All Questions
175 questions
1
vote
0
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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:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
2
votes
0
answers
67
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,399
2
votes
1
answer
111
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 6,215
2
votes
2
answers
88
views
Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
- 6,215
5
votes
2
answers
369
views
Markov process on a torus with prescribed invariant distribution
In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\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 ...
- 163
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 = \...
- 21
1
vote
0
answers
53
views
The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
0
answers
122
views
Slow points of diffusion processes
Let $W$ be a standard $d$-dimensional Brownian motion, and $X$ the solution to the SDE
$$dX_t = \mu(X_t) dt + \sigma(X_t) \, dW_t,$$
with $\mu$ and $\sigma$ Lipschitz continuous.
Given a (...
- 6,215
3
votes
1
answer
211
views
Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process
Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
- 41
1
vote
0
answers
193
views
Marcus-SDE to Itô-SDE
In the field of stochastic calculus, everyone knows the Itô and Stratonovich integrals, as well as the conversion from Stratonovich to Itô SDEs.
The Stratonovich integration has the particularity of ...
- 11
2
votes
0
answers
203
views
Time reversal of infinite-dimensional SDE
Consider the SDE $${\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\...
- 167
3
votes
2
answers
490
views
SDE driven by fractional Brownian motion
Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$
I am looking for references that ...
- 1,334
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
$$
X_t = \max( W_t, 0) - \...
- 109
2
votes
1
answer
173
views
Estimates on perturbation of drift of SDEs
Let $\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ be Lipschitz functions, of at-most linear growth; i.e. $\|\sigma(x)\|\lesssim \|x\|,\|...
- 364
1
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0
answers
134
views
Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
1
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0
answers
193
views
Stochastic volatility model question
Let suppose that $S_t$ is a process defined as:
$$ \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}$$
where the two Brownian motions have ...
- 393
1
vote
0
answers
102
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,904
3
votes
2
answers
339
views
Stability results for general linear stochastic ODE
I am interested in the following time-invariant multivariate SDE:
\begin{equation}
dx_i = \sum_{j} a_{ij} x_j\,dt + \sum_{j,k} b_{ijk} x_k \, dW_j
\end{equation}
Despite its simplicity the general ...
- 194
1
vote
0
answers
237
views
Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
- 11
2
votes
0
answers
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
- 167
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
2
votes
0
answers
95
views
Local martingale for a (two-dimensional) diffusion
Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
- 1,897
1
vote
1
answer
109
views
Phase space Brownian bridge
I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$
s.t. $x_0=0$ and $x_1=0$
where $dw_t$ is a Wiener process.
I am thinking about ...
- 13
7
votes
2
answers
613
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
- 199
2
votes
1
answer
392
views
Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
$(\Omega, \mathcal F, \mathbb P)$ be a probability space.
$B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
$X=(X_0^...
- 825
2
votes
1
answer
400
views
Existence of linear stochastic differential equation given solution
Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
- 29
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
2
votes
0
answers
201
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 21
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+...
- 573
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 ...
- 6,215
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 ...
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 ...
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
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
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
1
vote
0
answers
235
views
Two increasingly correlated Brownian motions and Williams decomposition
The Williams decomposition is
Let $(B_t-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_\infty^{-\nu}:=\sup_{t\in [0,\infty]}(B_t-\nu t)$. Then conditionally on $M_\...
- 5,474
1
vote
0
answers
156
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
0
votes
0
answers
468
views
The relationship between measurability and weak measurability
For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple
functions, measurability (the ...
- 113
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
4
votes
1
answer
218
views
Schauder basis of the Hardy space of semi-martingales
Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm
$$
\...
- 195
2
votes
1
answer
361
views
Is $g(t)=\mathbb P[\inf_{0\le s\le t}X_s>0]$ differentiable with respect to $t$?
Consider the SDE
$$dX_t =b(t)dt + a(t)dW_t,\quad \forall t>0,$$
with $X_0>0$ has a density function $\rho:\mathbb R_+\to\mathbb R_+$. Consider the probability $g(t):=\mathbb P[\inf_{0\le s\le t}...
user478492
2
votes
0
answers
50
views
Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient
Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...
- 1,334
0
votes
1
answer
277
views
Autocorrelation function of Itô process
I'm working with a time independent (vector) Itô SDE such as:
$$
dX = a(X) dt + b(X) dW.
$$
I've looked (numerically) at several examples and it seems that the autocovariance function $r_{xx}(\Delta t)...
- 309
0
votes
0
answers
97
views
Uniqueness of the solution to some SDE of state-dependent coefficient
This is a continuation of my question posted in Uniqueness of the solution to some SDE
Consider
$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
...
- 1,334
1
vote
1
answer
133
views
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
2
votes
1
answer
139
views
Search for conditions of the positive probability that a stochastic process never hits zero
Consider a stochastic process $X$ defined by
$$X_t:=1+\int_0^t b(s,X_s) \, ds+ W_t,\quad \forall t\ge 0,$$
where $(W_t)_{t\ge 0}$ is a standard Brownian motion. Suppose that $b:\mathbb R_+ \times \...
user420828
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[...
- 1,334