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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Decision theory references for advanced undergrad/early grad students?

I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
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Strong existence of solutions to SDE and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, ...
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When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $...
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Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
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Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
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Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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Question about Ito integration in SDE in Stochastic optimal control

Here is my question statement. I cannot understand the last equality. Let $U=[-1,1]$. \begin{equation} \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\...
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Trace term in the Itō formula

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(...
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GMM with full and diagonal covariances

I have Gaussian Mixture Model-- distribution with probability density function, that is a weighted sum of Gaussian probability density functions: \begin{equation} p(X)=\sum_{i=1}^k \omega_i\mathcal{N}...
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Solving Langevin equation

In a past exam paper that I am looking at, there is the following question: Given that the displacement, $\mathbf{x}$, of a particle in $3$-dimensional Brownian motion is given by: $$m\ddot{\...
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Find continuous stochastic variable $X$ with PDF $f_X = \frac{1}{x^2}$

Given the uniform stochastic variable $U$ defined on the interval [0,1]. Using $U$, define a continuous stochastic variable $X$ with probability density function (PDF) $$f_X(x) = \begin{cases} \frac{1}...
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A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
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Stochastic integral of $\mathrm{sin}(X_t+Y_t)\mathrm{cos}(X_t+Y_t)$ [closed]

As part of an exercise I am doing from stochastic analysis I need to compute the following integral: $$E(\int_0^x \mathrm{sin}(X_t+Y_t) \mathrm{cos}(X_t+Y_t)dt)$$ where $X_0=0$ and $Y_0=0$. Presumably ...
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Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
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stochastic calculus, stopping time, ito integral vector brownian motion

I'm referring to chapter 4, question 7 in Harrison's book 'Brownian Motion and Stochastic Flow Systems.' Problem In the setting of (9) let $f_{n}(x)=E_{x}[\int_{0}^{T}X_{t}^{n}dt]$. Use Ito's ...
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Poisson process and Heaviside function

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside's function: $$\...
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Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as $e^{m+\frac{1}{2}...
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Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
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what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
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Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
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How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
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Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
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Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv \int\...
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Solving a nonlinear scalar Ito SDE

I need to solve the SDE: $$ dX_t = (X_t)^3 dt + (X_t)^2 dW_t ; X(0)=1 $$ Now what I found is this is an SDE of the form: $$dXt =a(X_t)dt+b(X_t)dW_t$$ where $a(x) = \frac{1}{2} b(x)b′(x)$ Using the ...
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Equivalence of two Ito formulae

Let $X$ and $Y$ be two $1$-dimensional Ito processes. There are two Ito formulae for the product $X_tY_t$ given by $d\left(X_tY_t\right)=X_tdY_t+Y_tdX_t+d\left[X_t,Y_t\right]$ $d\left(X_tY_t\right)=...
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What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
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Determining distribution and therefrom probability

The problem is as follows: Assume that $V_1$ and $V_2$ are independent random variables with $V_1 \sim \chi^2(5), V_2\sim\chi^2(9)$. Find the value of $b$ such that: $$P[\frac {V_1}{V_1 + V_2} \lt b] ...
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To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable ...
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Solving the following SDE with a constant

Given is the stochastic differential equation: $\frac{dX(t)}{X(t)}=\mu+\sigma \theta dt+ \sigma dW(t)$, where $W(t)$ is the standard Wiener process and $X(0)=x_0$ I try to solve this by the Itos ...
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Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
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A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
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Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
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Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
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Existence and uniqueness of SDE, is the independence requirement needed?

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5: $\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you ...
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How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ϕ(x_0,t_0;x,...
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Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ dX_s,...
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stochastic differential equation exact solution

whats (is there) exact solution of (for) this sde? $dX_{t}=\mu X_{t}dt+\sqrt{\sigma X_{t}} dW_{t}$ and what's the distribution of that? thanks
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$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
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Integration by parts - Brownian motion and non-random function

Let $B$ be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function $\phi$ that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ...
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Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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Why an optional process could not be predictable?

We know that a predictable process is also optional (*). Why an optional process could not be predictable ? Why we cannot use the same arguments as the proof for (*) ?
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Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...
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1answer
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discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...