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.

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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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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-...
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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 ...
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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 ...
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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(\...
mathex's user avatar
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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 ...
mathex's user avatar
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2 votes
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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 ...
Akira's user avatar
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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 $n$-...
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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-\|...
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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 ...
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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 \...
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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 ...
optimal_transport_fan's user avatar
3 votes
1 answer
181 views

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 ...
Jean Daviau's user avatar
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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 $\sup_{t\in [0,\infty)}B_t-t$?
Tiancheng's user avatar
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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?
ABIM's user avatar
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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(�...
epsilon's user avatar
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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 ...
matilda's user avatar
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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 ...
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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 ...
SRB121's user avatar
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0 answers
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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 ...
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1 vote
0 answers
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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 ...
17miles's user avatar
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4 votes
1 answer
179 views

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 $d$-...
Akira's user avatar
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3 votes
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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 ...
Andrea Aveni's user avatar
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1 answer
352 views

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 ...
Nate River's user avatar
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7 votes
2 answers
436 views

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}\...
Peter's user avatar
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5 votes
1 answer
382 views

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 ...
Greyearl's user avatar
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 ...
Nate River's user avatar
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7 votes
2 answers
1k views

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 ...
Nate River's user avatar
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5 votes
0 answers
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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 ...
djr's user avatar
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1 answer
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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 ...
happy hello's user avatar
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 ...
Nate River's user avatar
  • 4,802
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 $\...
Syd Amerikaner's user avatar
3 votes
1 answer
144 views

Stochastic representation of Laplace equation with Neumann boundary condition

Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$. What if ...
user479223's user avatar
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4 votes
1 answer
100 views

Reflecting Brownian motion in disk

What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it? The transition density ...
Focus's user avatar
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2 votes
0 answers
54 views

Continuous-time Wold decomposition

I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line. I am aware of the classic result in the book from Rozanov, which ...
arknas's user avatar
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2 votes
0 answers
134 views

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\...
0xbadf00d's user avatar
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0 votes
0 answers
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Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$

The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by $$ \left\langle \frac{dB_H}{dt}, f \right\...
tsnao's user avatar
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1 vote
0 answers
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Heat kernel and estimates

In the article by Hairer-Labbe (A simple construction of the continuum parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
mathex's user avatar
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0 votes
1 answer
124 views

Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time

Consider the following stochastic integral: $$ X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s. $$ Is $X_t$ almost-surely non-negative? Using this answer, it seems that $$ X_t = \max( W_t, 0) - \...
oswinso's user avatar
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2 votes
1 answer
139 views

Estimates on perturbation of drift of SDEs

Let $\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ be Lipschitz functions, of at-most linear growth; i.e. $\|\sigma(x)\|\lesssim \|x\|,\|...
Math_Newbie's user avatar
1 vote
1 answer
225 views

Variance of Itô integral on general time interval

I have a question concerning the variance of the Itô integral on general time intervals, i.e. I want to calculate \begin{align*}\operatorname{Var}\left(\int_s^T f(t)dW_t\right),\end{align*} where $f: [...
LoyoL's user avatar
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-5 votes
1 answer
243 views

Calculus based on pdf [closed]

Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
Marius S.L.'s user avatar
2 votes
1 answer
216 views

Explicit solution to linear SDE with correlated Brownian motions

Let $W$ and $B$ be correlated one dimensional Brownian motions with constant correlation coefficient $r \in (-1, 1)$, that is, we have $d\langle W, B \rangle_t = r \, dt.$ We assume we have $B_0 = v$ ...
Nate River's user avatar
  • 4,802
1 vote
1 answer
220 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
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4 votes
1 answer
350 views

When are the transition densities of an SDE symmetric?

We fix $T>0$. Let $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $\sigma:[0, T] \times \mathbb{R}^d \rightarrow \mathcal{M}^\text{sym}_{d \times d}(\mathbb{R})$ be measurable and ...
Akira's user avatar
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1 vote
0 answers
189 views

Stochastic volatility model question

Let suppose that $S_t$ is a process defined as: $$ \begin{cases}dS_t = \mu S_t\,dt+m(v_t)\,dW^1_t\\ dv_t = \mu_v(v_t)\,dt + \sigma_v(v_t)\,dW^2_t\end{cases}$$ where the two Brownian motions have ...
NancyBoy's user avatar
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6 votes
0 answers
226 views

Global well posedness of $\phi^4_1$

We consider the $\phi^4_1$ model: $\partial_t\phi=\Delta\phi-\phi^3+\xi$ on $[0,T] \times \mathbb{R},$ where $\xi$ is a space time white noise. I know how to solve this equation locally on the torus, ...
mathex's user avatar
  • 387
4 votes
1 answer
231 views

Reference request: showing that solution of an Ito SDE stays bounded with positive probability

Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t \label{1}\tag{1},$$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \...
Fei Cao's user avatar
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1 vote
0 answers
93 views

Freidlin Wentzell for stochastic differential inclusions

Consider the SDI $$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$ Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
user479223's user avatar
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2 votes
0 answers
59 views

On the measurability of stochastic integrals

Let $S(t)$ be a $C_0$-contraction semi-group, $W$ is a cylindrical Wiener process in a separate Hilbert space $U$. Assume the following conditions: $$ \|F(t,u_1)-F(t,u_2)\|_{H}< C\|u_1-u_2\|_{H},~~...
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