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.
932
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3
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Weighted sum of standard Brownian bridges
Let $\{B_j\}_{j=1}^k$ be a sequence of Brownian bridges.
Let us consider $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (distribution or may ...
7
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0
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Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
1
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0
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Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?
Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
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0
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Derivative with respect to initial condition for the solution of an SDE
Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...
3
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0
answers
88
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Explicit example of drift $F$ so that the law of $F+B$ is not absolutely continuous with respect to $B$
Let $\mu_0$ be the law of Brownian motion on the space of continuous functions. If $\mu\sim\mu_0$ agrees on null sets then there is some progressively measurable $F\in W_0^{1,2}$ a.s. so that $\mu$ is ...
1
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0
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92
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$
I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
2
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0
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134
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Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
1
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1
answer
140
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Does the convergence of drifted Brownian motion imply the convergence of expectation?
Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\...
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Accuracy of the definition of the space-time white noise
We have seen that there exists $\overline{\xi}:\Omega \to \mathcal{E}'$ ($\mathcal{E}'$ is the space of tempered distributions) which usually replaces the space-time white noise (indexed by $L^2(\...
0
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0
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Martingale defined by an integral
Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
6
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Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
3
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Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
2
votes
1
answer
188
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Decay estimate of moment of an SDE
We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
3
votes
1
answer
181
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Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process
Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
2
votes
1
answer
349
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A mean field SDE with hitting time
Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.
$$\int_{(0,\infty)}\mu_0(dx)=1,$$
consider the mean field SDE :
$$dX_t = \mathbf{1}_{\...
3
votes
1
answer
381
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On stochastic integration
This questions has been asked on math.stackexchange
I have two questions on stochastic integration.
(1) Constructing the Ito integral, there is the following remark in Jacod/Shiryaev (page 46, 2nd ...
5
votes
1
answer
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Is this process strictly positive?
Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and $\...
4
votes
1
answer
553
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Random time change from Oksendal's SDE textbook
I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in ...
0
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0
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Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
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1
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394
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How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
1
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1
answer
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For fixed $f \in L^2$ and $T>0$, choose $g$ so that $ \mathbb{E}^x[g(T-\tau)\chi_{X_\tau=1}]=-\mathbb{E}^x[f(X_T)\chi_{\tau \ge T}]$
Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that
\begin{align*}
0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\...
2
votes
1
answer
500
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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,...
1
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1
answer
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Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?
Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
1
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1
answer
91
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Interchange the deterministic and stochastic integrals
We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
2
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0
answers
66
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Assumptions for uniform measure of SDE on manifolds
Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
0
votes
2
answers
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Find the distribution of maximum of $B_t-t$
Let $B_t$ be a standard Brownian motion. It is easy to show that $\sup B_t-t<\infty$ a.s. . The question is, can we determinate the distribution of $\sup_{t\in [0,\infty)}B_t-t$?
1
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1
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Autocovariance of time integrated Ornstein–Uhlenbeck process
$\newcommand{\Cov}{\operatorname{Cov}}\newcommand{\Var}{\operatorname{Var}}$if $X(t)$ is the Ornstein–Uhlenbeck process and $Y(t)$ the time integrated OU process I am trying to calculate the ...
9
votes
1
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Quadratic variation and predictable quadratic variation for martingales
Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...
2
votes
0
answers
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How to estimate the difference between two Ito diffusions?
Suppose $𝑏:\mathbb R^d \to \mathbb R^d, \sigma:\mathbb R^d \to \mathbb R^{d\times d}$ are measurable functions and satisfy
\begin{equation*} 2\langle 𝑥−𝑦,𝑏(𝑥)−𝑏(𝑦)\rangle +\|\sigma(𝑥)−\sigma(�...
1
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0
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On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$
Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
2
votes
0
answers
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Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
3
votes
0
answers
170
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Flow property for semimartingale driven SDE at a stopping time
Let $S$ be an $n$-dimensional semimartingale such that the SDE
$$dX_t = \sigma(X_t, t) \, dS_t$$
with $\sigma$ Lipschitz continuous admits a globally defined unique strong solution on $[0, T]$.
For $t ...
2
votes
0
answers
53
views
Autocovariance of harmonic oscillator in fluid (Langevin Equation)
I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
1
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0
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Piecewise Ornstein-Uhlenbeck process time integral
Let $X_t$ be a piecewise Ornstein-Uhlenbeck process with infinitesimal variance $\sigma^2$ and (piecewise) infinitesimal mean $\theta_1$ for $x<c$ where $c$ is a constant and $\theta_2$ for $x\geq ...
3
votes
1
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Constants in Meyer inequalities
Let us first recall Meyer inequality in the Malliavin calculus framework
$$\|\delta(u)\|_{L^p}\leq C_p\|u\|_{\mathbb{D}^{1,p}},\qquad \forall u\in \mathbb{D}^{1,p},$$
where $\delta$ is the skorohod ...
2
votes
1
answer
479
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Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
5
votes
1
answer
545
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Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
4
votes
1
answer
179
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Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sqrt 2 \, dW_t$$
where
$b$ is locally Lipschitz such that $|b(x)| \le C |x|^2$ for $x \in \mathbb R^d$.
$W$ is a standard $d$-...
3
votes
0
answers
70
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Monotone Characteristic Function
Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$.
What other properties must $X$ have in ...
6
votes
1
answer
352
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Is a martingale conditioned to be large a submartingale?
Let $X$ be a continuous time martingale such that $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Let $x \in \mathbb R$ be such that $\mathbb P(X_\infty \geq x) > 0$, and define the ...
5
votes
1
answer
382
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On the convergence of a martingale
Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by :
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$
and for $t\geq 0$, we ...
7
votes
2
answers
436
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Interpretation of second order term in Fokker-Planck equation
Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by
$$
\begin{align*}
\nabla^2\cdot G(x)
&=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...
0
votes
2
answers
291
views
Predictability of countably valued accessible stopping times on complete and cadlag filtrations
The following question is motivated by this part of the proof of Lemma 2 on page 107 of the book Stochastic integration and differential equations of Philip Protter.
Lemma 2. Let $T$ be a totally ...
23
votes
5
answers
3k
views
What phenomena are better modelled by SDE instead of ODE?
Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
1
vote
1
answer
708
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 $...
7
votes
2
answers
1k
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A curious martingale
Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely?
Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
5
votes
0
answers
76
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General questions on stochastic calc on manifolds
I've done coursework in differential geometry and stochastic calculus, but I haven't formally seen any connections between the two. I have read that both information geometry and stochastic ...
0
votes
1
answer
124
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Polar form of 2D Brownian motion
Consider two independent unidimensional Brownian motion $w_1$ and $w_2$.
What is the polar form of $(w_1,w_2)$?
If $r(t)$ and $\phi(t)$ satisfy $(w_1,w_2) = r(t)(\cos(\phi(t)),\sin(\phi(t)))$, how to ...
1
vote
1
answer
192
views
A representation formula for the expected value of a stochastic process at a random time
Let $X$ be a continuous stochastic process, and $\tau$ an almost surely positive random variable, not necessarily a stopping time with respect to the natural filtration $\mathcal F_t$ of $X$.
We write ...
3
votes
2
answers
237
views
Definition of weak conditional convergence of random variables
I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...