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

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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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Variance of interarrival time of events [on hold]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
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1answer
48 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
29 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
43 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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25 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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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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1answer
40 views

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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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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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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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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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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1answer
48 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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1answer
40 views

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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1answer
52 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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253 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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41 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
115 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
122 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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67 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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1answer
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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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68 views

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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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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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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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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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
47 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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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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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 ...
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Unique solution in differential equation

Given a functions g(t,T) and Q(t,T) such that $g(t,T) = - \frac{\partial}{\partial T} \ln Q(t,T)$, $Q(T,T) = 1 = Q(t,t)$, T>0 and $t \in [0,T]$ Does it follow that $Q(t,T) = exp(-\int_{t}^{T} ...
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Existence and uniqueness of strong solution of stochastic differential equation.

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
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How to show $t \mapsto E[Z|\mathscr{F}_t]$ is a.s. borel measurable.

I'm going through Revuz and Yor and am stuck at a technicality. Suppose $Z$ is bounded and $A$ is bounded increasing continuous with $A_0 =0$. The goal of the problem is to show $E[ZA_\infty] = ...
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Find the density of the random variable X(t)(Kolmogorov Forward equation)

Let $V(x) = x^2 / 2+ W(x)$ where $W(x)$ is a smooth function with compact support. Let $f$ denote the probability density. $f(x) = \frac{e^{-V(x)}}{\int e^{-V(x)}dx}$. Consider the stochastic ...
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Probability of going into an absorbing state

If I have a random walk Markov chain whose transition probability matrix is given by $$ \mathbf{P} = \matrix{~ & 0 & 1 & 2 & 3 \\ 0 & 1 & 0 & 0 & 0 \\ ...
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Convergence properties of the Ito integral

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
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How to interpret a special construction of random variables

Let $\left( \Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space and $\left(f_n\right)_{n=1}^{\infty}$ a sequence of independent and identically distributed random variables with ...
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Specifying transition probabilities for a Markov Chain

If I have a queueing model and I suppose at most a single customer arrives during a single period, but that the service time of a customer is a random variable Z with geometric probability ...
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1answer
46 views

Urn Problem-Determining the Transition Probability Matrix

I have two urns A and B containing a total of N balls. An experiment is performed where a ball is selected at random (all selections equally likely) at time t(t=1,2,...) from the totality of N balls. ...
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3answers
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Show that the average of n observations is equal to the expected value

Show that the average of n observations is equal to the expected value with the density function with index k is equal to the number of observations equal to k divided by the total number of ...
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1answer
68 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
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1answer
56 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
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77 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
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Tossing two dice with sum equal to 4?

Exercise: Throw two dice. Suppose that eye sum are 4. Calculate the resulting conditional probability that a) the first dice gave a 3 . b ) the second dice gave two or fewer eyes. c ) ...
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Levy Processes - triplet for compound Poisson process

I'm stuck on 2 problems with Levy processes. People says that they are simple, but I can't solve it. Can anyone provide step by step solution? 1. Show that gamma distribution is infinitely divisible. ...