This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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Diffusion Process Expectation Smoothness Condition

Consider a diffusion process on a sample space $\Omega$ $$dx_t = \mu(\omega,t)dt+\sigma(\omega,t)dB_t,\, \forall\omega\in\Omega$$ where $B_t$ is the standard Brownian motion on the filtration ...
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1answer
22 views

Strong solution SDE - independence of initial conditiion

I am currently studying the existence and uniqueness of strong solutions of SDEs of type $$\left[\begin{array}{l} \, dX_t=\mu(t,X_t)\,\mathrm{d}t+\sigma(t,X_t)\,\mathrm{d}W_t\\ ...
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47 views

Does this make sense?

Can I write this? Let $W_s$ be a Wiener process and let $x_s$ be a stochastic square integrable process adapted to the filtration generated by $W$. Is such an expectation nonsensical? And if not, how ...
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30 views

When does convergence in quadratic variation imply a uniform convergence or vice versa?

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ the quadratic variation of a path $x\colon [0,T]\to \mathbb{R}$ is defined by $$ [x]=\lim_{n\to ...
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0answers
19 views

Fractional moments of stochastic integrals

I want to bound the moments of stochastic integrals as $$E\left|\int_0^1 f(s)d L_s\right|^\alpha,\alpha\in[0,1],$$ where $(L_s)_{s\ge0}$ is a Lévy process with Gaussian part $\sigma^2$ and Lévy ...
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1answer
41 views

Non-linear SDE: how to?

$$ \newcommand{\mcl}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\avg}[1]{\langle#1 \rangle} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\bck}[1]{\left\{ #1 \right\}} ...
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0answers
32 views

Integrability condition of stochastic Fubini's theorem

This is a special case of stochastic Fubini's theorem for deterministic integrands: Let $f : [0,t] \times [0,t] \to\mathbb{R}$ be measurable. Assume that $$ \int_0^t \left( \int_0^t |f(r,s)|^2 dr ...
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31 views

Ito integral via simple process when the integrand is C^1

I have the following problem. Let $H_t$ be an adapted process with trajectories a.s. of class $C^1$ on $\mathbb{R}_{+}$. Compute using simple process $\int_o^t H_s d B_s$. My idea is to firstly set ...
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0answers
24 views

Stochastic calculus for continuous time Markov chains

I have absolved a course on stochastic analysis, i.e. integrals with respect to the brownian motion. Now I know that there is a theory of stochastic calculus for diskrete matringales, however I was ...
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15 views

Problems with finding marginal density from joint density function

For two absolute continuos stochastic variables I have that the joint density function is 8y if 0 I now have to calculate/ show what the marginal density functions are. I got the right answer for y ...
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1answer
11 views

Find E(X^-1) for stochastic variable

Let $X$ be a stochastic variable with density function: $f(x)=x\exp(-x)$ if $x>0$ and $0$ otherwise. Show that $E(X^{-1} )=1$. I believe I have to integrate but is it simple $x\exp(-x)$ I ...
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1answer
30 views

Check process is a martingale

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a ...
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1answer
55 views

Two Ito processes : are they a 2-dim Brownian motion?

I am stuck with the following problem : I have a Brownian motion $B_t$ and an Ito process $$X_t:=\int_0^t sgn(B_s)\ d B_s,$$ where $sgn(x)=1$ when $x \geq 0$ and $sgn(x)=-1$ when $x<0$. I have ...
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1answer
85 views

Use Ito's Formula to prove following identity

Again, I am not sure how the following works; Could someone please give me an almost stupidly detailed explanation of why/what is happening in the part below. First, the question itself; Q. $B_t$ ...
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17 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = ...
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31 views

Versions of Tanaka's SDE

Consider the following versions: $$dX_t=x_0+sgn(X_t)dW_t \tag1$$ $$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$ $$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$ SDE (1) is a classical example of SDE with ...
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0answers
35 views

Variance of a simple Ito integral

I am trying to apply Ito's lemma to compute variance of the following integral $X(t) = \int_{0}^t W(s)dW(s),$ where $W(t)$ is a Wienner process. Could you please check my calculations? $$E(X(t)) = ...
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3answers
194 views

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
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0answers
20 views

Left limit poisson process (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . How could I show that $\int_{0}^{t} N_{s-} dN_s =1/2 (N_t^2-N_t)$ Thank you
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34 views

Martingale (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . I want to verify that the process Y given by $Y_t = \int_{0}^{t} N_{s-} dM_s$ is ...
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1answer
14 views

Can I swap conditional expectation and limit

My problem is the following : let $B_t$ be a standard Brownian motion and $H_t$ a progressive measurable process such that $\mathbb{E}\left(\int_0^{+\infty} H_t^2\ dt \right)<+\infty$. Denote ...
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1answer
28 views

integral of square of Brownian motion

What is expectation of $$\int_0^t B(s)^2ds$$ where $B(s) is standard Brownian motion. Is the integral a well known random variable?
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63 views

Integral inequality of transformed integrand with second order stochastic dominance flavor

Let $f,g : [0,1] \rightarrow [0,1]$ be two functions such that for all $x \in [0,1]$ $\int_0^x f(t) dt \geq \int_0^x g(t) dt$ and $\int_0^1 f(t) dt = \int_0^1 g(t) dt.$ Can I conclude that ...
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78 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 ...
3
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1answer
28 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 ...
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1answer
68 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 ...
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0answers
23 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 $C^1$. ...
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1answer
23 views

Partial differential equations involving Feynman-Kac

I am working on solving the following pde on $[0,T]$; $$f_t(t,x) + 2tf_x(t,x) + \frac{t^4}{2}f_{xx}(t,x) = 0 \qquad f(T,x) = x^2 = h(x)$$ By Feynman-Kac, the solution is given by $$f(t, X_t) = ...
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23 views

Integrating Wishart density

I have several points $\textbf{s} = s_1,...,s_n$ which follow Wishart distribution. In one of my problem, I have to integrate this Wishart pdf over a ball of radius $r$ at origin in $\mathbf{R}^2$ ...
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19 views

Ito integral of continuous function is Gaussian process

Let $f\in C[0,T]$. Then $X_t := \int_0^t f(s) dB_s$ is a Gaussian process with independent increments, with zero mean and covariance $\mathbb{E}(X_s X_t) = \int_0^{s\wedge t} f^2(s) \ ds$. This can ...
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1answer
30 views

Why is the isometry of It\^o integral called so? [duplicate]

For functions $f$ satisfying appropriate (good) conditions, the following property is called to be isometric. $$ ...
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1answer
45 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 ...
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20 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? ...
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2answers
23 views

Fourier transform with the derivative of a function

I have to identify the Fourier transform, defined as $\widehat f(x)=\displaystyle \int_{\mathbb R} e^{-ixy}f(y) dy$ As a task, I have to calculate the the fourier transform of $g(x)= ...
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1answer
44 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 ...
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35 views

Show that $e^{-rt}E\Phi(S_T)=S_0N(d_+)-Ke^{-rT}N_{d_-}$

Show that $e^{-rt}\mathbb E[\Phi(S_T)]=S_0N(d_+)-Ke^{-rT}N_{d_-}$ where $S_t=S_0e^{(r-\sigma ^2/2)t+\sigma W_t}$ for $t\in[0,T]$ , $W_t\sim \mathbb N(0,t)$ and $N$ is the cumulative density function ...
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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, ...
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21 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 ...
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54 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 ...
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39 views

Conditional expectation and Variance

I have the interest rate model: $r(t)= x(t)+y(t)+\phi(t) $ $r(0)=0 $ $dx(t)⁼-ax(t)dt+\sigma dW_1(t) $ $x(0)=0$ $dy(t)⁼-bx(t)dt+\nu dW_2(t) $ $y(0)=0$ $(W_1,W_2) $Brownian (2 dimensions) ...
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27 views

An Ito integral is normal if the integrand is a deterministic function

Why is an Ito integral normally distributed if the integrand is a deterministic function? This is constantly used in many proofs, and I often take it for granted.
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1answer
67 views

Can the integral of Brownian motion be expressed as a function of Brownian motion and time?

Let $W_t$ be standard Brownian motion, and define $$ X_t := \int_0^t W_s ~\textrm{d}s. $$ The marginal distributions of $X_t$ are easy to write down (see here), but it doesn't seem possible to express ...
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66 views

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ ...
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0answers
29 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 ...
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16 views

Integrating R.V with respect to time

I would like to compute the time integral of a random variable $X(t)$ given by $\int_s^t X(u) e^{k u} du$ where $X(t)$ is a CIR square root process $dX_t = k (\theta - X_ t) dt + \sigma \sqrt{X_t} ...
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28 views

Construction of the Itō integral with (local) martingales as integrators

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$. $\xi_i$ be a real-valued random variable on ...
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38 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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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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93 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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0answers
30 views

Generalization of the Ito formula

I have a question concerning Ito’s formula for semimartingales with jumps. I am familiar with Ito’s formula in the following setting: Let $X_t=X_0+M_t+A_t$ be an $\mathbb{R}^d$-valued continuous ...