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

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2
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1answer
31 views

Closure of the set of elementary predictable stochastic processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t)_{t\ge 0}$ be a real-valued stochastic ...
2
votes
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 ...
1
vote
1answer
31 views

$\sin(W_T)$ 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, $X = W(T)$ ...
2
votes
1answer
42 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
2
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0answers
53 views

Absorbed brownian motion is a Markov process

I have been asked to prove that the Brownian motion absorbed at the origin is a Markov process. Formally, let $B_t^x$ be a Brownian motion originating from $x>0$ and let $\tau^x_0 = \inf\{t>0 : ...
2
votes
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 ...
3
votes
1answer
47 views

Ornstein–Uhlenbeck SDE.

I am trying to understand the solution to the following exercise, however it is kind of poorly written. Can someone please explain it to me? For $V = (V_t)$ the solution to the Ornstein-Uhlenbeck SDE ...
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2answers
26 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
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0answers
7 views

stationary process with discontinuous spectral distribution function

Let's say we have a zero mean stationary process $X_t$ with spectral distribution function $F$, then the autocovariance function of $X_t$ can be written as ...
3
votes
1answer
63 views

Stochastic continuity

Let $(X_t)_{t \in \mathbb{R}}$ be a square-integrable real-valued process with a continuous mean value function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ and a continuous covariance function ...
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0answers
9 views

Large Deviation Theory

Consider a differential equation of the form: $$dX_0 = f(X_\epsilon) dt$$ and it's perturbed form: $$dX_\epsilon = f(X_\epsilon) dt+ \epsilon dW(t)$$ It's well-known that if one assumes $f$ is ...
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0answers
21 views

Reflection principle application

I want to calculate the probability: \begin{equation*} P(W_4>2, \inf_{0\leq t\leq4} W_t >-1) \end{equation*} and $W$ is a Wiener process. I tried: \begin{equation*} P(W_4>2, \inf_{0\leq ...
0
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1answer
35 views

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 ...
0
votes
1answer
53 views

Application of Ito's formula

I have the following process: \begin{equation*} X_t= \exp \left(\int_{0}^{t}s \, dB_s-\frac{t^3}{6} \right), \end{equation*} where $B$ is a Browinan motion. My textbook asks to write Ito's formula ...
1
vote
1answer
13 views

Ito's formula for this stochastic differential - please explain this step?

Referring to those two lines, can someone please explain how those results were obtained? My understanding is, the following formula is being referenced: $$dV_t = dV(S_t,t) = \frac{\partial ...
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0answers
8 views

Finite-Difference Scheme for a Non-Linear PDE?

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
1
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1answer
35 views

Itô integral of an elementary process

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t,t\ge 0)$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t,t\ge 0)$ be a stochastic process on ...
0
votes
1answer
13 views

function of independent random variables

I have following question: If $X$ and $Y$ are independent, then are $g(X)$ and $g(Y)$ independent as well, for any real function $g$?
4
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1answer
46 views

Can we derive the PDE followed by a marginal transition probability density?

A pair of correlated stochastic processes follow the SDEs \begin{align} dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\ dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, ...
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0answers
41 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
2
votes
1answer
40 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even tough its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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0answers
16 views

How to calculate the variance of the argument of a complex number?

Given a number $s \in \mathbb{C}$ and the (Gaussian) variances of its components $\sigma^2(\Re(s))$ and $\sigma^2(\Im(s))$, how can I calculate the variance $\sigma^2(\arg(s))$ and the covariances ...
1
vote
1answer
14 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
0
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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 ...
2
votes
1answer
54 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
3
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0answers
29 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
1
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1answer
30 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
4
votes
1answer
87 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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0answers
18 views

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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votes
1answer
55 views

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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2answers
48 views

Ito Differential Equation example [closed]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
0
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0answers
40 views

Expectation of squared Ito integral

Let $\omega$ be a standard Brownian motion. How do you compute the expectation involving the square of an Ito integral: $ ...
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0answers
14 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
2
votes
1answer
31 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
1
vote
1answer
31 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
2
votes
1answer
17 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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0answers
22 views

Condition for the mgf of a stochastic integral to be finite

Fix $t>0$, let $B$ be a Brownian motion and let $\sigma$ be a previsible process such that $$\mathbb{E}\left[\text{exp}\left(\frac{1}{2}\int_0^t\sigma_s^2ds\right)\right]<\infty.$$ Then is ...
3
votes
1answer
39 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 ...
2
votes
1answer
38 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
1
vote
1answer
23 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
0
votes
1answer
19 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 ...
1
vote
2answers
34 views

System of Stochastic Diff Eq

How can I solve the system of stochastic differential equation $$dX_{1}=X_{2}dt+adW_{1}$$ $$dX_{2}=-X_{1}dt+bdW_{1}$$
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1answer
35 views

How do we apply Ito's lemma to a product of functions

In finance an optimal portfolio choice it is common to use some tools of stochastic calculus. Going through a book, I found the following statement, ...
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1answer
27 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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0answers
27 views

Application of Ito's Lemma to stochastic integrals

From my understanding, the Ito Integral is a random variable itself. Suppose we have $X_t=\int_0^t Z_udZ_u$. To find $dX_t$, I would think we can apply Ito's Lemma. However, how would the partial ...
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0answers
17 views

Proof: Sum of two independent gaussian vectors is a gaussian vector

I want to show that the sum of two independent gaussian vectors is a gaussian vector. We had, that a gaussian vector can be written as $X=A*Z+b$ where $A$ is a real matrix, $b$ is a real vector and ...
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0answers
11 views

Distribution of SDE numerically from Fokker-Planck.

I'm aware of some numerical methods related to SDEs such as Euler-Maruyama, Milstein etc. However, couldn't one also simulate the equivalent Fokker-Planck equation via finite element methods? This ...
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0answers
16 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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0answers
26 views

Is the expectation of an integral equal to the intergral of expectation?

I have to calculate the expectation of the integral between $t_0$ and $t_1$ of a random variable $S(t)$ can we say that: $E[\int^{t_1}_{t_0} S(t) dt]= \int^{t_1}_{t_0} E[S(t)] dt$?
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vote
2answers
51 views

Differential and Differential Equation - Difference in meaning?

I am a little confused, an exercise by a teacher has been set which says: For $X_t = 2e^{B_t}$ Where $B_t$ is brownian motion at time $t$. a) Find the stochastic differential $d(X_t)$ b) Find the ...