All Questions
Tagged with pr.probability stochastic-differential-equations
88 questions with no upvoted or accepted answers
1
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121
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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
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
1
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0
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235
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Two increasingly correlated Brownian motions and Williams decomposition
The Williams decomposition is
Let $(B_t-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_\infty^{-\nu}:=\sup_{t\in [0,\infty]}(B_t-\nu t)$. Then conditionally on $M_\...
- 5,474
1
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0
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156
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Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
1
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0
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157
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The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
1
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0
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248
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Regularity of Fokker-Planck equation
Consider solutions $\rho_{1,2}$ of the Fokker-Planck equation
$$\begin{cases}\partial_t \rho_i = \Delta \rho_i + \nabla \cdot (\rho_i \nabla \Phi_{1,2})\\
\rho_i(0,\cdot) = \rho^0 \end{cases}$$
for ...
- 223
1
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0
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54
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Conditions ensuring that conditional law of a process belongs to a given exponential family
Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
- 77
1
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78
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If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?
Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
- 309
1
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0
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166
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Are SDE adapted to the natural filtration?
Let $(B^H_t)_{t\in [0,T]}$ be a fractional Brownian motion. We consider the following SDE where $b$ and $\sigma$ are Lipschitz
$$X_t=x+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB^H_s.$$
When $H>1/2$, ...
1
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0
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57
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Choice of Banach space for stochastic processes
In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used:
$$
\sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p}
$$
and
$$
\mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p]^...
- 379
1
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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
1
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0
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276
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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
1
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0
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62
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Reference request for invariance principles
In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...
- 111
1
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61
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Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
1
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0
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80
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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 ...
- 5,405
1
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0
answers
73
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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
1
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0
answers
127
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Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE
Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...
- 167
1
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0
answers
134
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Moment Estimate
Let $\epsilon > 0$ be a small parameter and consider the following lemma.
Lemma. Let $B(t)$ be a bounded, continuous, $R^{n \times n}$-valued function defined on a time interval $[0,T]$ such that ...
- 31
1
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0
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90
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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
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0
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340
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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
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0
answers
249
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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$...
- 11
1
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57
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Matching Numbers in Ito McKean
Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...
- 163
1
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0
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260
views
Transforming reaction-diffusion equations to random walk processes
I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
- 53
1
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0
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29
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Usually trivial Excursion-type process
How Can i construct a stochastic process $X_t$ which has the property that:
$X_t \in [0,1]$ for all $t \in [0,T]$ and
$m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]...
1
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0
answers
102
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What is meant by local time of BM on the boundary $\partial D$?
I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
- 282
1
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0
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118
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Full version of Soucaliuc's research announcement "Réflexion entre deux diffusions conjuguées"
Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...
- 43
1
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155
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Convergence of approximate quadratic variation in $L^p$
For a diffusion $X_t$, I can set
$$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$
Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...
- 9,853
1
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0
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108
views
Cauchy Problem and stochastic representation for discontinuous initial data
Where can I read more about the Cauchy problem, i.e. solutions to
$$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$
for some elliptic differential operator $L$ where $f$ is not ...
- 237
1
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0
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119
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When the completed filtration of a process increases slowly
If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$
$$\mathcal{F}^{\...
- 19
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0
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120
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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
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0
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468
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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
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0
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97
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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
0
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0
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47
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Exit probability on a finite interval
I have a question about the estimate of the exit probability on a finite interval. Given a $q$ function bounded and continuous, given the following SDE
\begin{cases}
dX_s=(\beta-q(s))X_sds+\frac{1}{2}...
- 1
0
votes
1
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257
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Solving SDE with sign function in drift term?
Consider the following SDE with $X_0 = 1$,
$$
dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t,
$$
where $\operatorname{sign}(x) = \mathbb{1}\{x \ge 0\}$. How am I supposed to solve this SDE?
- 1
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0
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294
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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. ...
- 5,405
0
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0
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48
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Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 5,405
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0
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76
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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
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0
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57
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Parametric distribution where the parameter follows a diffusion process
I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...
- 1,902
0
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0
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153
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Embedding a martingale by SDE
Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...
- 1,835