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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Optional stopping theorem: different versions
$(X_k)_{k \in \mathbb{N}}$ is a sub-martingale, $R_1,R_2$ two stopping times such that $R_1 \leq R_2$.
The optional stopping theorem which I know is the following (from probability theory, ...
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Convergence of the probabilities that drifted Brownian motion with jump never hits zero (continuation)
This question can be seen as a continuation of my question at Convergence of the probabilities that drifted Brownian motion with jump never hits zero
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion ...
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Convergence of the probabilities that drifted Brownian motion with jump never hits zero
Let $X_t=2+t+W_t$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion. For every $n\ge 1$, set $X^n_t:=X_t-{\bf 1}_{t\ge n}$. Denote respectively
$$\tau:=\inf\{t\ge 0:~ X_t\le 0\}\quad \...
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Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$,
$$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
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Gronwall type lemma for an Ito process
For all $t\in \mathbb{R}$ let $h_t = \frac{1}{2} + \int_0^t v_s\cdot dB_s$ be an Itô process, where $B_s$ is a standard Brownian of $\mathbb{R}^d$ and $v_t$ an $\mathbb{R}^d$ valued adapted process, ...
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Where does the "mixing" occur in convex combination of Girsanov measures?
In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
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L2-closure of absolutely continuous stochastic processes?
Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
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Decomposition of reversed processes
Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...
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Almost supermartingale and a.s convergence
After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
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Bounding Brownian motion and an Ito process simultaneously
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
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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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How to find the "natural scale function" for more general stochastic processes?
In stochastic analysis, for an Ito diffusion $X_t$ such that $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$, we can exlpicitly compute a "natural scale function"
$$S(x)=\int^x\exp\left(-\int^y\frac{2\mu(...
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Schwartz regularity for the density of a stochastic process
Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$
It ...
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Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
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If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?
Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
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Laplace Equation for Brownian Motion [closed]
So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...
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On a property of resolvents associated with holomorphic semigroups
This question is about semigroup theory.
Let $E$ be a locally compact metric space, and $X=(X_t,t\ge 0;\,P_x,x\in E)$ be a Markov process on $E$. We assume that $X$ is symmetric with respect to $m$, ...
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Normalization of exponential in the context of Feynman integrals from a White noise perspective
I apologize in advance if this question is not suitable for MO (please let me know), but the fact is that since I am not familiar with the theory of Feynman integrals I don't know whether this is a ...
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For stopping times $\tau_k,\mathcal{F}_{\sup_{k \in \mathbb{N}^*}\tau_k}=\sigma(\bigcup_{k \in \mathbb{N}^*}\mathcal{F}_{\tau_k})$?
$(\tau_k)_{k \in \mathbb{N}^*}$ is a sequence of stopping times (taking values in $\overline{\mathbb{N}}$) for the filtration $(\mathcal{F}_n)_{n \in \mathbb{N}^*}.$ Let $\tau=\sup_{k \in \mathbb{N}^*}...
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English version on Dynkin's 1963 paper on stopping
I am looking for an English version of the following paper:
Е. Б. Дынкин, Оптимальный выбор момента остановки марковского процесса, Dokl. Akad. Nauk SSSR 150, 238-240 (1963).
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Differentiable approximation of Brownian diffusion with unbounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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1
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135
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Differentiable approximation of Brownian diffusion with bounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
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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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Is the topology generated by the convergence of finite-dimensional distributions metrizable?
Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
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Expectation of first exit time of a bounded set by a time-homogeneous Ito diffusion is finite
This is a question concerning Remark(i) under Theorem 7.4.1(Dynkin's formula) on Page 124, $\textit{SDE}$, by Oksendal.
It says that if $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$ is an $n$-dimensional time-...
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Is my quadratic variation derivative bounded?
Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
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Zeros of a non-degenerate bivariate Gaussian Process
Suppose $(X(t),Y(t))$ $t\in[0,1]$ is a bivariate Gaussian process. We can assume that each component is continuously differentiable, but not necessarily stationary, and that the covariance kernels of $...
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130
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Translation of Dellacherie's Capacités et Processus Stochastiques
I have been studying the Strasbourg school's general theory of processes from Dellacherie and Meyer's Probabilities and Potential, and I really like it. I have heard very good reviews about another ...
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Convolution of Wiener measure and measure on $W_0^{1,2}$
Let $F$ by a process adapted to the filtration of a standard Brownian motion. Suppose that the Doleans Dade martingale exists and is a martingale. $F$ is a measure on $W_0^{1,2}$, call it $\nu$. Let $\...
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242
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Associativity rule for integration against fractional Brownian motion
In Itô calculus, it is easy to construct an associativity rule. Namely, if $B_t$ is a Brownian motion and $M_t = \int_0^t X_s dB_s$ for suitable $X_t$, then we have the following associativity rule: $...
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390
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Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure
I'm trying to figure out the connections between two contructions of Gaussian measure.
Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
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406
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Forwards Feynman–Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
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Continuity of the random variable defining the occupation measure of a continuous Gaussian process
Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable
$$
X_\alpha = \lambda( \{t \; : \; Z(t) &...
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Local inverse bound of Cameron Martin and Banach norms
Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
2
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180
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Onsager--Machlup functional as the density across a mesh of discrete points
It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint ...
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Derivative of the function of random variable
Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...
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Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?
Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
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When is the dual infinitesimal generator of a S.D.E self-adjoint and negative definite?
Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be,
self-adjoint i.e $\int f ({\...
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1
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615
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Mean square derivatives and modifications
Suppose we have a stochastic process $X$ on $\mathbb{R}$. Suppose there exists a stochastic process $\frac{d X(t)}{d t}$ such that
$$\lim_{h\to0} \mathbb{E}\left[\left(\frac{X(t+h)-X(t)}{h}-\frac{d X(...
3
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2
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775
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Regarding sample continuity of Gaussian Processes
Suppose we have a Gaussian Process $X_t$ on $\mathbb{R}^n$ with mean function $m(t)$ and covariance function $K(t,s)$. Then is $X_t$ being sample continuous (i.e. the sample paths of $X_t$ are almost ...
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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\...
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70
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Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
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0
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586
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Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
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1
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407
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Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
1
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1
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243
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Time-Reversal of BSDE = SDE
Let $(Y,Z)$ be a solution the the BSDE on a stochastic base $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$:
$$
Y_t = \int_t^T f(s,Y_s,Z_s)ds + Z_t dW_t \qquad Y_T = \xi \in \mathcal{F}_T^W;
$$
...
5
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Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
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1
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461
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The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
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1
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127
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About deriving the Fokker-Plank-Smoluchowski equation of a (homogeneous) S.D.E
We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t ...
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1
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322
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Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
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2
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294
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Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...