Questions tagged [stochastic-calculus]
Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.
935
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Stochastic Integration via Skorohod Representation
I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...
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0
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Tanaka stochastic differential equation and Kolmogorov equation
Given Tanaka sde
$$dX_t=[a{\rm sign}(X_t)+b]dW_t$$
is there associated a diffusion process and so a Kolmogorov (Fokker-Planck) equation? What is this equation?
References answering the question are ...
- 1,659
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1
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A class of Ito integrals
I am currently working on stochastic processes and I have met a stumbling block in the Ito integral
$$\int_{t_0}^tdt'G(t')[dW(t')]^\alpha$$
with $\alpha\in\mathbb{R}$ and $\alpha>0$. Textbooks ...
- 1,659
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1
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Finding a stochastic differential equation as limit of a discrete stochastic process
I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm.
Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and
$X_{n+...
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3
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A non-degenerate martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions.
$W_t$ is
a standard Brownian motion.
Let $Y_t$ be a martingale given by
$$...
- 1,369
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1
answer
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White noise in Lie group
The matter is in the title.
Is there a means to define the white noise process in Lie group. A basic definition
link text
Question:
Can we replace $\mathbb{R}^n$ by a Lie group?
In fact, I would ...
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How to deal with the vector norm item as a denominator in this expectation?
Hello, everyone.
I want to calculate the expectation shown in the following formula, where $X$ follows a standard $d$-dimensional multi-variable normal distribution as $X\sim\mathbb{N}(\mathbf{0},\...
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1
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X-harmonic and mean value property
We know in elliptic equation theory(or related area) that harmonic function has mean value property. Roughly speaking, harmonic function function at point x is equal to its average on the spherical ...
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0
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341
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Observing drift of a Levy process
It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy ...
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How to calculate this expectation with logarithm?
If $\mathbf{x}\sim\mathbf{N}(\mathbf{0},\mathbf{I})$, and assume that $A$ is a symmetric positive definite matrix, how can I calculate the following two expectations where there is a logarithm in it?
$...
- 607
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2
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How to calculate this expectation where the random variable is restricted on a sphere? [closed]
Hello! I have a question about how to calculate the expectation of a quadratic form as follows, where $X$ is a random variable that uniformly distributed on the unit sphere:
$$
E_X[(\mathbf{x}^\top A\...
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1
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2
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Getting $B_t$ from its local times $L^x_t$
Hi
Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?
Using occupation time formula this would mean solving for some $f$ the following ...
- 1,304
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1
answer
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Joint law of the time integral of Brownian motion and its maximum
Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\...
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1
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Measure changes for gamma process
GENERAL THEORY
In his book Ken-Iti Sato ("Lévy Processes and Infinitely Divisible Distributions") provides the theory for measure change for Lévy processes in Theorems 33.1 and 33.2.
It can ...
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2
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Generalized Ito's formula
Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X_t) = F(X_0) + \...
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2
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Stochastic Green-Gauss Theorem
Is there a stochastic analog for the Green-Gauss theorem? I'm looking for an expression that relates the flux (or statistical moments of the flux) through a random surface to the divergence of the ...
3
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1
answer
802
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Stochastic integrals as honest martingales — exponential damping
We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
- 667
3
votes
1
answer
537
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Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
$...
- 667
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1
answer
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Positive martingale representation with jumps
I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question:
Martingale representation theorem for Levy processes
My best guess is (subject to ...
- 667
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0
answers
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Levy jump measure vs. Levy measure vs. sum of jumps
This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...
- 667
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1
answer
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Reachability for Markov process, continuous time
Let $X$ be a strong Markov process in the continuous time with a state space $\mathbb R^n$. Consider a reachability problem for this process, i.e.
$$
v(x):=\mathsf P_x(X_t\in A\text{ for some }0\leq t&...
- 1,605
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1
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Martingale representation theorem for Levy processes
Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
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Fractional Brownian motion and Laplacian
Having read this link on math stackexchange, I would like to submit to your wisdom the following questions.
Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian ...
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1
answer
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Time-dependent Markov process: infinitesimal generator
If one talks about homogeneous Markov diffusion
$$
\mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t
$$
with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is ...
- 1,605
2
votes
3
answers
603
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Stochastic Integrals and Cauchy Variables
I hope there is a straighforward literature-pointer here.
If I were interested in $\sum_{t=1}^{n} f(t) X_{t}$, where $X_{t}$ consists of independent normal random variables, I could approximate the ...
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4
votes
2
answers
880
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law of iterated logrithm
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is
a standard Brownian motion. By law of iterated logarithm, one has
$...
- 1,369
6
votes
1
answer
4k
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What does progressively measurable actually entail?
There is a definition that has always left nagging questions in my mind. To set it up, let $(\Omega, \cal{A}, (\cal{F}_t)_{t\geq 0}, P)$ be a filtered probability space. From Comets & Meyre's ...
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Karhunen–Loève approximation of Brownian motion and diffusions
The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:
$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$
where $Z_n \sim \mathcal{...
- 1,646
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2
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2k
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Weierstrass' function and Brownian motion
Is there a known connection between Weierstrass' function
$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$
and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass ...
- 1,646
1
vote
2
answers
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Martingale part of the discontinuous put payoff
I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?
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0
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Inverse Skorokhod Embedding Problem
The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.
Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ (...
- 1,304
1
vote
1
answer
1k
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infimum of a set of stopping times
Let $(Y^a: a\in \Lambda)$ be a set of random processes given by
$$Y^a(s) = \int_0^s \sigma^a(r) dW(r)$$
where $W$ is Brownian motion w.r.t. filtered probability space
$(\Omega, \mathcal{F}, P, \...
- 1,369
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1
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Local continuous martingale
Hi,
This is a relatively simple result with a simple proof. However, there are 2 things I don't understand:
Why is M a brownian motion?
How is I calculated ("Thus, we get...")?
Any insight would be ...
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1
answer
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The only continuous martingales with stationary increments are Brownian motions
I know that the above statement is true, but I can't demonstrate it.
It's a pretty powerful theorem, here is its mathematical formulation:
Theorem: The only continuous martingales with stationary ...
- 31
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votes
1
answer
4k
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Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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votes
1
answer
956
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Exact simulation of SDE
Consider a one dimensional SDE of the form $dX_t = \mu(X_t) dt + \sigma dW_t$, where $\sigma>0$ is a constant. Under mild regularity assumptions on $\mu(\cdot)$, one can exactly simulate ...
- 2,133
4
votes
1
answer
553
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Time-integral of a smooth, vector-valued function of a planar Brownian bridge
I'm looking for information on how to compute the distribution of the random vector
$$Z = \int_0^t f(B_s) ds$$
where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in \...
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1
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642
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square root processes with correlated deriving Brownian motion
$$dX = \kappa_x (\theta_x - X)dt + \sigma_x \sqrt{X} \,dW_x$$
$$dY = \kappa_y (\theta_y - Y)dt + \sigma_y \sqrt{Y} \,dW_y$$
$$dW_x dW_y = \rho\, dt$$
we know that $X$ and $Y$ are marginally ...
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votes
1
answer
388
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Homogeneous linear stochastic DE with noncommuting coefficients
The system I am studying can be reduced to a Stratonovich vector
stochastic differential equation
$dX = A X \; dt + \sum B_k X \circ dW_k$
with $W_k$, $k=1..m$ the Brownian motion in $m$ dimensions, ...
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votes
2
answers
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Change of measure Markov process
We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$.
I would like to find the conditions ...
- 1,605
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votes
3
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470
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comparing diffusions
Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility ...
- 2,133
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votes
2
answers
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Has anyone found a way to determine the invariant measure of a one-dimensional jump-diffusion?
I am working with a jump-diffusion on the unit interval, with absorbing endpoints, and I was hoping someone has found a way to determine the invariant measure, similar to that of an Ito process.
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1
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Change of space-time in Walsh's stochastic integral
One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
http://www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf (Wayback Machine)
For $U,V\in \mathcal{B}(\mathbb{R}\...
2
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0
answers
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Stochastic Optimal Control - Maximizing convex terminal costs
The theory of stochastic optimal control deals with the following problem:
Find $\quad\sup\limits_{u} \; \mathrm E[g(X^{(u,x)}_T)]$
where $X^{(u,x)}_t$ solves the following controlled SDE:
$dX_t=\...
- 305
4
votes
1
answer
778
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Results for Hitting Times of (Not Stationary) Ito Processes
Let $W_t$ denote the Wiener process and let
$$
dX_t = a(t, X_t) dt + b(t, X_t) dW_t
$$
be an one dimensional Ito SDE (stochastic differential equation, see Ito stochastic calculus).
A hitting time $...
- 1,544
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4
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A simple decomposition for fractional Brownian motion with parameter $H<1/2$
Background
Let $X = \{X(t):t \geq 0\}$ be a (standard, real-valued) fractional Brownian motion (fBm) with parameter $H \in (0,1)$, i.e., a continuous centered Gaussian process with covariance ...
- 1,468
6
votes
1
answer
382
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When is it possible to construct a joint law from its two-dimensional marginals?
My question is much more specific than the title:
Given a symmetric
distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi_1,\xi_2,\dots$ of random variables such ...
- 1,503
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votes
3
answers
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Do convex and decreasing functions preserve the semimartingale property?
Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
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3
votes
2
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Expectation of time integral of Wiener process
I am trying to calculate $E(\int_0^T {W_s ds})$, where $W_s$ is a standard Brownian motion.
Now two approaches I can think of:
1) Take a partition of $[0,T]$. Calculate $E(\sum {W_{t_i}(t_{i+1} - ...
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votes
2
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2k
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Brownian Motion Winding Number
Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...
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