All Questions
250 questions
2
votes
0
answers
385
views
Ito lemma for manifold semimartingales
I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...
- 5,405
2
votes
0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
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 ...
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
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
2
votes
0
answers
204
views
Onsager-Machlup function for special matrix-valued diffusion process
Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...
- 83
2
votes
0
answers
98
views
Non-existence for a sort of probability measures
We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...
- 257
2
votes
0
answers
288
views
The existence of stationary measures for certain Markov process
My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
- 121
1
vote
1
answer
655
views
Expectation of stochastic integral
Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation
$$ \left\langle \int^...
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 ...
- 6,223
1
vote
1
answer
380
views
Approximate an exponential martingale through its kernel
Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale
\begin{align}
M_t = \exp\left[\int_{0}^{t} h_s \,dB_s - \frac{1}{2}\int_{0}^{t} h_s^2\...
- 503
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
3k
views
using Feynman-Kac formula
I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
1
vote
2
answers
119
views
SDEs: Bounding the variance of a solution
I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it.
Let
$$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} \...
- 133
1
vote
1
answer
107
views
How to obtain this differential relation about moments of a stochastic process?
$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos.
...
- 825
1
vote
1
answer
201
views
A comparison principle for SDE
Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE
$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$
$$dY_t = \mu_Y (t, \...
- 6,223
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, ...
- 77
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&...
- 284
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 ...
- 133
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 ...
- 5,405
1
vote
1
answer
67
views
Combination of the Dirichlet and Cauchy problems, find the PDE by which $\mathbb{E}_x M(X_{\tau_D \wedge t})$ is met
$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...
- 11
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
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 ...
1
vote
2
answers
240
views
Solution to SDE conditional on high maxima of driving Brownian motion
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = X_t \, dW_t \;, \quad X_0 = 1 \;.$$
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
- 6,223
1
vote
1
answer
293
views
Time-Reversal of BSDE = SDE
Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$:
$$
Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;
$$
...
- 5,405
1
vote
1
answer
472
views
Can derivatives of 2 stochastic processes be multiplied?
We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is ...
- 2,246
1
vote
2
answers
789
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 $\mathcal{P}(\mathbf{F})$...
- 309
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 ...
- 93
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:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
- 5,405
1
vote
1
answer
435
views
How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
- 29
1
vote
1
answer
242
views
Non-commutative Ito Formula
Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication?
That is where
$$
\Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t
$$
but instead
$$
\Delta X_t = ...
- 5,405
1
vote
2
answers
2k
views
Deriving the HJB equation for exponential utility
I would like to derive the HJB equation for the following stochastic optimal control problem:
$ \Phi(t,x)=\sup_{h} E \left[\exp \left\{\gamma \int_t^T g(X_s,h(s);\gamma)\ ds \right\} \right]$
where ...
- 111
1
vote
1
answer
460
views
Reflected SDE with non-Lipschitz coefficients
I have an equation of the form:
$$dX_t=\mu(X_t)dt+\sigma(X_t)dZ_t+dL_t, \quad X_0=x_0\in (-\infty,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
- 315
1
vote
1
answer
164
views
Hilbert-Space Values SDE in terms of Basis
Suppose:
$$
dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
- 5,405
1
vote
1
answer
133
views
Reference for convergence of Hilbert-space valued SDEs
I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However ...
- 5,405
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
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...
- 1,649
1
vote
1
answer
238
views
Perturbation of a Bessel process of dimension 2
Bessel process of dimension 2 is defined to be solution of
$$
dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0
$$
where $B$ is a standard 1-dimensional Brownian motion.
$X$ can be viewed as the norm of a ...
- 121
1
vote
0
answers
32
views
$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 305
1
vote
0
answers
58
views
Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
1
vote
0
answers
53
views
The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
1
vote
0
answers
122
views
Derivative with respect to initial condition for the solution of an SDE
Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...
- 111
1
vote
0
answers
159
views
Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$
I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
- 73
1
vote
0
answers
193
views
Stochastic volatility model question
Let suppose that $S_t$ is a process defined as:
$$ \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}$$
where the two Brownian motions have ...
- 393
1
vote
0
answers
102
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,904
1
vote
0
answers
108
views
Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)
Let $X$ be the solution to some stochastic differential equation
$$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$
Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
- 1,399
1
vote
0
answers
237
views
Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
- 11
1
vote
0
answers
190
views
Eigenvalues/eigenfunctions of a diffusion generator
Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
- 63
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
$$
d X_t = ...
- 657
1
vote
0
answers
121
views
Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
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
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