All Questions
120 questions
1
vote
1
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133
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What are the optimal times to sample a process?
Let $X$ be a one dimensional Ito diffusion given by
$$X_t = b \,W_t$$
where $b$ is a constant, and $W$ is a standard Brownian motion.
Let $B$ be another Brownian motion independent of $W$, and define ...
- 6,215
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 128k
1
vote
0
answers
94
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Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
- 167
3
votes
0
answers
145
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Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
- 931
1
vote
1
answer
249
views
Is the integral against a Brownian motion conditioned to stay bounded a local martingale?
Let $W$ be a standard Brownian motion on a probability space $(X, \mathcal F, \mathbb P)$ let and $\mathcal F_t$ its natural filtration.
For $\varepsilon > 0, T \in [0, \infty)$ let $A_{\varepsilon,...
- 6,215
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 6,215
1
vote
0
answers
54
views
Conditions ensuring that conditional law of a process belongs to a given exponential family
Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
CommunityBot
- 1
1
vote
1
answer
337
views
Bessel process conditioned to stay positive
This question has also been asked on https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive
Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x&...
- 14.2k
8
votes
2
answers
3k
views
Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
- 6,045
1
vote
0
answers
78
views
If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?
Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
- 309
1
vote
1
answer
913
views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
- 247
1
vote
1
answer
508
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Divergence form degenerate pde and Feynman Kac
Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
CommunityBot
- 1
0
votes
1
answer
271
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Associativity rule for integration against fractional Brownian motion
In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
- 10.3k
1
vote
1
answer
82
views
Local inverse bound of Cameron Martin and Banach norms
Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
- 128k
3
votes
1
answer
202
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Onsager--Machlup functional as the density across a mesh of discrete points
It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...
1
vote
1
answer
512
views
Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
- 128k
1
vote
1
answer
293
views
Time-Reversal of BSDE = SDE
Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$:
$$
Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;
$$
...
- 1,172
4
votes
1
answer
509
views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
- 128k
0
votes
1
answer
341
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Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
- 6,045
0
votes
2
answers
313
views
Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...
- 128k
1
vote
0
answers
276
views
Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
3
votes
0
answers
235
views
Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
CommunityBot
- 1
1
vote
2
answers
788
views
When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$?
The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$...
- 1,989
2
votes
1
answer
495
views
Stochastic integral with respect to a random field
I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
- 10.3k
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
2
votes
0
answers
215
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What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...
- 93
1
vote
1
answer
209
views
What is the drift for a convex combination of Girsanov measures?
Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a ...
- 7,514
1
vote
0
answers
80
views
Large deviations estimate for arbitrary continuous function
Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
- 5,380
1
vote
1
answer
90
views
Probability that a stochastic flow is near $0$
Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
- 5,405
0
votes
3
answers
639
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Non-smooth Ito lemma for semi-martingales
Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...
- 63.9k
1
vote
0
answers
73
views
conditional expected value and in Stochastic differential equations
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
0
votes
0
answers
294
views
Malliavin derivative of Ito process
Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
- 63.9k
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 325
1
vote
1
answer
154
views
Is there solution to a backward stochastic differential equation with $yz$ in the generator?
Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...
- 11
3
votes
1
answer
345
views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 4,361
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
2
votes
1
answer
391
views
Is there an Itō formula for random functions in infinite-dimensions?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
CommunityBot
- 1
0
votes
0
answers
76
views
Ornstein-Uhlenbeck type process with thresholding
(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...
- 31
6
votes
2
answers
2k
views
Tanaka-Meyer formula
I have a simple question about Tanaka-Meyer formula, I am having difficulty applying it. Let $X$ be a continous martingale vanishing at zero. From Tanaka-Meyer formula it holds $$d|X_t| = sgn(X_t)dX_t+...
- 6,045
2
votes
1
answer
280
views
Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
- 6,045
3
votes
1
answer
83
views
Filtering Mixed Discrete and Continous
Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...
- 1,675
1
vote
0
answers
90
views
Onsager-Machlup Function of a Killed Diffusion Process
Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
- 213
1
vote
0
answers
340
views
Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
1
vote
0
answers
249
views
Ito's formula for jump diffusions
Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...
- 138
3
votes
1
answer
110
views
Sequence of diffusions
Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
- 724
2
votes
0
answers
96
views
Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
- 19.6k
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 19.6k
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
- 19.6k
2
votes
1
answer
387
views
Weak convergence of sum of log normal random variables
Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...
- 489
23
votes
1
answer
1k
views
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, ...
CommunityBot
- 1