martingale exists and is a martingale. $F$ is a measure on $W_0^{1,2}$, call it $\nu$. Let $\...
- 133
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0
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173
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When is the dual infinitesimal generator of a S.D.E self-adjoint and negative definite?
Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be,
self-adjoint i.e $\int f ({\...
- 2,246
2
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0
answers
140
views
Convergence of the probability that hitting times being infinity
Let $X^n=(X^n_t)_{t\ge 0}$ and $X=(X_t)_{t\ge 0}$ be RCLL (right-continuous with left limits) processes such that
$$\lim_{n\to\infty}X^n=X,\quad \quad \mbox{almost surely},$$
where this convergence ...
user128095
2
votes
0
answers
101
views
The Itō isometry for Riemannian manifolds
If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \...
- 5,407
2
votes
0
answers
237
views
Semimartingale decomposition and filtrations
In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using.
So, taking a toy example,...
- 341
2
votes
0
answers
137
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Kernel of the adjoint of the infinitesimal generator of Levy SDE
Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the ...
- 2,246
2
votes
0
answers
75
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Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?
Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...
- 2,246
2
votes
0
answers
138
views
On the difference between Malliavin derivative and Gross-Sobolev derivative
Let $W=C_0([0,1],\mathbb R^d)$ be the classical Wiener space equipped with $\mu$ the Wiener measure.
If $F:W\to\mathbb R$ is a cylindrical function of the form
\begin{align*}
F(w)=f(W_{t_1}(w),\cdots,...
- 515
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
2
votes
0
answers
172
views
Non-integer conditional moment of exponential functional of Brownian motion
Let $B_t$ be a standard Brownian motion.
I want to solve the following:
$$
\mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right],
$$
for some fixed $0<\...
- 284
2
votes
0
answers
215
views
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...
- 93
2
votes
0
answers
83
views
What is the "mode" of the Girsanov density?
Consider the Girsanov density $$\exp\left(\int_0^Tf(s,B(s))dB(s)-\frac12\int_0^Tf^2(s,B(s))ds\right)$$
Is there a notion of "mode" of this density?
For example, is there a continuous path $z(t)$ ...
- 93
2
votes
0
answers
41
views
If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...
- 167
2
votes
0
answers
250
views
SDE conditional expectation
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
- 159
2
votes
0
answers
196
views
Girsanov density as a functional on $C[0,1]$
I'll formulate the question via an example.
On $( C[0,1], \mathcal{C} )$, where $C[0,1]$ is the set of continuous functions on $[0,1]$ and $\mathcal{C}$ the Borel $\sigma$-algebra given by uniform ...
- 273
2
votes
0
answers
64
views
Convergence of gPC expansions for random variables in the total variation distance
Suppose that a random variable $Y$ can be written as $Y=g(Z)$, where $g$ is a function and $Z$ is a random variable. When $Z$ is a continuous random variable with finite absolute moments, we consider ...
user85801
2
votes
0
answers
58
views
Uniqueness of martingale problem for Levy type operator
Consider the following Levy type operator:
$$
L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d),
$$
...
- 411