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.
376 questions with no upvoted or accepted answers
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Taking limits in stochastic partial differential initial value problems
Background: A (stochastic) Cauchy problem I am interested in looks like this:
$$
(1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
- 657
2
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0
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220
views
How to judge the solution process of an SDE to lie on the sphere?
Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...
- 261
2
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0
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91
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Conformal mappings and diffusion processes with boundary condition
I have a question on a relation between conformal mappings and diffusion processes with boundary condition.
Let $D_1$ be a smooth simply connected domain of $\mathbb{R}^2 \cong \mathbb{C}$. This may ...
- 721
2
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0
answers
184
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Explicit formula for Neumann heat kernel
It is well-known that
$u(x,y,t)=(4\pi t)^{-n/2}(e^{-|x-y|/4t}+e^{-|x-y'|/4t})$, $x,y\in \mathbb{R}^n_+=\{x\in \mathbb{R}^n|x_n\geq 0\}$, $y'=(y_1,\dots,y_{n-1},-y_n)$, is Neumann heat kernel of $\...
- 307
2
votes
1
answer
803
views
On Riemann integration of stochastic processes of order $p$
Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
user85801
2
votes
0
answers
74
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Floquet stochastic process
Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
- 205
2
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0
answers
119
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An Incorrect Construction of the Ito Integral
Let $B_t$ be a Brownian motion defined on the interval $[0,T]$, with underlying (filtered) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$. Call a function $f:[0,T]\times\Omega\...
- 115
2
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0
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591
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Stationary distribution of overdamped Langevin dynamics
Consider the over damped Langevin dynamics: $d X_{t} = d B_{t} - \nabla U(X_{t}) dt $ on $\mathbb{R}^{d}$ where $B_t$ is a standard Brownian motion. On pages 29 and 30 of the following book
Royer,...
- 165
2
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0
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140
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Is there a distinct Ito-Sasaki version of Riemannian stochastic development?
Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...
- 319
2
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0
answers
55
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Differentiability of a stochastic process depending on a spatial parameter
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\...
- 167
2
votes
0
answers
135
views
Connection between deterministic and stochastic problems in PDEs
In the study of conservation laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)):
Deterministic Cauchy problem:
$$(1) \hspace{1cm} \...
- 657
2
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0
answers
119
views
reference request “A measure-theoretic approach to Malliavin calculus”
Apart from citations all over the internet, the following paper appears to be off-the-grid.
K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
2
votes
0
answers
119
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Rate of convergence of generalized polynomial chaos
Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...
- 31
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203
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Is martingale solution equivalent to weak solution for SDE driven by stable process
Consider the following SDE
$$
d X_t=b(X_t)d t+d L_t,
$$
where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by
$$
L=\Delta^{\alpha/2}+b\cdot\nabla.
$$
Is the ...
- 411
2
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0
answers
169
views
How can we show that a $Q$-Wiener process on a Hilbert space $U$ takes values in $Q^{1/2}U$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
$U$ be an infinite-...
- 167
2
votes
0
answers
74
views
Convergence of empirical measure in case of proliferation
I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:
I don't understand why he calls $u(t,x)$ a time dependent ...
2
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0
answers
34
views
limiting behaviour of integrated sign of small increments of a Gaussian process
For a Gaussian process $(Y_t)_{t\geq 0}$ with Hölder continuous paths, define a new process $(X_t^\varepsilon)_{t \geq 0}$ via
$$X_t^\varepsilon = \int_0^t \operatorname{sign}(Y_{s+\varepsilon}-Y_s) \...
- 617
2
votes
0
answers
61
views
Assertion of Local Martingale
I am currently reading a proof of the Feynman-Kac representation theorem. The main step in the proof is to consider an "interpolation martingale" which has the form $$M_s := \varphi(t-s, x+B_s)\exp \...
- 21
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answers
220
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Convergence of stochastic integrals with respect to Poisson random measure
It is well-known from e.g. Kurtz-Protter 1996 that if $(H_n,X_n)$ is a sequence of semimartingales converging a.s. to $(H,X)$ in the Skorokhod topology, then (under some measurability conditions) we ...
- 931
2
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0
answers
385
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Ito lemma for manifold semimartingales
I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...
- 5,405
2
votes
0
answers
194
views
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 ...
- 325
2
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0
answers
299
views
constructing brownian motion adapted to a given brownian motion
Suppose we are given a Brownian motion $\{B_t,\mathcal F_t^B\}_{t\in[0,1]}$. I would like to know if it is possible to construct another Brownian motion $\{W_t,\mathcal F_t^W\}_{t\in[0,1]}$ from $B$, ...
- 325
2
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0
answers
107
views
Markov chain approximates a fractional diffusion
Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
- 323
2
votes
0
answers
226
views
Geometric ergodicity of dynamical system
I'm working with dynamical systems defined by ODEs and SDEs, in this latter case gradient systems in particular, a special case of Ito diffusions.
I've read that under reasonable assumptions this ...
- 205
2
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0
answers
74
views
Literature/Book on counting processes
I seek literature that makes a rigorous treatment of counting processes. In particular im interested in a precise treatment of the conditional intensity $\lambda_t$ which is often informally defined ...
- 315
2
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0
answers
193
views
Is the Itō integral $\int_0^TΦ_t\:{\rm d}B_t$ the mean-square limit of $\sum_{i=1}^nΦ_{t_{i-1}}(B_{t_i}-B_{t_{i-1}})$ as $\max_i(t_i-t_{i-1})\to 0$?
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a right-continuous filtration on $(\Omega,\mathcal A)$
$B$ be a Brownian motion on $(\Omega,...
- 167
2
votes
0
answers
221
views
Boundary behavior for Ito diffusions
The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
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0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
2
votes
0
answers
96
views
Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
- 21
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votes
0
answers
126
views
Construction of a random variable
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
- 721
2
votes
0
answers
84
views
limit multiple integral
I want to know if $\lim_{T-> \infty}$ of this integral
$$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\
\times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...
- 31
2
votes
0
answers
75
views
Holomorphic solution to SDE
Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$.
Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...
- 5,405
2
votes
0
answers
204
views
Onsager-Machlup function for special matrix-valued diffusion process
Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...
- 83
2
votes
0
answers
98
views
Non-existence for a sort of probability measures
We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...
- 257
2
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0
answers
227
views
Strong law of large number for semimartingale
I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion.
Thanks
- 31
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0
answers
448
views
integrability of Brownian motion stopped at some stopping time
Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...
- 1,835
2
votes
0
answers
288
views
The existence of stationary measures for certain Markov process
My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
- 121
2
votes
0
answers
261
views
Asymptotics of Variable Drift Ornstein–Uhlenbeck Process
The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...
- 3,626
2
votes
0
answers
341
views
Deriving HJB equation (why $\frac{dZ_t}{dt}=0$?)
I am trying to derive the HJB equation in a stochastic setting. Let
me exemplify my problem with the simplest case where there is no control,
just one state variable. Assume the payoff is given by
$$
...
- 315
2
votes
0
answers
340
views
Question about the characteristics of semimartingales
Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$....
- 1,835
2
votes
0
answers
71
views
Existence of 1-1 mapping/homeomorphism
Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants $0<\lambda&...
- 1,399
2
votes
0
answers
646
views
Fundamental theorem of calculus for iterated stochastic integrals
I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h \...
- 105
2
votes
0
answers
144
views
a generalization of Monge-Kantorovich Problem
I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on $\...
- 1,835
2
votes
0
answers
134
views
Supermartingale inequality on a particular event
Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
- 23
2
votes
0
answers
370
views
Cameron-Martin like RKHS
Hello,
I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product.
What is the RKHS ...
- 310
2
votes
0
answers
361
views
Computing a density function for the integral of a stochastic process, given its transition function
$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
- 693
2
votes
0
answers
113
views
Does this series stopping times marching forward?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider stochastic differential equation
$$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$
Note that, the above SDE has a strong non-negative ...
- 1,399
2
votes
0
answers
498
views
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},\...
- 607
2
votes
0
answers
291
views
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
2
votes
0
answers
313
views
Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk
Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
- 599