All Questions
Tagged with stochastic-processes stochastic-calculus
693 questions
2
votes
1
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236
views
Self-adjointness of generator and semigroup of an SDE
$
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- 835
1
vote
0
answers
31
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:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 305
0
votes
1
answer
51
views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 5,405
0
votes
0
answers
42
views
Bound on the radon-nikodym derivative between two stochastic processes at a time point
I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution:
\begin{align*}
dX_t &= b(t, X_t)...
2
votes
0
answers
41
views
Approximate the adjoint generator of the discretization of an SDE
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$;
$p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$
$(X_t)_{t\ge0}$ denote ...
- 167
2
votes
1
answer
111
views
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
- 6,195
2
votes
2
answers
88
views
Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
- 6,195
6
votes
1
answer
133
views
Coupling/Ordering of Brownian bridges
Suppose I have two 1D Brownian bridges $(B^{(1)}_t,t\in [0,1]),(B^{(2)}_t,t\in [0,1])$, one from $0$ to $0$ and one from $x$ to $y$ where $x,y \geq 0$. Is there a neat way to show that there exists a ...
- 228
0
votes
0
answers
76
views
When we should integrate on both side over a SDE?
Maybe I am quite stupid, I am quite confused about, when we should use ito formula to solve SDE and when it is appropriate to integrate directly to get the solution?
Specifically, let us consider the ...
- 9
4
votes
0
answers
122
views
Finiteness of the moments of the Malliavin derivative of the stochastic heat equation
I am studying section 2.4.2 from Nualart's book "The Malliavin calculus and related topics" on the stochastic heat equation. I have some questions on the validity of some estimates for the ...
- 61
3
votes
1
answer
218
views
Pathwise linearization of diffusion processes
Let $W$ be a standard $n$-dimensional Brownian motion, and $X$ the diffusion process given by the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $\mu: \mathbb R^n \to \...
- 6,195
4
votes
1
answer
111
views
Scaling of stopped Hölder norm of Brownian motion
I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.
For fixed $T>0$, self similarity ...
- 153
2
votes
0
answers
82
views
Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
- 135
4
votes
0
answers
113
views
SPDE Renormalization
some SPDE (in higher dimensions) can only be interpreted in a "renormalised" sense. For example considering $\Phi_2^4$ on $\mathbb{R}_+\times \mathbb{T}^d$ the solution is defined as the ...
- 573
2
votes
0
answers
42
views
Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
2
votes
0
answers
89
views
Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
- 61
4
votes
1
answer
315
views
Impulse signal detection
Notation: Here $\mathcal Y_t$ denotes the natural filtration of the process $Y_t$, and $\{\cdot\}$ denotes the fractional part of a real number.
This question concerns detecting the presence (or ...
- 6,195
4
votes
1
answer
107
views
Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...
- 73
3
votes
0
answers
50
views
Does double stochastic integral have exponential moments?
Consider $W=(W_1,W_2):[0,1]\to \mathbb{R}^2$ a planar Brownian motion, and $W'$ a second one, independent from the first.
Let
$I=\int_0^1\int_0^1\log (|W-W'|^{-1}) \, \mathrm{d} W_1 \, \mathrm{d} {W_1}...
- 131
2
votes
0
answers
61
views
Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
4
votes
0
answers
328
views
Convergence to unique stationary distribution for SDEs and Markov processes
I am interested in understanding the behavior of solutions to stochastic differential equations (SDEs) and continuous-time Markov processes with constant coefficients. Specifically, I would like to ...
- 807
3
votes
0
answers
54
views
Unique weak solution of an SDE for a general initial distribution
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- 835
0
votes
0
answers
36
views
Interpretation of Lévy process with signed Lévy measures
Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
- 393
3
votes
1
answer
209
views
Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?
Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...
- 6,195
5
votes
1
answer
205
views
Continuity dependence and convergence of the renormalized $\Phi^4_2$ model
This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.
Again, we are interested in the local behavior of the $\Phi_2^...
- 573
1
vote
1
answer
67
views
Combination of the Dirichlet and Cauchy problems, find the PDE by which $\mathbb{E}_x M(X_{\tau_D \wedge t})$ is met
$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...
- 11
0
votes
0
answers
101
views
Simulation of Markov processes with exponential timestepping
Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way:
Choose an initial ...
- 167
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
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- 835
5
votes
2
answers
369
views
Markov process on a torus with prescribed invariant distribution
In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\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
2
votes
1
answer
86
views
Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
- 41
2
votes
0
answers
89
views
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
5
votes
1
answer
774
views
Best textbooks/resources for "advanced" probability theory?
When I say "Advanced Probability", I mean for a person acquainted with the measure-theoretic foundations of probability theory, that wants to learn about Stochastic Processes from there, in ...
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
277
views
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
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- 835
2
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0
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66
views
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?
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- 835
2
votes
1
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246
views
Does $X_t$ with $t>0$ admit a density?
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- 835
2
votes
1
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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
2
votes
1
answer
126
views
Constructing Wiener process on a given probability space
This is just a short question, and may be to basic, but:
is there a way to construct a sequece of independent wiener processes on a given probability spaces?
- 163
2
votes
0
answers
95
views
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
0
votes
0
answers
33
views
Can the optimal stopping problem be expressed in another form by strong Markov property?
$X_t$ is a strong Markov process in $(\Omega, \mathcal{F},\mathcal{F}_t,\mathbb{P})$. $\tau$ is a stopping time, $T>0, \mathbb{E}_x(\cdot)=\mathbb{E}(\cdot|X_0=x)$. By Markov property, $\mathop{\rm{...
- 11
0
votes
0
answers
81
views
White noise: a tempered distribution version of the stochastic convolution
Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...
- 573
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
133
views
A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
1
vote
0
answers
99
views
Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
- 620
2
votes
0
answers
44
views
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
7
votes
0
answers
151
views
Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
- 439
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
2
votes
1
answer
216
views
Decay estimate of moment of an SDE
We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
- 835