All Questions
Tagged with stochastic-processes brownian-motion
219 questions
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Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$
$$ dB(t) = r(t) B(t) dt$$
Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
- 203
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1
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2k
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Brownian motion and its maximum and its minimum
Let $W_u, 0\leq u \leq t$ be Brownian motion.
Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$.
The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
- 29.7k
3
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1
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902
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Exercise on a hitting time for a Brownian Motion
I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ...
- 203
3
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1
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281
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Covariation of the stochastic integral and the Wiener process
Let$^1$
$T>0$
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
$(e^n)_{n\in\mathbb N}$ be an ...
CommunityBot
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5
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1
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523
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Scaling of First-passage times for Random Walk on integer lattices
Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...
CommunityBot
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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 \...
- 19.6k
7
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1
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277
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A Converse of the Skorokhod Embedding Theorem
I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds:
Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
7
votes
1
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1k
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Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
- 984
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1
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110
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Equivalence of tail sigma-fields between two triangular arrays of Brownian motions
Let
$$\{B^{(i)}_t : 0 \leq t \leq T, i \geq 1\}$$
be a sequence of i.i.d. standard Brownian motions. For each $n,$ let $V^{(n)}_t, \,0 \leq t \leq T$ be a continuous bounded process adapted to the ...
CommunityBot
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2
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0
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194
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forward Ito integral
Forward integral is introduced by Francesco RussoPierre Vallois as a generalization of Ito integral. For simplicity, let $B$ be a standard Brownian motion and let $\phi$ be a measurable process. The ...
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1
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715
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"Brownian motion" without assuming continuity of path at origin of state space
This question is inspired partly by this question Any reference on Brownian Motion continuity. In this post, the author asked if the following three axioms can define a Brownian motion without ...
- 24.8k
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1
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250
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Azema's martingale and quadratic covariation
Given a filtered space $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ supporting a Brownian Motion $B$, where the filtration $\mathcal F$ is the augmented Brownian filtration, the Azema's martingale is ...
- 325
2
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1
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172
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Neat little proof using local time from Ito McKean
This is a cool little result, the proof of which uses the machinery of local time. On p. 72, Prob 1 asks to show that $\int_0^1 dt/x(t)$ exists, where $x(t)$ is a continuous time brownian motion.
In ...
- 1,989
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1
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528
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Any modern/recent version of Ito & McKean?
This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
CommunityBot
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653
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Explicit martingale representation for a Brownian bridge
Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
- 341
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68
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Brownian motion on a $\mathbb{Z}$-cover
Let $(M,g)$ a smooth closed Riemannian manifold with non trivial first homology group $H^1(M,\mathbb{R})$. Any element of $H^1(M,\mathbb{Z})$ will define a riemannian $\mathbb{Z}$-cover of $M$ ...
- 66.3k
3
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0
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101
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A dependent and discrete version of the Komlós-Major-Tusnády theorem
The well-known Komlós-Major-Tusnády approximation gives sharp speed of convergence of a uniform empirical process to a Brownian bridge. Here I am considering how to approach a similar problem with ...
- 66.3k
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1
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613
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2 Random Walkers on 2d square lattice, Torus
I am looking for the probability that two random walkers initially at different sites, meet at step t if they are moving on a 2-dimensional torus(Square Lattice)
Any help would be appreciated.
CommunityBot
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3
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0
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186
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When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?
When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...
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608
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weak convergence of the solutions to stochastic heat equation
$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...
CommunityBot
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1
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1
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179
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Quartiles of local time of brownian bridge at origin
Let $(L_t)_{0\leq t \leq 1}$ be the local time at $0$ of a brownian bridge. Let $(T_l)_{0\leq l \leq L_1}$ be its generalized inverse (as in $T_l := \inf\{t\geq 0 : L_t \geq l\}$). What is the joint ...
- 6,045
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1
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460
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Reflected SDE with non-Lipschitz coefficients
I have an equation of the form:
$$dX_t=\mu(X_t)dt+\sigma(X_t)dZ_t+dL_t, \quad X_0=x_0\in (-\infty,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
- 6,045
2
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1
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879
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Existence of solution for reflected SDE
I have an equation of the form:
$$dX_t=\mu(X_t)X_tdt+\sigma(X_t)X_tdZ_t+dL_t, \quad X_0=x_0\in (0,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...
- 6,045
3
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1
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1k
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Strong solution for geometric brownian motion with varying drift and volatility
I have an equation of the form:
$$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$
I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the ...
- 315
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161
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Continuity of solution map to Stratonovich Integral
For paths $x:[0, T] \rightarrow \mathbb{R}^n$, the Stratonovich integral along a one form $\omega$ on $\mathbb{R}^n$ can be defined by
$$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\...
- 9,853
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1
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2k
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How to calculate the PSD of a stochastic process
This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.
Say we have a stochastic process described by a ...
CommunityBot
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4
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1
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404
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Weighted global Holder property for Brownian motion paths
It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
\sup_{t,s\in[0,1]}\frac{|W_t-W_s|}{|t-s|^{\...
- 14.2k
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1
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149
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Brownian motion increments
We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too.
Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$
and I seek another Brownian ...
- 54.2k
3
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1
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933
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Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)
This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
CommunityBot
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2
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2
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1k
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Total absolute variation of brownian motion, with different sampling rates
Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed.
Let's compute the total absolute variation when sampling period = $\delta$ is fixed:
$$V(\delta) = \sum_{i=0}^{N-...
- 587
5
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2
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724
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Brownian motion in $\mathbb{R}^n$, probability of hitting a set
Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
- 29.7k
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1
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2k
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Blumenthal and Kolmogorov 0-1 law
Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...
- 29.7k
7
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2
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984
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Brownian motion in $n$ dimensions
Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
- 2,703
6
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1
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374
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Large deviation for Brownian path on $[0,\infty)$
It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...
- 7,514
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$X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]
What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations ...
- 11
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0
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77
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Law of motion when initial condition is perturbed
We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
$$dX_t=\mu(t,X_t)...
- 1,533
2
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1
answer
3k
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Time Change of a Brownian motion
We know that for if $X$ is a stochastic integral of the form below -
$X_t = \int_0^t v(s,\omega) db(s,\omega)$.
then we can use time change formula to claim that
$X_t = W_{\alpha(t)}$ where $W$ is ...
- 3,085
3
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1
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604
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Brownian bridge on a Lie group as a stochastic differential equation
Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
- 17k
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2
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3k
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Quadratic variation for discrete Martingale
Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...
- 11
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1
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106
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Defining a brownian bridge indexed by angle
I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin (centroid)...
CommunityBot
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4
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1
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447
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Area enclosed by Brownian motion (without winding number)
The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
CommunityBot
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1
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4k
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Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
- 29.7k
1
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2
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440
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$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?
I am trying to prove the following Lemma, which seems intuitive, but I still have doubts:
Lemma
Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...
- 237
1
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1
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971
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Integration of independent Brownian motions
I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where $\tilde{...
- 11
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0
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57
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References for symmetric α-stable process (SSP) for $a>2$
Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...
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2
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436
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Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds
I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...
- 2,281
3
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0
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75
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What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?
The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$.
1)So I was wondering how can one compute $P_{x}(T_{B_{r}(...
- 5,474
0
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0
answers
233
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Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A
In other words, the probability that Brownian motion stays within $A^{c}$.
What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...
- 5,474
3
votes
1
answer
156
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Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)
I am trying to formulate the measure of event
$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,
where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...
CommunityBot
- 1
2
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1
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960
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Branching Brownian Motion and the KPP equation
I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
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