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

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1answer
34 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 ...
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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? ...
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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 $g$?
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1answer
29 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 ...
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1answer
38 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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2answers
115 views

Deriving Black Scholes using CAPM

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from ...
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0answers
40 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 ...
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14 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 ...
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1answer
83 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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1answer
53 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 ...
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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 ...
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1answer
13 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 ...
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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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1answer
90 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
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0answers
28 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 ...
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1answer
70 views

Cadlag process integration

Let $A,B$ be non-decreasing cadlag processes such that $A_0 = B_0 = 0$ and limits $A_\infty = \lim_{t \to \infty} A_t$ and $B_\infty = \lim_{t \to \infty} B_t$ are finite. I am trying to prove that ...
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2answers
322 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
38 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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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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2answers
46 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 $y = e^{w(t)}$
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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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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 ...
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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 ...
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1answer
22 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 ...
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1answer
30 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 ...
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1answer
62 views

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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0answers
24 views

Fixed Point involving limits of Integrals

Assume $\omega$ is a random variable with a p.d.f $f(\omega)$. There is a function $\lambda(\omega):[0,1]\rightarrow[0,1]$ such that $\int_0^1\lambda(\omega)f(\omega)d\omega=\bar{\lambda}$ with ...
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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 ...
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1answer
16 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 ...
3
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1answer
38 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
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1answer
37 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 ...
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2answers
33 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
26 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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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 ...
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1answer
34 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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0answers
47 views

Solving an expectation related to CIR process

I encounter the following question Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the ...
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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
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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25 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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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 ...
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0answers
37 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
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16 views

Stochastic Calculus for Pure math

Is there any use of taking Stochastic Calculus for pure math career (not financial math)? I am undergraduate and I've already taken courses in Calculus, Analysis and Real Variables. Thanks!
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1answer
154 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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1answer
79 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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33 views

Jump-Diffusion Process: How to calculate the expectation of integral of S(t)

Having a jump-diffuion process $S(t)$ and the transition density $f_{dS(t)}(x)$. How can I calculate the Expectation of the integral of $S(t)$ between two instants $t_0$ and $t_1$? $S(t_0)$ is ...
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22 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
3
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1answer
51 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
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1answer
45 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 ...