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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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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Why do people who work in stochastic analysis and geometry tend to work in sub Riemannian geometry?

There is a rich theory of diffusions on manifolds. Every time I see someone who studies diffusions on manifolds, it seems like they study the sub Riemannian setting. I get that this is more general ...
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Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} \widehat{g}(\xi)}{|\xi|^{d-2[\...
Abdelmalek Abdesselam's user avatar
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When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^...
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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; ...
Tom Alberts'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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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
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
0xbadf00d's user avatar
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Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
Sergio Almada's user avatar
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Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
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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 ...
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Uniform bound for the occupation time of a diffusion

Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$. Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions. Suppose the ...
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Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. ...
Ceeerson's user avatar
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Skorokhod' representation theorem: What is (are) possible filtration(s) on the common probability space?

I asked this question on math.stackexchange at https://math.stackexchange.com/questions/1941142/skorokhods-representation-theorem-what-is-the-filtration-on-the-common-probabi and haven't received ...
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Second order calculus and rough paths

In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form $$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$ where $X$ is a semimartingale on a manifold $M$...
Matthias Ludewig's user avatar
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Stochastic calculus for several inputs

In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic ...
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Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link. G–Brownian motion has a very rich and interesting ...
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Integrating a Bessel Bridge

Preliminaries An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
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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 $\...
kenneth's user avatar
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stochastic control / geometric mean

Consider the following problem: Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
Bernard 's user avatar
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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 ...
Grzenio's user avatar
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A notion of SDE via the martingale representation theorem

$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
Emily's user avatar
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Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define $$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
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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 ...
Fei Cao's user avatar
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Log Sobolev inequality for Wiener space

I am reading https://arxiv.org/pdf/1003.1649.pdf and saw equation 10.2.3 that said that on Wiener space $$E\left[f^2\log\left[\frac{f^2}{E[f^2]}\right]\right]\leq 2 E[|\nabla f|_H^2],$$ where $\nabla$ ...
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Occupation time of SDE

Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation $$ X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
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Pedestrian proof of Gaussian chaos for order-two polynomial?

Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
combNightmare's user avatar
4 votes
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327 views

Martingale polynomial functions

If $B_t$ is a Brownian motion then using Hermite polynomials one can find that $$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$ are martingales. If $X_t$ is a diffusion $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
Kate 's user avatar
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Exit time of a stochastic process defined by a SDE

Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation \begin{align*} \...
nabla's user avatar
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Compactness of semigroups of one-dimensional diffusions

I have a question about semigroups of one-dimensional diffusions. Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as $$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$ It is ...
sharpe's user avatar
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Definition of the Stratonovich integral in Hilbert spaces

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a (standard, real-...
0xbadf00d's user avatar
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Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
user94063's user avatar
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73 views

Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...
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Expectation of running maximum of diffusion processes

Let $X$ be a one-dimensional Ito diffusion $$X_t=x+ \int_0^t b(X_s)ds + \int_0^t \sigma(X_s)dW_s,$$ where $b,\sigma$ satisfy the usual Lipschitz continuity and linear growth conditions. Define the ...
epsilon's user avatar
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324 views

Lyapunov function of exponential growth for existence of a solution of an SDE

Let $$dX_t = a(X_t) dt + b(X_t) dW_t$$ be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...
Fisher's user avatar
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474 views

Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem $\mathbf{Theorem}$ To each ...
user35237's user avatar
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Tail for the integral of a diffusion process

I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x), $$ and $X$ is a diffusion ...
Mawaki's user avatar
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Dynamic programming principle (DPP)

In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
kenneth's user avatar
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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-\|...
0xbadf00d's user avatar
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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 ...
user479223's user avatar
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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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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
3 votes
0 answers
94 views

Dealing with noise that is white in time, colored in space numerically

I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
Yhtomit's user avatar
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Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?

In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
Martin Geller's user avatar
3 votes
0 answers
90 views

Martingale problem for regime switching SDE

Let $\mu$ be a continuous time Markov process switching between two possible values $a, b \in \mathbb R$ with time homogeneous transition rates. Consider the one dimensional SDE $$dX_t = \mu \, dt + ...
Nate River's user avatar
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3 votes
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A path with zero increments and positive area

I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
Martin Geller's user avatar
3 votes
0 answers
119 views

Joint law of two stochastic integrals with respect to the same Brownian motion

Let $a:\mathbb R_+\to [1,2]$ be "smooth". Given a standard Brownian motion $W$, define for $t\ge 0$ $$X_t:=\int_0^t\frac{1}{a(s)}dW_s \quad \mbox{and}\quad Y_t:=\sup_{0\le u\le t} \int_0^u a(...
GJC20's user avatar
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Density of invariant measure of stochastic differential equation

I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
Oleg's user avatar
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3 votes
1 answer
167 views

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 ...
yassine yassine's user avatar
3 votes
0 answers
232 views

Probability of a particle surviving forever

Consider a particle whose position is driven by the following equation: $$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$ where $y>0$, $0<C<1$...
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