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Questions tagged [stochastic-analysis]

For questions about stochastic analysis or stochastic calculus, for example the Itô integral.

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Convergence sequence of Gaussian random variables as mesh grid goes to zero

Consider a compact interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})...
Grandes Jorasses's user avatar
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Is a Gaussian process with covariance $C''(0)=0$ allowed?

Let $y(t)$ be a zero mean, homogeneous isotropic Gaussian process: $$ \left<y\right>=0 \qquad,\qquad \left<y(t_1)y(t_2)\right>=C(t_2-t_1) $$ where $\left< \dots \right>$ denotes ...
Sal's user avatar
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Whether we can get energy estimate $d\|X\|^2+2\|X\|_{H^1}^2 dt=2(X, B dW) +{\rm Tr}(BB^*)dt$ of a weak solution of stochastic differential equations? [closed]

Set an probability space $(\Omega,{\mathcal F},P)$ with filtration ${\mathcal F}_{t,t \ge 0}$, and seperatable Hibert space $H$ and $U$ a self adjoint, sectoral,densely defined operator A on H with ...
shanlilinghuo's user avatar
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Transition probability density of non-intersecting Brownian bridges are independent of h function from Doob-h transform?

I've been trying to derive the transition probability of a $2$-dimensional Brownian motion $B = (B^{(1)}_{t}, B^{(2)}_{t})_{t \geq 0}$ conditioned to stay in the Weyl chamber and also conditioned to ...
tornt's user avatar
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One question about exponential martingale inequality at a paper of Ann. of Math.

I saw a strange inequality about martingales: If $\operatorname{M}\left(s,t\right)$ is an continuous $L^{2}$ martingale start at $s$, $\left[\operatorname{M}\right]\left(s,t\right)$ is its quadratic ...
shanlilinghuo's user avatar
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2 answers
39 views

Expectation of Solution to SDE, logarithmic extension of Vasicek

I have the solution to the following model $$dr(t)=r(t)(\eta-a\log r(t))dt+\sigma r(t)dW(t)$$ which, through the Vasicek model and a change of variable, I found to be $$r(t)=\exp[Y(0)e^{-at}+\frac{\...
mtcicero's user avatar
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Show that $S(\omega) := \sum _{i=1}^{Y(\omega)} X_i(\omega)$ is a square-integrable random variable in $L^2(\Omega,\mathcal A, P)$

Let $n \in \Bbb N,M \geq 0$ be constants,$( \Omega,\mathcal A,P)$ be a probability space, $Y : \Omega \to \{0,1,\dots ,n\}$ a random variable and $X_1,\dots, X_n : [-M,M]$ random variables with: (i) ...
gagamaga's user avatar
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Can a random field of Ito diffusions spend positive time in a Lebesgue measure 0 set?

Suppose I have a family of Itô diffusions governed by the following SDEs: $$dX_t(x) = b(t,x, X_t(x))dt + \sigma(t,x,X_t(x))dW_t $$ with $X_0(x) = h(x)$. Suppose $x \in \mathbb{R}^d$, $X_t(x)$ is $\...
qp212223's user avatar
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"Heuristic" vs. Rigorous Ito's Lemma

Assuming $X_t$ is a standard Brownian motion and $t$ is the time variable, I have learned to derive Ito's lemma for a function $F(X_t, t)$ using the following results (below, $X_t=X(t)$, and I ...
Jan Stuller's user avatar
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1 answer
106 views

Probability that a probability will be less than a certain value

Suppose I have a nonnegative random variable $X$ and I don't know its expected value, but I know that its expected value is less than or equal to $a$ with at least probability $p^*$. i.e, $\mathbb{P}(\...
curiosity's user avatar
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Best book for stochastic analysis on manifolds

I know there is a book by Emery on Stochastic calculus on manifolds but is there any other book/online notes with an easier exposition of the subject ? Any reference will be highly appreciated.
Soumya Ganguly's user avatar
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Converging PDEs as hydrodynamic limits of converging particle processes

Suppose that some interacting particle process $\{X^i_t\}_{i=1}^n$ has a hydrodynamic limit which is a PDE $$\rho_t = L \rho + f\quad \quad \quad (1) $$ where $L$ and $f$ are possibly nonlinear. By ...
900edges's user avatar
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How do we use strong Markov property to get the formula

Suppose $X(t)$ is a homogeneous Markov process and satisfies the strong Markov property. Assume that $U\subset\mathbb{R}^{n}$ is an open and bounded set, denote the boundary of $U$ by $\Gamma$, $\tau_{...
R-CH2OH's user avatar
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Extending Law of Large Number for Square integrable Martingale

Suppose we have a square integrable martingale $\{M_n\}$ with $\lim_{n\to\infty}M_t=\langle M\rangle_n=\infty$ a.s. Now, if we have a non-decreasing function $g:[0,\infty)\to[0,\infty)$, and also $$\...
Arbitor Lunae's user avatar
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Is a randomly restarted Markov process again Markov?

Let $E$ be a topological space and $(X_t)_{t\ge0}$ be a right-continuous time-homogeneous Markov process with transition semigroup $(P_t)_{t\ge0}$ on a filtered probability space $(\Omega,\mathcal A,(\...
0xbadf00d's user avatar
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Proof that excessive function are also regular (super-harmonic)

In page 117 of Shiryaev's book optimal stopping rules, he claims that the excessive functions are also regular under some condition and state the proof is analogous to another proof, but I fail to ...
Stocavista's user avatar
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Skorokhod topology vs. weak topology [closed]

Let $\mathbb{D}([0,T],V)$ be the collection of the cadlag(right-continuous and has left limit) functions on $[0,T]$ taking values in a Hilbert space $V$. The space $\mathbb{D}$ is equipped with the ...
George's user avatar
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3 votes
2 answers
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What is the transition semigroup of toroidally wrapped Brownian motion?

More generally, let $(B_t)_{t\ge0}$ be an $\mathbb R^d$-valued Lévy process and $W:=\iota(B)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (the floor function is applied ...
0xbadf00d's user avatar
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2 votes
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If $X$ is a time-homogeneous Markov process, is $\tilde X:=X-\lfloor X\rfloor$ Markov as well?

Assume $(X_t)_{t\ge0}$ is an $\mathbb R^d$-valued time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ on a probability space $(\Omega,\mathcal A,\operatorname P_x)$ ...
0xbadf00d's user avatar
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1 vote
1 answer
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Expression of a stochastic process at a stopping time: $X_{\tau}$

Let $(X_t)_{t \in [0,\infty)}$ be a real valued stochastic process, let $\mathcal{F}$ be a filtration on $[0,\infty)$. Let $\tau$ be a stopping time of $\mathcal{F}$. Is it possible to give an ...
Fran712's user avatar
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1 answer
103 views

Quadratic variation of the square of Brownian motion

Let $B_t$ be the Brownian Motion. Find the quadratic variation of a martingale $ M_t = B_t^2-t$. My solution: By Ito's formula for $f(t, x) = x^2-t$, we know $$d(B_t^2-t) = 2B_t dB_t$$ thus $\langle M ...
nessy's user avatar
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3 votes
1 answer
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Prove a Characterization for Localization of Stochastic Processes

In Exercise 3.4.16 in Stochastic Calculus and Applications by Cohen and Elliott, it states: Let $C$ be a set of processes such that if $X$ is a process with $X^T, X^S \in C,$ for $S,T$ stopping times,...
Yung-Hsiang Huang's user avatar
5 votes
1 answer
63 views

Showing bounds of Stochastic Process

Suppose that we have the SDE: $$ dZ_t = 2Z_t(1-Z_t)dt + 4Z_t(1-Z_t)dB_t $$ With $Z_0 = \frac{1}{3}$. How can I show that $0 \leq Z_t \leq 1$. I have tried solving the 'alalogous' differential ...
Lehmann's user avatar
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Limit of expectation of function of stopping time

Let $\tau_{n}$ be a stopping time such that $P(\tau_{n} \rightarrow \infty) = 1$, and consider the following limit: $$\lim_{n\rightarrow \infty} E(e^{t\wedge \tau_{n}}f(X_{t\wedge \tau_{n}}))$$ Can I ...
Ethan Davitt's user avatar
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25 views

A problem of measurability of stochastic kernel

I want to use the stochastic kernel theorem to prove the existence of the following measure: I have a probability space $(\Omega,\mathcal{F},P)$ and and infinite-dimensional measure space $(A,\...
houssem agili's user avatar
2 votes
1 answer
74 views

Quadratic covariation of martingale transforms to simple processes.

Let $X$ and $Y$ be simple processes, that is $X_t=\sum_{n=0}^\infty\xi_n{1}_{(t_{n},t_{n+1}]}(t)$ for a uniformly bounded sequence $(\xi_n)_{n\in\mathbb{N}}$ of random variables so that $\xi_n$ is $\...
Daan's user avatar
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2 votes
0 answers
23 views

The probability of first passage within finite time

Let $\left(X_t\right)_{t\ge0}$ be a L'evy process with $X_0=0$ which drifts to $-\infty$. Let $T>0$. For $a>0$, define $\tau_a=\inf\{t\ge0: X_t>a\}$. What kind of assumption on $X$ should be ...
user377704's user avatar
2 votes
0 answers
48 views

Can every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be extended to $C^2(\mathbb{R}^d)$?

When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In ...
Jesen's user avatar
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1 vote
1 answer
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May the sum of Wiener processes be a Wiener process?

May $X_t = W_t + \tilde{W}_t$ be a Wiener process, if $W_t, \tilde{W}_t$ are Wiener processes? I know that: $X_t$ may be not a Wiener process, e.g. in case $W_t = \tilde{W}_t$. If $W_t$ and $\tilde{...
Sergei Nikolaev's user avatar
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0 answers
15 views

Prove $(g_i ◦ X_i) , i ∈ \{1, . . . , n\}$, are independet random variables (Check my proof please)

Let $(Ω, \mathcal A, \Bbb P)$ a probability space, $n ∈ \Bbb N$ and let $(R_i, \mathcal R_i), (S_i, \mathcal S_i), i ∈ \{1, . . . , n\}$,measuring spaces. Let $X_i: Ω → R_i, i ∈ \{1, . . . , n\}$, ...
gagamaga's user avatar
3 votes
1 answer
100 views

Probability that Brownian Motion takes value in an $L^2$-Ball

Suppose $W:[0,1]\times \Omega \to \mathbb{R}$ is a standard Brownian motion on the unit interval. With $L^2[0,1]$ denoting the space of real-valued square-integrable functions with standard norm $\|\...
LostStatistician18's user avatar
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0 answers
24 views

Variant of Gronwall’s inequality

Gronwall inequality typically is used to bound a function $u(t)$ if it satisfies $u(t)\leq \alpha(t)+ \int_{0}^{t}\beta(s) u(s)ds $ with the condition that $\beta$ is non-negative. I want to use these ...
Tiramisu's user avatar
1 vote
1 answer
50 views

On the integrand of the infinite dimensional stochastic integral

Let $W$ be a $Q$-Wiener process taking values in a Hilbert space $U$, $U_0:=Q^{\frac{1}{2}}U$ be the reproducing kernel Hilbert space of $W$. In [Da prato and Zabczyk,2014], it states that the ...
George's user avatar
  • 124
2 votes
0 answers
42 views

Linearizing/Approximating Nonlinear Stochastic Differential Equations

I'm working with a multivariate stochastic differential equation (SDE) of the form \begin{equation} \mathrm{d} \boldsymbol{x} = \boldsymbol{f}(\boldsymbol{x},t) \mathrm{d}t + \boldsymbol{G}(\...
kjc93's user avatar
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0 answers
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On the definition of Gelfand triple

Gelfand triple is a useful concept in functional analysis and stochastic PDEs. In chapter 4 of [Rockner and Liu, 2015], it states that: Let $H$ be a Hilbert space, and $V$ be a reflexive Banach space ...
George's user avatar
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5 votes
0 answers
84 views

How to solve the SDE $\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t)$?

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.4, the author claims that the problem $$\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t),X(0)=...
R-CH2OH's user avatar
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0 answers
47 views

Karhunen–Loève theorem expansion random variables

In the Karhunen–Loève theorem's statement the random variables in the expansion are given by $$Z_k = \int_a^b X_te_k(t) \: dt$$ $X_t$ is a zero-mean square-integrable stochastic process defined over a ...
Carlo C's user avatar
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4 votes
0 answers
53 views

Does a Poincaré inequality for a Markov process $X_t$ with invariant measure $μ$ infer a convergence rate of $\frac1t\int_0^tf(X_s){\rm d}s$ to $μf$?

Let $(X_t)_{t\ge0}$ be a time-homogeneous shift-ergodic Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ and invariant measure $\mu$. One implication of a Poincaré inequality is a $L^2$-...
0xbadf00d's user avatar
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1 vote
0 answers
30 views

For a Markov process, does a larger spectral gap imply faster convergence to the invariant distribution?

Say I have two time-homogeneous Markov processes $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ which both admit $\pi$ as their unique invariant measure. Let $A$ and $B$ denote their generators on $L^2(\pi)$ and ...
0xbadf00d's user avatar
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0 votes
1 answer
80 views

Solve SDE $dX_t = X_tW_tdt + dW_t,$ [closed]

I was working on question 7.5d in Applied Stochastic Analysis by by Eric Vanden-Eijnden, Tiejun Li, and Weinan E, which is given as $$dX_t = X_tW_tdt + dW_t, X_{t|t=0} = X_0, t\in \mathbb{R}_+$$ ...
HHHHHHHH's user avatar
1 vote
0 answers
38 views

Finding Cameron-Martin space of process

The question Find Cameron-Martin space of known process was not answered, but I found it rather interesting. It is well-known that Cameron-Martin space of Wiener measure is space $W_0^{2,1}$ (see ...
GeoArt's user avatar
  • 139
4 votes
0 answers
60 views

Ratio of Expected Hitting Times of Brownian Motion with Drift

Suppose $W_t$ is Brownian motion and consider the following two stopping times: $$\tau_a \equiv \inf \{t \ge 0 : W_t + at \ge b(t) \} \wedge T$$ and $$\tau_{-a} \equiv \inf\{t \ge 0: W_t - at \ge b(t)\...
qp212223's user avatar
  • 1,662
3 votes
1 answer
50 views

Proof that an Ito diffusion start with 0 will be positive immediately

Let $X_t$ be a Ito diffusion satisfying the SDE $dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$ with $\mu$ and $\sigma$ being Lipschitz and $X_0=0$. Assume that $\sigma(x)>0$. Can we prove that $\tau=0$ almost ...
Stocavista's user avatar
5 votes
0 answers
140 views

Expected number of "survivors" of independent Brownian motions on the line

[Edited since originally posted] I'm trying to find a closed form for the expected number of "survivors" of $n$ independent standard Brownian motions in $\mathbb{R}$ up to time $t > 0$ ...
Jacob Shkrob's user avatar
1 vote
2 answers
97 views

Potential density of killed brownian by local time

Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(-...
JY0's user avatar
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0 answers
25 views

Bernstein von Mises approximation of posterior density

Is there a version of the Bernstein von Mises theorem that allows us to perform a Gaussian approximation of the posterior density (assuming such a density with respect to the Lebesgue measure exists)? ...
ExcitedMathematician's user avatar
1 vote
1 answer
62 views

Proving martingale properties

I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc. I have two proofs that I'm now sure how to handle. Problem 1. Prove that an ...
eMathHelp's user avatar
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0 votes
1 answer
41 views

Showing Variant of Exponential Martingale is a super-martingale

Let $B_t$ be a standard Brownian motion, and let $a_t$ be progressively measurable and so that $a_t \in [0,1]$ almost surely. It's well known that $e^{\theta B_t - \frac{1}{2} \theta^2 t}$ is a ...
Alan Chung's user avatar
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0 votes
1 answer
26 views

Express a stochastic process in terms of another

Let Y and Z be two random variables on $(\Omega,\mathcal{F},P)$, it is well-known that $\sigma(Z)\subset \sigma(Y)$ is equivalent to that there exists some Borel measurable function such that $Z=f(Y)$....
tfatree's user avatar
  • 359
1 vote
1 answer
38 views

Expected value exists because it has an integrable lower bound?

I'm trying to understand an argument my prof made: Given $h:\mathbb R \rightarrow \mathbb R$ convex, we look at $E[h(X)]$. The expected value exists because $h(X)$ is lower-bounded by [some] $l(X)$ ...
mathematics-and-caffeine's user avatar

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