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

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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
25 views

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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1answer
31 views

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
18 views

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
24 views

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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15 views

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
32 views

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
22 views

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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47 views

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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1answer
51 views

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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31 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
34 views

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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0answers
8 views

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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19 views

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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1answer
34 views

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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14 views

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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50 views

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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1answer
36 views

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 $M= X \cdot W$. ...
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1answer
49 views

Proof of martingale representation theorem monotone class argument

Martingale representation theorem for reference: Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is ...
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26 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. ...
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28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
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How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...
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1answer
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Application of Ito's formula to log and exponential

Let $X$ be a strictly positive continuous semimartingale with $X_0 = 1$ and define the process $Y$ by $$ Y_t = \int_0^t \frac{1}{X} dX - \frac12 \int_0^t \frac{1}{X^2} d \langle X \rangle. $$ Let the ...
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1answer
33 views

Infinitesimal Random Variable

I have been very confused by the idea of infinitesimal random variables, namely letting $\{Z(\omega,t)\}_{t\in\mathbb{R}}$ be a stochastic process. What do we mean by $dZ$. Is this meant by ...
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10 views

How to decompose a Markov Modulated Poisson Process (MMPP)

I have two questions to ask here. The superposition of two independent MMPPs is also a MMPP. How to calculate the rate of a new burst and the rate of requests within one burst if these two MMPPs are ...
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0answers
14 views

Expectation of maximum and minimum in problem in stochastic programming

I am working through *Introduction to Stochastic Programming" and am having trouble following one step. The authors Birge and Louveaux state one problem as a second stage expected value function ...
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1answer
144 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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65 views

Proving the identity $P( X + Y = a)= \int_{-\infty}^{\infty} P( X + y = a)f_Y(y) \, \text{d}y $

Suppose $\lambda_1, \lambda_2, a \in \mathbb{R}$ and $X,Y$ are random variables. If it is needed, I can assume that $X$ and $Y$ are independent. I want to show, that the identity ...
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49 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
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Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
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54 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...
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263 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
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1answer
42 views

Computation of a stochastic integral with respect to a local martingale

I am trying to compute the stochastic integral $$\int_{(0,t]}\mathbb{1}_{[a,b)}(s)dM_s$$ where $0 < a < b< \infty$ are constant and $M$ is a continuous local $L^2$-martingale. I am guessing ...
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1answer
117 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
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2answers
132 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
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Which books would you buy (Stochastic analysis Physical mathematics PhD)? [closed]

I have a soft question for you, any help will be appreciated. I have a large availability of money to buy mathematical text books (graduate level). I'm looking for suggestions with these topics: ...
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68 views

How can I show/solve this equation?

I need help to prove the following equation. $X_n$ is an iid random variable, with: $$\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$$ Show: ...
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33 views

Explosion time of $dX_t=X_t(adW_1+bdW_2)$

I found in Karatzas & Shreve (1991), $dX=\sigma(X_t)dW_t$ cannot explode. But what about $dX_t=X_t(adW_1+bdW_2)$? Here $W_1$ and $W_2$ are independent. Feller's test for explosion seems to work ...
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left limit of filtration

Let $N(t) = I_{(X\le u, \delta = 1)}, X = min(T,C), \delta = I_{(T\le c)} $ $F_s = \sigma \{ N(u), I_{(X\le u, \delta = 0)} , 0\le u \le s\rbrace$ and $F_{s^{-}}$ = $ \sigma \{ \cup_{(u<s)} ...
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Derivation of Kolmogorov Forward Equation

By Ito's formula we have that for any suitable function $v(t,x)$, $$ v(T, X_T) = v(t,X_t) + \int_t^T\left( v_s(s, X_s)+ b(s, X_s)v_x(s,X_s)+\frac{1}{2}\sigma^2(s, X_s)v_{xx}(s, X_s) ...
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22 views

Stratonovich SDE and generator in divergence form

Let $a:\mathbb{R}^d\rightarrow\mathcal{S}_{d\times d}$ be a smooth map that takes its values in the space of $d\times d$ symmetric matrices and suppose there exists a $d\times d$ matrix valued ...
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1answer
51 views

Malliavin derivative of a Lebesgue integral.

Let $X_t$ a random process such that its Malliavin derivative is well defined for all $t$. Then I have read that : $D_s(\int_0^t \! X_u \, \mathrm{d}u)=\int_s^t \! D_s(X_u) \, \mathrm{d}u.$ What I ...
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Preservation of the Markov Property for SDEs

Let $X$ be a continuous Markov process on $\mathbb{R}^d$ that is also a semimartingale. Let $V=(V_1,...,V_d)$ be a collection of suitably nice vector fields on $\mathbb{R}^d$ such that there exists a ...
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1answer
33 views

Finding mean and variance of a population problem

A population beings with a single individual. In each generation, each individual in the population dies with probability $1/2$ or doubles with probability $1/2$. If I let $X_n$ denote the number of ...
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73 views

Find the Stationary Distribution of an infinite state Markov chain

A Markov Chain on states 0,1,..... has transition probabilities $P_{ij}=1/(i+2)$ for j=0,1,....,i,i+1. I'm supposed to find the stationary distribution. So do I take the limit as n goes to ...
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2answers
41 views

Limiting Distribution of a Markov Chain

I'm having trouble understanding how to find a limiting distribution. If I have a Markov Chain whose transition probability matrix is: $$ \mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 & ...
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20 views

p-variation of semimartingales

Does every (particularly continuous) semi-martingale have bounded 2+$\epsilon$-variation for all $\epsilon>0$? Note that I am not asking, whether they have finite quadratic variation - that is ...
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1answer
52 views

Predictable Stochastic Processes

I tried to understand intuitively what a predictable stochastic process is (in particular, what is "predictable" about it), but found the definition via the measurability with respect to a certain ...
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17 views

Showing which classes are recurrent and which are transient

If I have a Markov chain on states {0,1,2,3,4,5} $$ \mathbf{a} = \matrix{~ & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1/3 & 0 & 2/3 & 0 & 0 & 0 \\ ...
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1answer
49 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...