Questions on the calculus of stochastic processes, or processes that have a random component.

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6
votes
0answers
78 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
5
votes
0answers
65 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...
1
vote
1answer
22 views

Quadratic variation of a Poisson process where time depends on a function F

Let $(N_t)_{t \geq 0}$ be a Poisson process of rate $1$. Then the quadratic variation of $N$ is itself (so $\langle N\rangle_t=N_t$). For a suitable function $f$, what would be the quadratic variation ...
3
votes
1answer
26 views

Application of the Clark-Ocone's Formula to $\mathbb{1}_{S_t > K}$

At page 291 of Nonlinear Option Pricing by Julien Guyon and Pierre Henry-Labordère, the Clark-Ocone's Formula is applied to $\mathbb{1}_{S_t > K}$. I do not get how to get from the second to the ...
0
votes
0answers
23 views

Is $\tau(\omega )= \infty \forall \omega \in \Omega$ a stopping time?

My guess is yes since $\{\infty \leq t \}=\phi \in \mathcal{F}_t, \forall t \geq 0$ where $\phi$ is the empty set which is always in the sigma algebra. Am I right?
1
vote
0answers
16 views

Superposition of two Renewal Processes

Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} \frac{N_1(t)}{t}$, where $N_1(t)$ is the total number of renewals of $P_1$ ...
1
vote
1answer
21 views

Increasing the Rate of a Renewal Process

This problem is a dual question of "Splitting a renewal process". Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} ...
1
vote
0answers
20 views

Independence of two random variables 2

Assume we put points on the two dimensional $x-y$ plane according to Poisson distribution, consider $r_1$ is the location of the closest point to origin and $r_2$ is the location of the second ...
4
votes
1answer
55 views

Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
0
votes
0answers
15 views

Does the law of an Itô process $X_t$ at each time has a density? [duplicate]

Let $X$ be a the strong solution of the SDE on $\mathbb{R}$ $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ $$X_0 = x$$ where $b,\sigma \colon \mathbb{R} \to \mathbb{R}$ are Lipschitz functions. Dose the law ...
3
votes
1answer
55 views

Splitting a renewal process

This is a follow-up question of the question "When superposition of two renewal processes is another renewal process?". How can we split a renewal process $P$ into a renewal process $P_1$ and ...
1
vote
0answers
14 views

Which research groups use stochastic processes and/or stochastic differential equations in computer graphics/vision?

Some research groups use stochastic differential equations for mathematical image processing. Which research groups do use stochastic processes in general and/or stochastic differential equations in ...
2
votes
0answers
16 views

Ito formula for a function of class $C^1$

Can the Ito formula be applied with a $C^1$ function if the second order terms vanish ? For example, let $g(t)$ be a function of class $C^1$ and define $F(x,t)=xg(t)$ which is also of class $C^1$. ...
0
votes
1answer
39 views

Ito's formula for Poisson process

Suppose ($Y_t$) is a rate 1 Poisson process, and consider the jump process $Z_t=Y_{\int_0^tf(X_s)ds}$ for some non-negative process $X_s$. What would be the quadratic variation of $Z$, and how would ...
1
vote
0answers
129 views

What is the probability a random walk crosses a line before another?

Let $n \geq 0$, $X_n$ be a random walk, where $X_{n+1} = X_n + 1$ with probability $p$, and $X_{n+1} = X_n - 1$ with probability $1-p$. $X_0 = 0$ Let $l_n, r_n$ be a sequence of integers, where for ...
1
vote
1answer
53 views

Almost Surely Finite Stopping Time Inequality

Assume $\tau$ is a $\mathcal{F}_n$- stopping time such that there exists a positive integer $m$ and some $\epsilon>0$ such that for all $n$ $$\mathbb{P}(\tau\leq n+m \,\, \vert \mathcal{F}_n) ...
1
vote
0answers
18 views

What is the conditional distribution of $X_3$ where $dX_t = adt + dW_t$?

Suppose $X_t$ satisfies the stochastic differential equation $dX_t = adt + dW_t$ ($a$ is a constant). What is the conditional distribution of $X_3$ given the values $X_0=0$, $X_1=-1$, $X_4=2$, and ...
1
vote
1answer
19 views

Solve and prove uniqueness of the SDE $dY_t = tY_tdt + e^{t^2/2}dB_t$ without using the general linear SDE formula

Let $(B_t)_{t \in [0,T]}$ be standard brownian motion, and let $(Y_t)_{t \in [0,T]}$ be a stochastic process in $(\Omega, \mathscr F, \mathbb P)$. Without using the general linear SDE formula, solve ...
0
votes
1answer
24 views

How do I read this equation: $ Pr(T < t \mid y) = \int_{0}^{\infty }\int_{\Omega }k(x)p(t, x\mid y)\,dx\,dt \ $?

How do I read this equation (especially the left side) in terms of a Continuous Markov Process model? $$ Pr(T < t \mid y) = \int_{0}^{\infty }\int_{\Omega }k(x)p(t, x\mid y)\,dx\,dt \\ $$ Where $ ...
0
votes
0answers
14 views

Numerical stochastic ODE: finding E[X(t)] etc at discrete points

So if this is trivial, please tell me I'm an idiot. I guess, given a continuous time stochastic process, one can numerically approximate it at discrete points. Nothing too extreme. I'm new to this ...
0
votes
0answers
67 views

How to find derivative of this intergral?

Let $\alpha(t)\in\{0,1\}: 0\leq t\leq 1$ be a two state continuous Markov chain with the generator (https://en.wikipedia.org/wiki/Continuous-time_Markov_chain) $$ Q=\begin{bmatrix} -\alpha & ...
3
votes
1answer
15 views

$Y(t)=W^2(t)\cdot e^{aW(t)}$, find $dY$

The answer given in a textbook is $dY=e^{aW}[t+\frac{1}{2}a^2W^2+2Wa]dt+ae^{aW}W^2dW$, but mine is $dY=e^{aW}[1+\frac{1}{2}a^2W^2+2aW]dt+e^{aW}(2W+aW^2)dW$. Which is correct? My solution: ...
1
vote
1answer
42 views

Definition of stochastic integral, square integrable function

Hello I have a question about Stochastic integral. Let $X=(X_{t})_{t \geq0}$ be a Brownian motion started at $0$. I know the following fact: Let $(\varphi(t))_{t\geq0}$ be a progressively measurable ...
1
vote
2answers
54 views

Continuity of probability measures for a process

Let $(B_t)_t$ be a Brownian motion, then I am given a stopping time $\tau_s:=\min(\inf\{t \ge 0; B_t=a\}, \inf\{t \ge s; B_t=b\}; \inf \{t \ge 0;B_t=c\}),$ where $a<0<b<c.$ Now, I want to ...
5
votes
1answer
70 views

To confirm the Novikov's condition

I have a question about Novikov's condition. Let $L$ be a local martingale such that either $\exp \left(\frac{1}{2}L \right)$ is a submartingale or $E[\exp\left(\frac{1}{2} \langle L,L \rangle_{t} ...
0
votes
0answers
19 views

Why is the stochastic integral $\int_0^t \nabla u(B_s)\cdot dB_s $ a local martingale?

This is from Durrett's book Stochastic calculus: a practical introduction. I don't understand the last sentence in the picture. Could anyone help explain why the first term is a local martingale? ...
0
votes
0answers
18 views

Can we prove martingale convergence theorems and inequalities without relying on discrete results?

I'm preparing a stochastic calculus course and I'm trying to structure my thoughts on the subject before writing everything in $\LaTeX$. I would like to prove Doob upcrossing inequality, the standard ...
0
votes
0answers
22 views

Derive the Black-schole formula by solving the BSDE.

Consider the FBSDE as following: $$ \left\{ \begin{aligned} dS_t &=S_t\mu d_t+S_t\sigma dW_t \\ dX_t &=(rX_t+\frac{\mu-r}{\sigma}Z_t )d_t+Z_tdW_t \\ S_0 &=s\\ X_T &=(S_T-k)^+ ...
0
votes
1answer
33 views

Ito's formula and Brownian motion

Let $a \in R$,$B=(B^1,B^2)$ a brownian motion. $$X_t=e^{B_t^1}\left(\int_0^te^{-B_s^1}dB_s^2+a\int_0^te^{-B_s^1}ds\right)$$ Show there is a brownian motion $\beta$ such that $$X_t=\int_0^t ...
0
votes
1answer
67 views

Distribution of stopping times

I encountered the following question in my research: Let the diffusion process $\{X_t\}_{t\ge 0}$ be governed by $$d X_t=s(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $s>0$, and $B_t$ is ...
4
votes
0answers
63 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
1
vote
0answers
18 views

Expectation of exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion. Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...
1
vote
1answer
67 views

Uniqueness in law associated to nonlinear SDEs

I do not understand the following when reading a paper on Propagation of Chaos, written by A.S.Sznitman: Consider an $n$- dimensional process $X$ satisfying the following SDE: $$ dX_t = b(t, ...
1
vote
1answer
30 views

Convergence of stochastic processes and integral

Let $X_n(t)$ and $X(t)$ are bounded stochastic processes ($|X_n(t)|<C$ and $|X(t)<C|$) such that for each $t\in[0,T]$ $$ X_n(t) \to X(t) \text{ as } n\to\infty \text{ almost surely }. $$ ...
0
votes
0answers
16 views

Box calculus for sequential differential

A shorthand rule of thumb for Ito calculus is the Box calculus where one assumes that $dtdW^{(i)}=0$ and $dW^{(i)}dW^{(j)}=\delta_{ij}dt$ where $dW^{(i)}$ and $dW^{(j)}$ are increments in two ...
0
votes
0answers
21 views

Identify a probability distribution with coordinate transformation

I have a problem with this task: We have a random variable $X:\Omega \rightarrow \mathbb R^2$, which is uniformly distributed on $K:= \{(x_1,x_2) \in \mathbb R^2 : \sqrt{x_1^2+x_2^2} \le 1 \}$ Now I ...
1
vote
0answers
41 views

Evaluating integral with respect to brownian motion

I am attempting to integrate $$ \int _{0}^{t} \sin(s) dW_s $$ whereas $W_s$ is brownian motion, in some sense a normal random variable with mean 0 and variance $s$. I looked around in stack ...
1
vote
1answer
33 views

Solve SDE $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$

Solve the SDE given by: $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$. My attempt Following the hint of my professor: suppose $X_t = e^{\gamma t} g(W_t)$. Then we ...
0
votes
0answers
13 views

Exponential, supermartinagles

I have a question about super-martingale and martingale convergence theorem. Fix $T>0,\lambda>0$. For a continuous local martingale $X=(X)_{t \in [0,T]}$, we can define \begin{align*} ...
1
vote
1answer
9 views

Symmetrical function of Brownian Motion

Let $W_t$ be Brownian motion. Using software, I can compute $E[e^{\beta t} \sin{(\gamma W_t})] = 0$. Could one void this computation with a clever symmetrical argument. That is: Since $sin(t)$ is an ...
0
votes
0answers
29 views

Generalized Ito's lemma

I have the following quantity: $$ g(t)=(f(t))^{M_{t}}, $$ where $M_{t}$ is a jump process neither Markovian nor Levy and $f(t)$ is a positive, increasing but limited, right-continuous function. How ...
0
votes
0answers
16 views

right continuity of martingales

I am stuck with the following problem : let $M_t$ be a right continuous martingale. Show that the map $t \to M_t$ is right continuous from $\mathbb{R}^{+}$ to $L^1$. I know that, fixed $\omega$, the ...
0
votes
0answers
22 views

what does “span” mean in measure theory?

I have a random variable $X \in L^1(\Omega, F,P)$ and I am required to study "the family of random variables of the type $\mathbb{E}\left[X | G \right]$,where $G$ spans the set of all sub-$\sigma$ ...
0
votes
1answer
23 views

Can I swap limit and expectation?

I know that a random variable $X$ is integrable, that is $\mathbb{E}\left[|X| \right]<+\infty$. Can I apply the dominated convergence theorem and state that $\lim_{a \to +\infty} \mathbb{E} (|X|\ ...
3
votes
0answers
47 views

Stopping times and hitting times for cadlag processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable $\tau_{C}:=\inf\{t\in ...
1
vote
0answers
15 views

Compute $E(Y_t^2)$ with $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$

Consider the process, $Y_t =\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} dW_{u})ds$. To compute the variance of this process, I need to compute $E[(\int_{0}^{t}(e^{-as} \sigma \int_{0}^{s} e^{au} ...
0
votes
0answers
24 views

Complete (not heurestic) proof of Ito lemma?

Where can I find a formal and complete proof of Ito lemma. I found a few of them but all are "heurestic" type like on Wikipedia, operating on $dX_t$ notation. Not really proofs. Thank you for any ...
2
votes
0answers
54 views

Brownian motion in $2$ dimensions on the plane

Consider a $1$ dimensional Brownian motion of a particle starting at $0$. Then, we know that the probability that the particle reaches the point $x$ at a time $\geq t_0$ is given by ...
0
votes
0answers
18 views

Difference between Fokker-Planck approach and SDEs

This may be a weird question to ask. Consider a Ito SDE: $dX_{t} = \mu(X_{t},t)dt + \sigma(X_{t},t)dW_t$ This is known to be formally linked to a Fokker-Planck equation for a probability density ...
0
votes
0answers
21 views

Modify process to semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...