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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A comparison principle for SDE

Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE $$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$ $$dY_t = \mu_Y (t, \...
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Calculate asymptotic value of an integral of exponential function

In the existing literature [From the Schrödinger problem to the Monge–Kantorovich problem], below the Example 3.4, the author claimed that using Laplace method, we can obtain $$ \int_0^{\infty}e^{-z^p/...
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Solution of SDE with time power law singular diffusion

I was wondering if anything could be said at all about the well-psedness of the following time-inhomogeneous singular diffusion SDE: \begin{align}d X_t&=\sigma(X_t,t ) d W_t , \qquad t\geq 0, ...
Mr_3_7's user avatar
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How is the definition of the stochastic integral w.r.t. an Ito process consistent?

I have the following definition for an Ito process: For $a(\omega, t), b(\omega, t)$ real valued, adapted stochastic processes that respectively satisfy the conditions $$ P(\int_0^t \vert a(\omega, s)...
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The relationship between measurability and weak measurability

For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple functions, measurability (the ...
Guomin Liu's user avatar
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217 views

Solution to SDE conditional on high maxima of driving Brownian motion

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE $$dX_t = X_t \, dW_t \;, \quad X_0 = 1 \;.$$ For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
Nate River's user avatar
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Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral

Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
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The stochastic parallel transport as a limit of piecewise geodesic parallel transports

Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
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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{...
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Characterization of Brownian motion: processes with right-continuous paths

I am looking for a reference with a proof for the following fact: If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
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A quadratic optimization problem involving Brownian motion

I wonder how to solve the following quadratic optimization problem? $$\min_{f:[0,1]\to \mathbb{R}} \int_0^1 f(t)^2dt-2\int_0^1 f(t)dB_t$$ where $B(t)$ is standard Brownian motion. Intuitively, the ...
Zuofeng Shang's user avatar
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Distribution of local martingale is absolutly continuous to that of the Brownian motion?

Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
null's user avatar
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For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?

Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields $$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
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Construction of a Markov process with prescribed local behavior and state-dependent jump distribution

Let $(E,\mathcal E)$ be a measurable space $\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
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Uniqueness of the solution to some degenerate SDE

Consider the one-dimensional stochastic differential equation: $$dX_t = {\bf 1}_{\{X_t>0\}}\big(b(t,X_t)dt + a(t,X_t)dW_t\big),\quad \forall t>0,$$ or equivalently $$dX_t = b(t,X_t)dt + a(t,X_t)...
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Time derivative of relative entropy in this setting

I was reading the following article : https://arxiv.org/pdf/2005.13097.pdf and a question came up. In page 30 in the proof of Lemma 16, when taking the time derivative of the KL divergence, there is ...
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Lyapunov function utility in stochastic optimal control

The article Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching by (X.Mu, Q.Zhang) studies necessary and sufficient conditions on near-optimal controls. In both ...
Hamdiken's user avatar
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What is famous mistake made by Feller?

I heard "Feller made a famous mistake in 1954 and fixed by A.D. Wentell in 1959" from one lecture. There is no further explain what is that mistake? Is there someone know it? Is it ...
Fractional analysics's user avatar
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When is the mode of a stochastic process a better statistic than the mean?

This is a soft question. I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces. ...
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Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...
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Probability that a geometric Brownian motion with additional determinstic drift ever hits zero

Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE $$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$ where $\mu, \sigma, C, k > 0$ are constants, ...
Nate River's user avatar
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Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
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How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?

Let $E$ be a $\mathbb R$-Banach space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$-...
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Another large noise limit

Note: Here all processes take values in $[0, 1]$. Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant. Let $X$ be the solution to the SDE $$dX_t = \sigma X_t \, dW_t$$...
Nate River's user avatar
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Continuation : Does the density of a stopped drifted Brownian motion vanish at zero?

Let $$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$ where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and ...
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What is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ ...
Fractional analysics's user avatar
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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{...
Lochend's user avatar
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Where does the extra term in the density of a diffusion with respect to $c B(t)$ come from?

It is well known that for the diffusions \begin{align*} dX&=f(X)dt+&cdB\\ dY&=&cdB \end{align*} the density of the law of $X$ with respect to the law of $Y$ is \begin{align*} \frac{d\...
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Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$

Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse. It is evident that $$ \mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p, $$ where $X\sim N(0,1)$. Is ...
Fancier of Mathematica's user avatar
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Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality?

I have a question on Lawler – Notes on the Bessel process, on page 4. Let $X_t$ be one-dimensional Brownian motion, and we want to use $N_t$ as a measure-changing (local) martingale, defined as $$N_t=\...
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Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?

I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it. Before I make a very ...
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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 ...
Nate River's user avatar
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On the growth of sample paths of Gaussian random fields

Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random ...
S.Z.'s user avatar
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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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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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Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$

Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
dohmatob's user avatar
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What is the Cameron-Martin norm associated to $X(t)=\int_0^t B(s) ds+B(t)$?

The process $X(t)=\int_0^t B(s) ds+B(t)$ is a centered continuous Gaussian process. Therefore it defines a Gaussian measure on $C[0,T]$. Therefore there is a Cameron-Martin space with Cameron-Martin ...
user341290's user avatar
2 votes
1 answer
299 views

A large noise limit

Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion. Denote $Y := \int_0^1 f(t) \, dW_t$. For each $\varepsilon > 0$, consider the conditioned random ...
Nate River's user avatar
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On the Lipschitz constant of $\Gamma$

Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
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Choice of stochastic integral picking the forward point in Riemann sum approximation and reversibility?

Consider the standard Riemann sum approximation of a stochastic integral (w.r.t Brownian motion for example) which is given by \begin{align} \int_0^t \sigma(X_s) \circ_{\lambda}dB_s \approx \sum_{i=1}^...
almosteverywhere's user avatar
3 votes
1 answer
239 views

Intersection of Brownian motion and finite variation process

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration. Question: Denoting by $\mathcal L$ the ...
Nate River's user avatar
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What are the optimal times to sample a process?

Let $X$ be a one dimensional Ito diffusion given by $$X_t = b \,W_t$$ where $b$ is a constant, and $W$ is a standard Brownian motion. Let $B$ be another Brownian motion independent of $W$, and define ...
Nate River's user avatar
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Preservation of variance for log-normal variables under change of measure

Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
Jan Stuller's user avatar
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1 answer
189 views

Filtering the period and amplitude of a sine wave corrupted by noise

Let $W$ be a standard Brownian motion and $\mathcal F_t$ its natural filtration. Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F_t$. Let $Y_t$ be the process $$Y_t :=...
Nate River's user avatar
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Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
defenestrator's user avatar
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193 views

Large deviation for empirical median

I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov ...
rime's user avatar
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When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
ABIM's user avatar
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2 votes
1 answer
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Does higher volatility of SDE imply lower probability of staying positive?

Given two SDEs $X^1$, $X^2$ : $$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$ where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
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The different quotient of the first exit time with respect to the initial state

Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that $$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \...
mnmn1993's user avatar
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3 answers
292 views

Is $p_t(y,x)$ the kernel of the time reversal of the diffusion $X$, for $p_t(x,y)$ the kernel of $X$?

Short version. If $X$ is a diffusion with generator $L$ and the Lebesgue measure is invariant for $X$, then $L^*$ has no term of order zero and it corresponds to another diffusion $X^*$. Denoting by $...
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