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Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Ito's formula for Ito diffusions

An Itô diffusion is a process $X_t$ such that $dX_t = U_t dB_t + V_t dt$, where $B_t$ is standard Brownian motion, $U$ and $V$ are bounded, adapted processes. Itô's lemma states that if $f \in C^2(\...
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Quadratic Variation and Covariation of Poisson Processes

I need to campute the quadratic variation for a compound poisson process defined as $X_{t}=\sum_{i=1}^{N_{t}}{Y_{i}}$. I also know that the compensated poisson process can be expressed as $X_{t}-\...
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linearity of stochastic differential equations

If $dX(t) = a_1(t)dt + b_1(t)dW(t)$ $dY(t) = a_2(t)dt + b_2(t)dW(t)$ Can we say that $d(X(t)+Y(t)) = (a_1(t)+a_2(t))dt + (b_1(t)+b_2(t))dW(t)$ ?
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Supremum of a general Gaussian Process

I have a stochastic integral of the form \begin{align*} X(t) = \int_0^t h(v) W(v) dv \end{align*} where $W(v)$ is the standard Brownian motion and $h(v)$ is a positive, integrable function. While $X(t)...
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Two different answers when integrating with respect to Brownian motion

Consider the integral with respect to Brownian motion $$\int_{0}^{t}s \ dB_{s} \ . $$ A textbook I am reading uses integration by parts to rewrite the above integral as $$tB_{t}-\int_{0}^{t}B_{s} \ ...
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Adding a linear drift to a diffusion

Consider a multidimensional diffusion in $\mathbb{R}^d$ whose generator is given by: $$Gf(x)=\langle b(x), Df(x)\rangle+\frac{1}{2}\operatorname{Trace}(Q(x)D^2f(x)),$$ where $b(x)\in\mathbb{R}^d$ and ...
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Ito integral of $\int_0^t \exp(B_s^2)\,dB_s$ [on hold]

I am self studying Itô calculus. The question is using Ito formula, find $\int_0^t \exp(B_s^2)\,dB_s$. I need a hint on what $g(t,x)$ I should choose.
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Estimation for $\log$ expression with two variables.

Consider arbitrary $\alpha\in[0,1)$, $p\in(\frac{1}{2},1)$. Does following estimation $$\log\Bigg(\bigg(\frac{1+\alpha}{p}\bigg)^p\bigg(\frac{1-\alpha}{1-p}\bigg)^{(1-p)}\Bigg)\le \log(2)$$ hold? Is ...
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Why all stochastic integrals are taken with respect to Brownian motion?

I feel like we could integrate with respect to any process with continuous paths. Can we expand Ito calculus to other processes?
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Stratonovich Differential: $Y_t\circ dB_t( dB_t) = Y_t\circ dt = Y_t\, dt$?

I have the following where $B_t$ is standard Brownian Motion $$Y_t\circ dB_t( dB_t)$$ I assume I can convert it to $$Y_t\circ dB_t( dB_t) = Y_t\circ dt$$ Then I further assume I can do this: $$...
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Hadamard Derivative of time transformed stochastic process

Given a continuous time stochastic process $X(t)$, we can define the functional transformation, $$f(X)(t) = (X(t))^2 - 2X(t)$$ and evaluate the Hadamard derivative. Given a transformation on the real ...
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Are we able to determine the distribution of a general stochastic integral?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $M$ be a real-valued continuous local $\mathcal F$-martingale on $(\...
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Difference between weak ( or martingale ) and strong solutions to SDEs

Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$ ...
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Calculating $\mathbb{E}(|B_t|^{-2})$ for 3-dim. Brownian motion

Ok so I'm given a standard 3-dimensional Brownian motion $B(t) = (B_{1}(t),B_{2}(t),B_{3}(t))$, function $f(x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}$ and the process $A(t) = f(B(t))$ and $t \in [1;\infty)...
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$E[\frac{1}{X}]$ for $X\sim\Gamma(n,\theta)$

Consider iid random varibales $(X_i)_{1\le i\le n}$ with $X_i\sim Exp(\theta)$ for $1\le i\le n$ and $\theta\in(0,\infty)$. Then we have $$\sum_{i=1}^nX_i\sim \Gamma(n,\theta)$$ with density function ...
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What does a 'pathwise' solution mean

I have not done very much stochastic calculus but know the basics. (Ito formula, Ito lemma, Stochastic Integral construction, the properties of B.M, Some properties of the stochastic integral, I have ...
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What is cylindrical Brownian Motion / Wiener Process

I have been given some reading on the Krylov-Bogoliubov Method for constructing invariant measures. An SDE in Hilbert space H is introduced as $$d(X)=b(X)dt + \sigma(X)dW $$ Where W is the ...
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On Martingale representation and continuous versions of the martingales.

I have troubles understanding the comment just before the following theorem, If a processes has a version which is continous or equivalently can be written as a Brownian integral, why is it that "...
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Indistinguishable Continuous Version of Martingale

Assume a filtered probability space, where the filtration is the augmented filtration generated by a Brownian motion. More precisely, define the set $$ \mathcal{N} = \{ A \subset \Omega \mid \exists ...
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A Question about the Fixed Point Method for SDEs

We assume a filtered probability space satisfying the usual conditions. Let $\mathbb{H}^2$ denote the space of $dt \otimes P$ square integrable and progressively measurable processes with the norm $$ |...
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1answer
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Showing independence of increments of a stochastic process

The textbook on stochastic calculus I am now reading says that if $X\colon [0,\infty)\times\Omega\rightarrow\mathbb R$ is a stochastic process such that $X(t)-X(s)\sim N(0,t-s)$ for all $t \geq s \...
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Space of Adapted Continuous Process is a Banach space

We assume a complete filtered probability space with right continuous filtration. Let $\mathbb{S}^2$ be the space of adapted and almost surely continuous processes $X = (X_t)_{t \in [0,T]}$ with ...
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Convergence using Laplace transform

Consider the sequence $(Y_n)_{n\ge 1}$ of iid and integrable random variables, which are not almost surely constant. Let $\delta>0$ and for all $\lambda\in(-\delta,\delta)$ it holds $$L(\lambda):=...
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Proof of Burkholder-Davis-Gundy inequality in the Almost Sure blog

I am currently reading the famous almost sure blog by George Lowther about stochastic calculus. I am currently reading the section about the Burkholder-Davis-Gundi-Inequality (BDG inequality). At the ...
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$ lim_{n \to \infty}\int_a^b|f(s)-G_nf(x)|^pds = 0$

Let $f \in L^p$ and define $G_nf(s) := \sum_{i=0}^{n-1}f_iI_{[t_i,t_{i+1})}(s)$, where $f_0 = 0, f_i = \frac{n}{b-a}\int_{t_{i-1}}^{t_i}f(s)ds$, $a = t_0 < t_1<...<t_n = b$, $t_{i+1}-t_i = (b-...
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1answer
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Probability for more general sumprocess

Consider $(X_n)_{n\ge1}$ iid with $$P(X_1=1)=p,\quad P(X_1=-1)=1-p$$ for some $p\in(0,1)$, $a,b\in\mathbb{Z}$ with $a<0<b$, the sum process $S_n:=\sum_{i=1}^nX_i$ and the stopping time $$T_{a,...
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Is there a process that is not square-integrable but still gives rise to a martingale when integrated w.r.t. Brownian motion?

When an adapted process $X$ satisfies $\int_0^TX_t^2dt<\infty$ a.s. but not $E\int_0^TX_t^2dt<\infty$, the stochastic integral $\int_0^tX_sdB_s$, $0\le t\le T$, is only guaranteed to be a local ...
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Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
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What is the difference between a global solution and strong solution of an SDE?

Can someone guide me to know if there is a difference between a global solution of a SDE and a strong solution? I'm a little bit confused.
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How to calculate the Covariance between two stochastic integral

How to calculate the covariance of the integral of Brownian motions: $$\text{Cov}(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t)$$ I know the answer: $$\int^{t_1\wedge t_2}_0\sigma^2(t)dt.$$ If ...
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stochastic ordering of counting processes/vectors

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...
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Check whether the following process is martingale with respect to Brownian filtration.

Is $X(t) = t^2W(t)-2\int_{0}^{t} sW(s) ds$ a martingale? I was thinking to use Ito integral to prove this but couldn't prove the term left on other side is a martingale.
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Stochastic ordering question

Let $N_1(t)$ and $N_2(t)$ be two random processes with $t\geq 0$ such that $\mathbb{P}(N_1(t)\geq b) \geq \mathbb{P}(N_2(t)\geq b) $ for all $b \in \mathbb{R}$. Can I conclude the following?: $$\...
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A question about Malliavin derivatives of Lebesgue integrals of processes

I have been attempting Exercise 2.2.1 from David Nualart's The Malliavin Calculus and Related Topics. The problem, on page 107, goes like this: Let $\sigma,b\in\mathscr{C}^1(\mathbb{R})$ with bounded ...
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Definition of $X_T$

Let $X$ be a stochastic process and $T$ a stopping time. Then one forms the random variable $X_T$. I have a quite vague question: apparently $X_T$ will be quite different if we replace $X$ by one of ...
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Variance spectrum of sde

I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal. Consider the Ito SDE. $dX_t=−λX_tdt+σdB_t$ Now state the variance spectrum of $\{X_t\}$ and ...
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How to calculate the probability of node X being connected over other nodes with node Y?

I'm struggling with the following szenario... There is a network of 1.000 nodes. Each of them is connected directionally to exatly 50 other nodes. How can I calculate the probabilities: 1) A random ...
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SDE existence and uniqueness

I am working on the following problem in a course using "Stochastic Differential Equations" by Oksendal. Consider the two coupled It SDEs. $dXt=−λX_tdt+σdB_t$ $dYt=−\sin Y_tdt+sX_t\cos Ytdt$ Where ...
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Find $E[X(t)^2]$ given the partial differential equation… where $B(t), t\geq 0$ is a Brownian motion.

Find $E[X(t)^2]$ given, $dX(t) = (\frac{4(1+t)^2 - X(t)^2)}{8X(t)})dt + \frac{1}{2}X(t)dB(t)$, where $B(t), t\geq 0$ is a Brownian motion. I think I must apply $It\hat{o}$'s formula to $X(t)^2$ but I ...
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Definition of predictable process

I am trying to understand the notion of predictable process. Let $(Ω,F_t,P)$ be a filtered measure space, satisfying the usual condition. Things starts with the predictable $\sigma$-algebra ${\mathcal ...
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Monotone class theorem, bounded processes vs. caglad processes

I struggle understanding the monotone class theorem and hope you can help me. The monotone class theorem states: Let $\mathcal M:=\{f_\alpha ;α\in J\}$ be a set of bounded functions, such that $...
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1answer
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Replacing a scalar with a random variable

Let $X,Y,Z$ be independent random variables. If for all $a\in \mathbb{R}$ $$P(X>a) \geq P(Y>a),\,\,\,\,\,(1)$$ can we conclude $$P(X>Z) \geq P(Y>Z)?\,\,\,\,\,\,\,(2)$$ I am a little ...
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43 views

On the stochastic exponential in stochastic calculus

Define $E(X)_t=\exp(X_t-\frac{1}{2} \langle X \rangle _t)$ where $(X_t)$ is an adapted continuous semimartingale. Then it is trivial that this is a continuous semimartingale and is the unique solution ...
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Standardised Brownian Bridge integral

Let $Z(\lambda)$ denote a standardised Brownian Bridge $Z(\lambda)=(W(\lambda)-\lambda W(1))/\sqrt{\lambda(1-\lambda)}$. As $\sup_{\lambda \in [0,1]}Z(\lambda)\rightarrow \infty$, $\int_{0}^{1}Z(\...
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Expected value of running minimum of BM

I need to calculate the following: $E[1_{\{Max_{0 \leq r \leq t} B(r) \ge 2\}}(-1 - min_{T_{2} \leq r \leq t}B(r))^{+}]$, where + means maximum of $0$ and the value in parenthesis. My attempt: The ...
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$E[|X|]$ for gamma distributed random variable $X$

Let $X$ be Gamma distributed with density $$f_\theta(x)=\frac{\theta^3}{2}x^2e^{-\theta x},\quad\theta\in(0,\infty).$$ Now I want to calculate $E[|X|]$, i.e. $$\int_{-\infty}^\infty|x|\frac{\theta^3}...
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Understanding Itô's lemma

I am new to stochastic calculus and I am learning it during off-hours, so I am full of doubts that maybe someone more expert may dispel. Let's say I get the usual geometric Brownian process, where $...
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Product of two processes Differential

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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43 views

Some problems on OU process

From Marc Yor's Continuous Martingale and Brownian Motion Page 38, we know that the process $X_t=e^{-\lambda t}B_{e^{2\lambda t}}$ is an OU process, where $B_{t}$ is an one-dimensional standard ...
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When can a Gaussian Process solve an SDE?

Considering an SDE of the form: $$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t$$ ... (where $W_t$ is a Weiner process) is there a set of necessary and sufficient conditions on the structure of the ...