Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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Proving existence of Itō Integral

Here's an extract from some Continuous Martingales notes I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that ...
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31 views

What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...
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Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
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29 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
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Doleans measure for local martingales

I came across the following question in my textbook and something in it doesn't quite make sense to me. Let $M$ be a local $L^2$ martingale. Then $X,Y \in \mathcal{L}(M,\mathcal{P})$ are ...
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68 views

An application of Itô's lemma

I found this question in a past exam for a course on Financial Economics. Given the function $f(t,x)$, let $F(t,x)$ be a function such that $∂F/∂x = f$. (a) By writing Itô’s formula in ...
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Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
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2answers
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Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
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55 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...
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What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [closed]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?
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Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] ...
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A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
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Examples of predictable processes

I am asked to prove that the following processes are predictable. I am used to looking at stochastic processes as sequences of random variables (by fixing time) or as a collection of paths (by fixing ...
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Distribution for Arithmetic Mean of n Geometrically Distributed Random Variables

For the evaluation of an algorithm I implemented for work, I need to find the distribution function for the arithmetic mean of $n$ independent, geometrically distributed random variables. Let ...
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Smoothness requirement for Stratonovich Integral

Every place I've seen defines the Ito formula for the Stratonovich integral as $df(X_t) = f'(X_t) \circ dX_t$ for $f \in C^3(\mathbb{R})$ and $X_t$ brownian motion, while the Ito integral only ...
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Right-continuous process is measurable with respect to product measure.

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{X_t\}_{t\geq0}$ be a collection of real random variables such that the map $t\mapsto X_t$ is right-continuous. Show that the map ...
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How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
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1answer
80 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
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1answer
36 views

Eigenfunctions of an operator using Laguerre Polynomials

I am trying to find the eigenfunctions of the following operator: $$\mathcal{L}f=(-\gamma x+\frac{\mu}{x})f_x+\mu f_{xx}$$ I know that I must somehow use Laguerre polynomials, the solutions to the ...
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1answer
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$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
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2answers
21 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...
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Infinitesimal generator of a semigroup

I know that if $\{T_t, t>0 \}$ is a conservative Markov semigroup on E, and $f \in D(A)$ has an absolute maximum in x then $Af(x) \le 0$. Where $D(A)$ is the infinitesimal generator of $T_t$. I ...
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Machine Learning and Probability/Stochastics

Main question: What connections are there between machine learning and stochastics (Probability theory, analysis, processes, SDEs)? Background: I've just been accepted into a master's programme for ...
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Applying Picard-Lindelöf iteration to a stochastic integral equation

Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$ I want to show that by using Picard ...
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Tower Property for Expectations and Stopping Times

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered proability space with $X\in L^1(P)$ and two stopping times $S$ and $T$. Show that \begin{equation*} ...
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2answers
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A question on measurability of stochastic process

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{X_t:t\geq0\}$ be a collection of real-valued random variables with index set $[0,\infty)$. Show that the mapping $t\mapsto X_t$ is ...
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1answer
41 views

What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
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1answer
61 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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Passing Expectation into Series (specifically Sine)

I want to show that this is true: $${ \mathbb{E}\big[\sin X_t \big]} = \sum_{n=0}^{\infty} \frac{(-1)^{n}{ \mathbb{E}\big[ X_t^{2n+1} \big]}}{(2n+1)!}$$ ($X_t$ is a Brownian Motion). By linearity I ...
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What is the integral of a family of diffusion processes? [closed]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...
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What is the integral over independent Wiener processes [duplicate]

This is actually similar to a question I posted yesterday, but with a fundamental difference which is not allowing me to solve my problem. Here is the question: Let $s \in [0,1]$ and define a ...
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1answer
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Is the integral of Ito processes still an Ito process?

Let $s \in [0,1]$ and define diffusion processes, $$dS(s)_t = \mu(s) dt + \sigma(s) dW_t$$ The question is if the following make sense, $$ \int_0^1 dS(s)_t ds = \int_0^1 \mu(s) ds dt + \int_0^1 ...
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1answer
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Proving a probability mass function to be $\le 1$

Assume $\Omega = \mathbb{N}_0$ and $k > 0$. Prove that $f(\omega) = e^{-\lambda} \cdot \frac{\lambda^{\omega}}{\omega !}$ is a mass probability function. Showing $f \geq 0$ is trivial as well as ...
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Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
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1answer
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Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
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1answer
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Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
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Variance estimation of a diffusion process

The framework of this question is a 1 dimensional diffusion process, defined ny the following equation: $dx_t=adt+bdw_t$ Where $w_t$ is a standard berownian motion and and $a$ is a constant drif ...
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1answer
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Checking if $B_t^3 $ and $3tB_t$ are martingales?

$$\mathbb{E}[ B_t^3 - 3tB_t + 3B_t | \mathcal{F}_s]$$ $$\mathbb{E}[B_t^3 | \mathcal{F}_s] - 3\mathbb{E}[t B_t | \mathcal{F}_s\}$$ $$\mathbb{E}[(B_t^3 - B_s^3 + B_s^3) | \mathcal{F}_s] + [ not \space ...
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1answer
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Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
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Filtrations and Sigma-Algebras and Stopping Times

In a previous post Filtrations and Sigma-Algebras I asked the question: $\textbf{Previous Question:}$ Let $\Omega=\{1,2,3\}, \mathcal{A}=\mathcal{P}(\Omega)$ and $P(\{\omega\})=\tfrac{1}{3}$ for each ...
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Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
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50 views

Exponential Martingales - Properties

This question relates to the exponential martingale, \begin{align} Y(t) = \exp\left(-\int_{0}^{t} \lambda(s)\,dW(s) - \tfrac{1}{2} \int_{0}^{t} \lambda^2(s)\,ds \right) \end{align} and specifically ...
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1answer
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How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
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prove $\sum\limits_{t=m+2}^n \sum\limits_{k=m+1}^{t-1} a_k \cdot X_{1,t-k} \cdot X_{2,t} = O_p(n^{1-\nu}) $ for $n \longrightarrow \infty$

Here are the preconditions required for the Lemma I have to prove: Let $X_{i,t}$ and $Y_{i,t}$ be random variables such that $E[X_{i,t}]^2 < \infty$ and $E[Y_{i,t}]^2 < C_1 \cdot \epsilon^k$ ...
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Wiener measure of smooth function in space of continuous function.

How do we show that the Wiener measure of class of smooth functions in $C[0, \infty)$ is 1?
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Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
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Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
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Justifying a step in proving $M_{S\wedge T} = \mathbb{E}[M_T | \mathcal{F}_S ]$

$S,T$ are stopping times and $M$ is a (right) continuous martingale. My lecturer set this as an exercise and I am given a solution(essentially split $M_T = M_T \mathbf{1}_{S≤T} + M_T ...
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Applying the martingale representation theorem

I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. ...