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

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23 views

Proof of the stochastic Fubini's theorem

I am trying to prove the Stochastic Fubini's theorem which is an exercise of An Introduction to Stochastic Calculus Applied to Finance. Let $(W_t)_{t\in[0,T]}$ be a Brownian motion and $H(t,s)$ has ...
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21 views

Assumptions of Exponential Family with possibly a Counterexample

In U.Küchler "Exponential Families of Stochastic Processes" 1997 [p.19-20] one consider a class of probability measures $P:=\{P_{\theta}:\theta \in \Theta\subset \mathbb{R}^{k}\}$ on ...
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35 views

Bessel Process and Brownian motion

Let $\beta_s$ be a Bessel process, i.e. the positive solution to the SDE $$\beta_s = B_s + (n-1) \int_0^t \frac{1}{\beta_s} \mathrm{d}s,$$ where $B_s$ is a one-dimensional Brownian motion. Let $U_s$ ...
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1answer
25 views

Integrating w.r.t. the pushforward measure

Let $X,Y : \Omega \rightarrow \mathbb{R}$ be independent r.v.'s and $f$ continuous. Then $A \subset \Omega$ $\int_{A} f(X,Y) dP = \int_{(X,Y)(A)} f(z) dP_{X,Y}(z) = \int_{(X,Y)(A)} f(x,y) dP_X(x) ...
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24 views

Stochastic integral and weak integral

Can the stochastic (Skorokhod) integral be seen as a special case of the weak of Pettis integral with the Banach space which win integrate into chosen appropriately?
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7 views

Is the value of the discrete Green's function on a box independent of the position of the points within the box?

I currently contemplate over the discrete Green's function on a box and am trying to gain an intuition for its behaviour. Consider the box $B := \{-N,\cdots,-1,0,1,\cdots,N\}^2$ and let $(X_i)_{i \in ...
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24 views

Conditional expectation unique?

Let $A \subset B$ be a subalgebra. This means that $E(X|A)$ is also $B$ measurable. Now, if I can show that $E(X|B)$ is also $A$ measurable, does this imply that the conditional expectations are the ...
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1answer
116 views

Problem 3.24 of “Brownian Motion & Stochastic Processes” by Karatzas and Shreve - Submartingales and stopping times

I'm doing the problem 3.24 of Brownian Motion and Stochastic Processes by Karatzas and Shreve. There is two specific parts troubling me, I need some help to see what to do. Here is the problem: ...
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53 views

How to calculate analytical solution to fokker planck equation in finite domain?

I am looking for the solution to the following fokker planck equation. $\frac{\partial f(x,t)}{\partial t}= (k_1 x - v)\frac{\partial f(x,t)}{\partial x} + (k_2 x + D) \frac{\partial^2 ...
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31 views

How does the sample space remain constant in filtered

I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like ...
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9 views

Principal Component Analysis in a stable framework

are you familiar with stable distributions. It is denoted by $S_{\alpha}(\sigma,\beta,\mu)$ where $\alpha$ is the tail index, $\beta$ is the skewness, and $\sigma$ and $\mu$ are the location and scale ...
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1answer
59 views

Completeness of Probability Distribution as a Measure.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space and $X:\Omega \to \mathbb{R}^n$ be a $\mathcal{F}$-measurable function, i.e., it is a random variable. Then, in the book ...
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73 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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36 views

Symmetric random walk about $y=2$.

Consider a simple (symmetric) random walk $p=q=\frac{1}{2}$ and $(X_n)_{n\geq 0}$ with $X_0 = 0$. Using the reflection principle, find the probability that $X_{12} = -4$ and $X_1 < 2$, $X_2 < ...
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15 views

Distribution of a Brownian bridge

I am self studying some stochastic calculus material and come across this question to show that the distribution of $P(W(s)\in dy|W(t)=x)$ with $W(0)=0$ is normal with mean $\frac{s}{t}W(t)$ and ...
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1answer
49 views

Continuous path of stochastic processes

In my probability book a stochastic process is defined as a measurable map $X: \Omega \rightarrow S^T,$ where $S^T$ is equipped with the sigma algebra of cylinder events. Our professor mentioned that ...
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16 views

Ito's Lemma $\ln(S_T / S_t)$

Assume that $dS = \mu d_t + \sigma d_z$, where $dz \sim N(0,\delta t)$ When I apply Ito's lemma to $\ln(S)$ I obtain that $d\ln(S) = [\mu-\frac{1}{2}\sigma^2]dt + \sigma d_z$ Now I thought that ...
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1answer
47 views

Prove that the first hitting time $\tau_x:=\inf\left\{t\ge 0:B_t=x\right\}$ of a Brownian motion is almost surely finite

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $$\tau:=\inf\big\{t\ge 0:B_t\in\left\{a,b\right\}\big\}$$ for some $a<0<b$. I want to prove, that $\tau$ is almost surely finite. Let ...
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0answers
10 views

Can we find the boundary condition of a function of diffusion process?

given $dx_t=b(t,x_t)dt+\sigma(t,x_t)dW_t$, if it is in one dimensional case, one can use Feller non-explosion test to see if $x_t$ attains a particular boundary. How about $f(x_t)$, $f$ is any ...
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1answer
47 views

Brownian motion / ito's formula

Little help is needed Can I use geometric Brownian motion here? The question I get: $Let z=(z_t)$ be a one-dimensional standard Brownian motion and define the process $y = ( y_t )$ by $y_t = z_t^2 − ...
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1answer
34 views

Obtaining martingales from Poisson process

All processes here are continuous. Suppose we have a Poisson process $(N_t)_{t\geq 0}$ with parameter $\lambda > 0$ and adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$. Fix $u\in\mathbb{C}$, ...
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1answer
46 views

Karatzas and Shreve - Problem 3.3.19

I'm struggling here to solve this problem, but with no success. I was able to prove a $\implies$ b, but the next implication is troubling me. In the book, they give a solution, but I think there ...
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1answer
65 views

Do I need topology to study stochastic process?

So far I dealt with probability from a very intuitive point of view, like guessing frequencies etc. But while studying stochastic process (particularly with application to finance), I came across ...
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1answer
16 views

Uniform Integrability and Supermartingal Process

Suppose $\{X_t, \mathcal{F}_t: 0\leq t<\infty\}$ is a right-continuous and nonnegative supermartingal. I want to show that $\lim_{t\to\infty}X_t(\omega)$ exists a.e. and that $\{X_t, \mathcal{F}_t: ...
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29 views

Correspondence between multi-dimensional Brownian motion and harmonic functions

Let $U\subseteq\mathbb R^d$ be a bounded domain. A continuous function $u:\overline U\to\mathbb R$ is called harmonic $:\Leftrightarrow$ $$u(x)=\frac 1{|\partial B_r(x)|}\int_{\partial B_r(x)} ...
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18 views

Please can you give me an example of a stochastic process that is a Markov process but not a martingale? [duplicate]

Please can you give me an example of a stochastic process that is a Markov process but not a martingale ?
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25 views

Covariance of two stochastic integrals

Consider the stochastic integral $\int_{0}^{1}J(r)M(r,\lambda) dr$ where $J(r)$ is a demeaned Ornstein-Uhlenbeck process and $M(r,\lambda)=W(r,\lambda)-\lambda W(r,1)$ a Brownian Sheet, independent of ...
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25 views

Brownian motion (conditional density)

Suppose $(B_t)_{t\geq0}$ is a Brownian motion. (1) For $0\leq s<t$, give the conditional law: $B_{(t+s)/2}|B_s=x, B_t=y$. (Hint: start looking for coefficients $a,b\in R$ ...
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1answer
41 views

Modified stochastic processes

I am looking for conditions such that a process $(X_t)_t$ where the $X_t$ are $\text{iid}$ such that there is a process $(Y_t)_t$ satisfying $P(X_t=Y_t)=1$ and $t \mapsto Y_t(\omega) \text{ is ...
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1answer
64 views

Borel $\sigma$-algebra of continuous functions

Let $B(C(T,\mathbb{R}))$ be the Borel sigma algebra of continuous functions mapping from the compact metric space $T$ to $S$ defined by the canonical metric $\|\cdot\|_\infty.$ Now I was wondering ...
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20 views

Condition for covariance of Ornstein-Uhlenbeck processes

I am considering this Ornstein-Uhlenbeck process: $$X_t=e^{-at}X_0+e^{-at}\int_{0}^{t}e^{as}\mathrm{d}W_s$$ in which, $a$ is a constant, and $W_s$ is a standard Brownian motion. Expected value of ...
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52 views

Product of two Ornstein Uhlenbeck processes : conditional distribution

Let $X(t)$ and $Y(t)$ be two independent OU processes (each with some fixed correlation time-scale), and let $S(t) = X(t)Y(t)$. Then is there a expression for the conditional distribution of $X(t)$ ...
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1answer
28 views

Why for a continous local martingale ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$

Why for a continous local martingale $(M_t)_{t\in \mathbb{R}_+}$ ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$ where $(B_u, u \geq 0)$ is Brownian motion . ...
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1answer
15 views

predictable process in continuous times.

If $X_t$ is a stochastic process in $(\Omega,\mathcal F,\mathcal F_t)$ which satisfy usual conditions. Can we get $X_t$ is a predictable process if $X_t$ is adapted to $\mathcal F_{t-}$? since ...
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1answer
81 views

Using Fubini's Theorem in Stochastic Calculus

In basic calculus: 'Fubini's theorem' allows us to switch order of integration in double integrals without changing the bounds provided we are integrating over a rectangle. From here: If the area ...
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2answers
40 views

How to obtain this result using Ito's Lemma?

My book writes that when: $dx = a(x,t)dt + b(x,t)dz$ $x' := x + dx$ Then using Ito's lemma: $E[F[x+ \Delta x, t + \Delta t \mid x)] = F(x,t) + [F_t(x,t) + a F_x + \frac{1}{2} b^2 F_{xx}] ...
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1answer
14 views

Proving linearity of a subspace of $L^2$

I'm reading p.79 of Steele s Stochastic Calculus and Financial Applications. It defines a space of functions $\mathcal{H}^2$ as the space of all measurable adapted functions such that ...
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0answers
38 views

Why isn't this stochastic integral trivial?

I have a stopping time $\tau$ and a stochastic process $f$. Then the following equation is true: \begin{equation} \int^{t\wedge\tau}_{0}f(s)dW(s)=\int^{t}_{0}f(s)\chi_{[0,\tau]}(s)dW(s) ...
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0answers
18 views

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$ and $E\{\mid \int_{\alpha}^{\beta}f(t)d\omega (t)\mid^2|\mathscr{F}_\alpha ...
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0answers
24 views

Solving a Stochastic PDE with two variables in time

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} ...
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1answer
19 views

laplace transform of smirnov density, i.e. how to calculate this integral?

I am trying to figure out how to perform the following computation. The objective is to compute the laplace transform of the smirnov density. The lecture notes I've seen online state that ...
2
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1answer
25 views

SDE solution with $\mu_t$ a bounded integral

Consider the following SDE: $$ dX_t = X_t (\sigma dW_t + \mu_tdt), \text{with } \mu_t \text{ a bounded integrable function of time} $$ The way I would solve this would be to use that: $$ X_t = ...
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1answer
12 views

question about martingale inequality (similar to the condition of submartingale convergence)

if $X_n$ is a submartingale which can be decomposed as $X_n=M_n-N_n$, where $M_n$ is a martingale, and $N_n$ is a non-negative supermartingale. Can we get the conclusion that $\sup_n\mathbb E ...
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1answer
26 views

Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid \int^{\tau}_{0} f(t)d\omega(t)\mid^2=E[\int^{\tau}_{0} f^2(t)dt]$.

Suppose $f \in L^{2}_{\omega} [0, \infty]$, and $\tau$ is a stopping time such that $E[\int^{\tau}_{0} f^2(t)dt]<\infty$. Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid ...
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26 views

Geometric Ornstein-Uhlenbeck process

Let the Geometric Ornstein-Uhlenbeck process be defined as: $dV_t=\theta(\mu-V_t)V_tdt+σV_tdW_t$ Does anyone know of a solution or a reference for where a solution may be found? Thanks!
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1answer
33 views

Why writing $[X,Y]_t$ as $dX_t dY_t$ is so called “abuse of notation”

Why writing $d[X,Y]_t$ as $dX_t dY_t$ or $[B]_t$ as $\int_0^tdt$ is so called "abuse of notation"? Is it because $[B]_t \rightarrow \int_0^tdt$ a.s. but they are not equal?
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0answers
13 views

Moments of the integrated Bessel process

I am trying to compute the moments of the integrated and the integrated-inverse Bessel process. For simplicity, if $X_t$ is a BES$(d)$ assuming $d>2$, I am trying then to compute $$\mathbb E ...
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0answers
23 views

Show that $\mathbb{E}Y_t^3 =0$

Let $\{W_t\}_{t\ge0}$ be standard Wiener process. Show that process $\{Y_t\}_{t\ge0}$ defined below $$Y_t:=\int_0^t W_s ds$$ satisfies the condition $\mathbb{E}Y_t^3=0$. Firstly, I calculate ...
3
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1answer
221 views

Integral of a Gaussian process

Let $(\Omega,\Sigma,P)$ be a probability space and $X: [0,\infty) \times \Omega \to \mathbb{R}$ be a Gaussian process (i.e. all finite linear combinations $\sum_i a_i X_{t_i}$ are Gaussian random ...
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0answers
20 views

What does it mean by totality of borel cylinder set?

I understand what cylinder set is, but what does it mean by totality of cylinder set? I encounter this term in stochastic book quite often but I do not get the idea quite well. Does the totality mean ...