All Questions
Tagged with stochastic-processes brownian-motion
219 questions
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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 168
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0
answers
24
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Relationship between transition density function and local time
Assume the local time is $L(t,y)$ and we know $P_x(L(t,y) \in d\tau)$ where $P_x$ denotes the probability measure for a stochastic process starts at $x$. Can we then derive the transition density ...
- 11
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1
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202
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Independent stationary increment process but with finite propagation speed
Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability ...
- 807
6
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1
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133
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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
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157
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Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
- 640
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61
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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
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127
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A "resampling identity" for the Bessel(3) process
I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
- 10.3k
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307
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PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
- 620
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133
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A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
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99
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Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
- 620
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148
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Brownian motion reflected at a trailing barrier
Let $X_t$ be a Brownian motion with positive drift starting at 0. The process with reflection at fixed barrier $b<0$ (sometimes called a "regulated Brownian motion") is:
\begin{equation}
\...
- 21
3
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1
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312
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Laplace transform of Brownian motion functional
Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity
$$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
- 228
3
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180
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Are the paths of the Brownian motion contained in a suitable RKHS?
Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...
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462
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On the convergence of a martingale
Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by :
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$
and for $t\geq 0$, we ...
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211
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Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
- 6,155
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143
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Stochastic braids
I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
- 2,152
4
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143
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Reflecting Brownian motion in disk
What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it?
The transition density ...
- 177
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2
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489
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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
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134
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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
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1
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159
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Measurability of two hitting times at the stopped $\sigma$-algebra
Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times
$\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
- 503
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95
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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
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66
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Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$
Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times
$$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$
for strictly ...
- 5,474
2
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1
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273
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If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine
We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$
Therefore $u-u(...
- 573
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1
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168
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
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545
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Each diffusion SDE is associated to a *unique* family of transition kernels
I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$.
How can I prove that there exists a unique family of transition ...
- 1,149
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58
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Locality and restriction properties for self-avoiding and loop-erasing random walks
This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa
I ...
- 541
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100
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Characteristic exponent after Girsanov transformation
Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be
$$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$
Now I want to apply a Girsanov ...
- 245
3
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225
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Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier
Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$.
Can we ...
- 177
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2
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613
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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
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93
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Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?
Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions
$$
\chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1)
$$
where $\chi(t)$ ...
- 297
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282
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Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:...
- 1,389
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1
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291
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Joint distribution for sticky Brownian motion
$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by
\begin{gather*}
dX_t=1_{[X_t\neq 0]}dB_t\\
L_t(X)=\int_0^t 1_{[X_s=0]}ds,...
- 5,380
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3
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1k
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"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?
If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale).
If one uses a ...
- 59
1
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2
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238
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Converse Cameron-Martin theorem for shifts by adapted processes
Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$.
Given a $C[0, 1] $ valued random ...
- 6,155
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469
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Derive the solution of the diffusion equation from the solution of a random walk
Summary
The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
- 274
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103
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Brownian motion hitting open set starting from its boundary
Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$.
Which conditions on $U$ are necessary and ...
- 177
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422
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Regularity of translations for Brownian motion
Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$.
...
- 319
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163
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Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
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2
answers
1k
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Brownian bridges as conditioning
Brownian bridges are interpreted as Brownian motions conditioned to start and end at given points. However, I have not seen a source that makes this precise, though this may be due to my own lack of ...
- 6,155
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2
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131
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Density of $W_t$ assuming it stayed above a line $L$
Let $W_t$ be a Wiener process with $W_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).
Assume that $W_t$ stayed above $L$ up to time $T$. What is ...
- 2,788
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0
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255
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Distribution of "occupation times" of Brownian Motion
Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set.
I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
- 245
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1
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186
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Upper left Dini derivative of Brownian motion at a hitting time
Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by
$$D^-W_t := \limsup_{h \to 0^-} \frac{W_{t+h} - W_t}{h}.$$
Fix $a > 0$, and define the stopping time $\tau$ by
...
- 6,155
2
votes
1
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392
views
Full version of Cameron Martin theorem for Brownian motion
I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement:
Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[...
- 6,155
4
votes
2
answers
455
views
Converse of Itô's formula
Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$
Prove that $f$ is ...
- 573
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1
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323
views
Integrated square difference of Brownian bridges
I am doing some work with measuring the distance between distributions, and someone pointed out to me that I should look into calculating the integrated squared difference of two brownian bridge ...
2
votes
1
answer
294
views
What is the quadratic variation of $W(B(t))$?
Let $W$ be a two sided real valued Brownian motion. Let $B$ be a one sided Brownian motion independent of $W$. Consider the process $X(t)=W(B(t))$. Is the quadratic variation finite and if it is, what ...
- 1,904
3
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1
answer
655
views
Forgery theorem: the Brownian motion stays close to any curve with positive probability
In a paper I am reading the authors claim that, if $B$ is a standard BM in $\mathbb{R}$ and $f\in C([0,1],\mathbb{R})$, then for any $\epsilon>0$
$$
\mathbb{P}(\sup_{t\in [0,1]}|B_t-f(t)|<\...
- 1,149
1
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0
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103
views
Continuity of Wiener measure on open balls
Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...
- 6,155
4
votes
1
answer
350
views
Lebesgue differentiation theorem at a stopping time
Let $W$ be a standard Brownian motion, and $\mathcal F_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F_t$ and integrable with respect to $W$.
Question: Is it true ...
- 6,155
1
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0
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89
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Comparison of the numbers of particles surviving forever
Consider two $N\text{-}$particle systems as follows : for $1\le i\le N$,
$$X^i_t=1+\int_0^t(b+\phi^i_s) \, ds+W^i_t \quad\mbox{and} \quad Y^i_t=1+ct+W^i_t,\quad \forall t\ge 0,$$
where $c>b>0$ ...
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