All Questions
120 questions
4
votes
1
answer
2k
views
Expected value of a stochastic integral expression
I am wondering if the following expression can be processed a bit analytically,
$$
E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right],
$$
where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
CommunityBot
- 1
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\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:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
2
votes
1
answer
111
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 3,669
2
votes
0
answers
85
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,215
4
votes
1
answer
143
views
When does an Itô diffusion give a semigroup on $L^2$
I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$.
I have a time-homogeneous Itô diffusion of the form
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t$...
CommunityBot
- 1
2
votes
2
answers
88
views
Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
- 493
5
votes
1
answer
188
views
Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
- 1,904
5
votes
0
answers
412
views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 573
2
votes
1
answer
311
views
Conditional expectation w.r.t. filtration of Brownian motion as a continuous map of its paths
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Itô process $dX_t = \...
- 12.9k
1
vote
1
answer
435
views
How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
CommunityBot
- 1
5
votes
2
answers
369
views
Markov process on a torus with prescribed invariant distribution
In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1,$$ where $\sigma:\mathbb R^d\to\mathbb R^{d\times k}$, $$b:=\frac{\Sigma+U}2\nabla\ln p+...
- 167
1
vote
1
answer
144
views
Ornstein Uhlenbeck process with discontinuous drift
This question is a modified version of this unanswered question asked on MSE, which mainly concerns an Ornstein-Uhlenbeck process with discontinuous drift on $\mathbb R^n$(for simplicity let $n=2$ for ...
- 8,992
1
vote
0
answers
95
views
A stochastic optimal control problem with filtering-like dynamics
I want to extend the following stochastic optimal control problem with randomized feedback control to the continuous time case
\begin{align}
\text{minimize}\quad \mathbb{E}_{\mathbb{H}}&\bigg[\...
- 303
1
vote
0
answers
53
views
The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
3
votes
1
answer
750
views
Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined
Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
CommunityBot
- 1
2
votes
0
answers
203
views
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 $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 1
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
3
votes
1
answer
211
views
Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process
Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
CommunityBot
- 1
2
votes
1
answer
534
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
- 63.9k
6
votes
1
answer
684
views
Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
CommunityBot
- 1
3
votes
1
answer
525
views
Malliavin differentiability of solutions to SDEs
In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
CommunityBot
- 1
0
votes
1
answer
154
views
Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time
Consider the following stochastic integral:
$$
X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s.
$$
Is $X_t$ almost-surely non-negative?
Using this answer, it seems that
$$
X_t = \max( W_t, 0) - \...
- 109
2
votes
1
answer
173
views
Estimates on perturbation of drift of SDEs
Let $\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ be Lipschitz functions, of at-most linear growth; i.e. $\|\sigma(x)\|\lesssim \|x\|,\|...
- 869
1
vote
0
answers
193
views
Stochastic volatility model question
Let suppose that $S_t$ is a process defined as:
$$ \begin{cases}dS_t = \mu S_t\,dt+m(v_t)\,dW^1_t\\ dv_t = \mu_v(v_t)\,dt + \sigma_v(v_t)\,dW^2_t\end{cases}$$
where the two Brownian motions have ...
1
vote
0
answers
102
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,904
1
vote
0
answers
237
views
Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
- 11
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
- 5,474
5
votes
1
answer
531
views
Riemannian metric induced by a stochastic differential equation
Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
CommunityBot
- 1
2
votes
1
answer
392
views
Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
$(\Omega, \mathcal F, \mathbb P)$ be a probability space.
$B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
$X=(X_0^...
- 6,215
0
votes
0
answers
120
views
Predictability of the mild solution of a SPDE
Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
- 5,474
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
4
votes
1
answer
351
views
Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain
Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...
- 5,474
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 321
2
votes
1
answer
139
views
Stochastic inverse
Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...
- 5,474
1
vote
1
answer
739
views
Joint law of a standard Brownian motion and its local time at a nonzero level
Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
CommunityBot
- 1
2
votes
1
answer
204
views
Comparing diffusion processes in different metrics
I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$.
Is there a way to apply ...
1
vote
1
answer
604
views
Is there an inverse Lamperti transformation for diffusions?
The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...
- 5,474
1
vote
0
answers
121
views
Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
0
votes
1
answer
272
views
Change of measure formula for the Föllmer process
While reading a preprint Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process I came across the following SDE in Section 3:
$$d X_t=d B_t+\nabla \log P_{...
- 136
7
votes
1
answer
249
views
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(...
- 203
0
votes
1
answer
206
views
Stochastic invariant subset
Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...
CommunityBot
- 1
1
vote
0
answers
156
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
1
vote
1
answer
107
views
Law of OU process with time-dependent dynamics
Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
- 115
0
votes
0
answers
468
views
The relationship between measurability and weak measurability
For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple
functions, measurability (the ...
- 113
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
3
votes
2
answers
271
views
For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?
Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields
$$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
- 708
2
votes
0
answers
187
views
Time derivative of relative entropy in this setting
I was reading the following article : https://arxiv.org/pdf/2005.13097.pdf and a question came up.
In page 30 in the proof of Lemma 16, when taking the time derivative of the KL divergence, there is ...
- 41
0
votes
2
answers
187
views
Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$
Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
- 6,853
0
votes
2
answers
207
views
Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality?
I have a question on Lawler – Notes on the Bessel process, on page 4. Let $X_t$ be one-dimensional Brownian motion, and we want to use $N_t$ as a measure-changing (local) martingale, defined as $$N_t=\...
- 662
4
votes
0
answers
259
views
Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$
Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
- 6,853