The stochastic-analysis tag has no wiki summary.
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Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
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1answer
37 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
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1answer
36 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
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1answer
36 views
Integral: Is there a closed form?
I wonder whether there is a closed form or way to compute explicitly:
$$\int_0^t e^{\alpha s} dB_s$$
where $\alpha$ is just a real number and the integral is in the Itô sense.
Thank you very much!
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0answers
19 views
Solve a special non-linear Backward SDE
It is straigtforward to solve a linear Backward SDE. i.e.
$dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.)
How can I solve $dY_t=Z_tdW_t+ ...
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0answers
23 views
Densities of r.v in stochastic analysis
I have several exercises to solve and there are two which I somehow do not manage to solve...
We consider $W=\{W_t:t\geq0\}$ a standard B.M. issued from zero, for $a\in \mathbb{R}$, ...
2
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1answer
74 views
Example Martingale not UI
I'm looking for an example of two stopping times $\sigma\leq\tau$ and a martingale $M$ that is bounded in $L^{1}$ but not uniformly integrablem for which the equality ...
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0answers
24 views
Variance & Expectation
$X$ is a random variable with values in the set of natural numbers and the Generating function G. In Addition: $t(n) = P(X>n)$.
Let $F$ be the generating function of the sequence $\{t(n): n \ge ...
2
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1answer
83 views
Haar basis on $L^2(0,1)$ - proof?
I have the following problem.
We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$,
$$f_{j,n}(t)=\left\{ ...
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1answer
31 views
Generating function of a random variable
I've got the following problem:
Give the generating function of the random variable $X$ whose mass function is defined by:
$$f(m) = P(X=m) = (m+1) p^2 (1-p)^m,$$ where $m$ is a positive integer ...
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35 views
Intuitive meaning of the generator of a Brownian motion $L=\frac{d}{\lambda(dx)}\frac{d}{dx}$
For a standard Brownian motion $B_t$, the generator is
$$
L_B=\frac12 \frac{d}{dx} \frac{d}{dx},
$$
we say that $B_t$ is a diffusion with canonical scale the Euclidean space, and speed measure the ...
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0answers
36 views
The identity of two parameters derived via conditioning arguments
Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors $\theta_1, ...
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1answer
85 views
Why is the following function not càdlàg?
I have constructed the following function but I can't see why it is not càdlàg on $[0,1]$:
$$f(x)=\begin{cases}
1, & ...
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0answers
24 views
Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$
I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
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1answer
23 views
Stochastic Processes Question
Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
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1answer
409 views
Stochastic Diff Eq SDE
enter link description here
Consider the following SDE
$$d\sigma = a(\sigma,t)dt + b(\sigma,t)dW $$
The Forward Equation (FKE) is given by
$$\frac{\partial p}{\partial t} = ...
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2answers
215 views
Ito Isometry and quadratic variation
Here is a confusion regarding stochastic integrals. Let
$Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
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1answer
60 views
How can I prove it is a martingale when there is a jump process
Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale.
Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $
$\tau_i$ ...
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1answer
47 views
One stochastic integrability problem
On a lecture notes, there is a following arguement:
To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
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1answer
46 views
Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$
I'm working on this problem:
Given a solution $X_t$ to the SDE
$$dX_t=dB_t+b(X_t) dt$$
where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
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1answer
29 views
Defining an equivalent measure starting from a continuous local Martingale
Suppose we have continuous local martingal $L$ given. We define $Z=\mathcal{E}(L)$, the stochastic exponential of $L$. I am interested in finding some condition such that $Z$ defines a density, i.e. I ...
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0answers
81 views
How do I derive the Gaussian Mixture distribution of an Ito Integral?
I have a question about the distribution of an Ito Integral. Consider the integral
$$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$
where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
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1answer
70 views
Joint Convergence and Donsker's Theorem
I have a question about joint convergence results derived from an FCLT (i.e., a Functional Central Limit Theorem). To motivate my question, consider the following setup:
Let $y_t$ be a random walk
...
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1answer
182 views
Is continuous L2 bounded local martingale a true martingale?
I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...)
I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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0answers
29 views
How do you convert an infintesimal generator of a Markov process to a transition function?
Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
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2answers
57 views
Prove this equation
I'm taking a course on stochastic analysis. I'm stuck on the very first problem of the lecture notes:
$\lim_{n \to \infty} \left(1+\frac{\lambda}{n} + o(n^{-1})\right)^n = \exp(\lambda)$
Prior to ...
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1answer
46 views
Clarke Ocone representation formula
Let $(B_t)_{t}$ a Brownian motion and $F \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then we know by Itô's representation theorem that there exist a process $X$ such that $$F=\mathbb{E}F+\int_0^T X_s ...
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2answers
204 views
Area enclosed by 2-dimensional random curve
Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
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2answers
50 views
Question Regarding Poisson and probability.
i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions.
Given Question:
At a subway station, eastbound trains ...
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0answers
57 views
Representation theorem for continuous process of finite variation
There is a martingale representation theorem
If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that
$$
M_t = M_0 + \int_0^t ...
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0answers
50 views
Initial Conditions for Finite Difference of PDE
I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab.
Specifically, I'm trying to show that the regular 1-Dimensional ...
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103 views
Levy's characterization of Brownian motion
Consider two processes:
$$A(T) = A(t)e^{(r-\frac12 \sigma_A^2)(T-t)+\sigma_A (W_A(T)-W_A(t))}$$
$$B(T) = B(t)e^{(r-\frac12 \sigma_B^2)(T-t)+\sigma_B (W_B(T)-W_B(t))}$$
where $W$ is a Wiener process ...
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1answer
53 views
Covariation of Wiener processes, $\langle W_1,W_2\rangle_t = \rho t$.
I'm wondering why this is true: $\langle W_1,W_2\rangle_t = \rho t$.
Where $W_1$ and $W_2$ are standard Brownian Motion.
I know that $\langle W_1,W_2\rangle_t = 0.5\big[ \langle W_1 + W_2 \rangle - ...
2
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1answer
234 views
Expectation of exponential martingale and indicator function.
Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$.
I want to evaluate
$$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
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1answer
65 views
Readings necessary to understand Ito Integrals?
I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
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0answers
126 views
Explaining Ito formula to an analyst
From the point of view of analysis, what is Ito formula?
Is it an integral by substitution, or, a radon-nikodym derivative?
Define the probability space
$$
\left(C\left(\Bbb ...
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1answer
30 views
Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$
Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true?
$$X(t)^a > K \iff X(t) > K^\frac1a$$
The context of the above is that I want to ...
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29 views
Prove that sinc function is desired interpolator when given condition holds.
The process $X_t : t \in \mathbb{R} $ is bandlimited with $S_X(\omega) = 0$ for $|{\omega}| > \omega_c$. Show that if:
$X_t = \sum_{n=-\infty}^{\infty} X_{nT}p(t-nT) $ $(m.s.)$
where ...
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41 views
Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation
Let $B$ be a standard Brownian motion, and,
$$
X_t=e^{\int_0^t f(B_s)ds},
$$
for some function $f$.
What are the condition on $f$ for $X_t$ to be of finite variation?
Let $Y_t=\int_0^t f(B_s)ds$, if ...
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0answers
126 views
Stochastic integral: Interchanging the order of expectation and integration
Let $B$ be a standard Brownian motion and
$$
X_t=\int_0^t f_s ds+\int_0^t g_s dB_s,
$$
where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$.
Is it true that
$$
...
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0answers
64 views
Local martingale iff each component is a local martingale?
This is probably an easy question:
A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
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1answer
44 views
Condition for existence of a stochastic differential equation
With $B$ a standard Brownian motion, write
$$
dX_t=f_tdt+g_tdB_t.
$$
What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists?
I think ...
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1answer
173 views
Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$
We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion.
However, is the following identity true? Also, why or why not?
$\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
2
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0answers
33 views
Finite p-th variation implies zero-valued q-th variation.
The Question: Let $X$ be a continuous process, and suppose $0 < p < q$.
Prove the case $V_t^p(X) < \infty \implies V_t^q(X) = 0$.
Definitions:
The standard setup.
$\Pi := ...
1
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1answer
92 views
Demonstrate that every martingale is a local martingale.
The Original Question: Demonstrate that every martingale is a local martingale.
Attempt at a Solution:
Consider the standard setup of this problem: $\mathscr{F}_t$ is the filtration that satisfies ...
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0answers
68 views
What kind of process is a locally bounded process?
Definition of locally bounded process is on http://planetmath.org/encyclopedia/LocallyBoundedProcess.html
On that website, it says any discrete-time predictable process is locally bounded. How can I ...
2
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2answers
82 views
Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$
Is there a way to solve
$$
\partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0?
$$
This appeared as a condition for
$$
X_t=u(t,B_t)e^{\int_0^tv(B_s)ds}
$$
to be a martingale.
With $B$ a ...
0
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1answer
299 views
Applying Ito formula to the Brownian bridge
Let $B$ be a standard Brownian motion and
$$
W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s
$$
be a Brownian bridge.
Calculate $dW_t$.
To apply Ito formula define
$$
f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s
$$
...
0
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1answer
54 views
another version of doob inequality
please consider the following problem:
Let $(M_t)_{t\geq 0}$ be a continuous and positive submartingale and $S_t=\sup_{0\leq s\leq t}M_s$. Please prove that for any $\lambda>0$ we have
$$\lambda ...
0
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1answer
73 views
How to show Wiener measure induces basic properties of Brownian motion?
page 19 of http://www.math.tifr.res.in/~publ/ln/tifr64.pdf gives a defintion of Wiener measure Ft1,t2,..,tk.
But how can we show it is a probability measure and it satisfies the consistency condition ...
