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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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)= \frac{32}{1875}...
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1answer
46 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 \sqrt{1+...
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0answers
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 ...
1
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1answer
70 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, X_t,...
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26 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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66 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) 1-...
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0answers
45 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
72 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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0answers
105 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\{ \begin{array}{...
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0answers
33 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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0answers
17 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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0answers
30 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 $(\Omega,\...
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0answers
42 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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0answers
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
122 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
31 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 ...
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0answers
45 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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0answers
42 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) \end{...
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0answers
22 views

How can I formally arrive at solution of “deterministic SDE”

Let $dX_t=\mu X_t dt+\sigma X_t dW_t$. We know that this is a shorthand for integral equation: $X_t=X_0+\int_0^t\mu X_s ds + \int_0^t\sigma X_s dW_s$ Now: what if our equation looks like this $dX_t=...
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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 \}=E\{\...
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0answers
28 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
31 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 \int^{\tau}_{...
1
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1answer
38 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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24 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
13 views

Which point is the correct one when integrating w.r.t a discrete martingale?

first post here so I will try to explain as well as possible. In Durrett's book of Brownian Motion and Martingales, he uses the following example: $$ X_t = \begin{cases} 0, t<T \\ \xi, t\geq T \...
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2answers
60 views

Why is a stochastic integral w.r.t a martingale always a local martingale?

In my course on stochastic calculus, the professor mentioned that stochastic integral w.r.t a martingale always a local martingale? How can I rigorously show this? I know that when integrating wrt to ...
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2answers
365 views

Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
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0answers
61 views

Strong Markov property of Ito Diffusion - why must the stopping time be a.s. finite ? (Oksendal 6th edition p117 )

I am reading the proof of the Strong Markov property for Ito diffusions In Oksendal 6th edition p117 Theorem 7.2.4, and I do not understand where the fact that the stopping time has to be almost ...
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2answers
36 views

Ito's Lemma Simple Application

I need help applying Ito's Lemma to show a given result. $B_t$ is standard Brownian motion $dS_t = 0.4 S_tdt + 0.5 S_tdB_t$ I need to find $dlog(S_t)$ I am told it is $(0.4-1/8)dt + 0.5 dB_t$ ...
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1answer
49 views

what's the usage of purely discontinuous martingale in stochastic integral?

Recently I'm reading Jacod's Limit Theorems for Stochastic Process ,chapter 1 and I'm confused with the general stochastic integral for semimartingales. $H$ is locally bounded predictable process. ...
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1answer
91 views

Motivation behind Ito integral

Today my professor introduced the Ito integral as a way to make sense of $$\int \sigma(u) \cdot "noise"du$$ where noise is modeled as Brownian motion. He then said: With Riemann integrals you ...
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1answer
26 views

Verifying the identity $E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du$

I am doing the following exercise: The thing I am struggling with is the identity given in the hint: $$ E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du $$ I am unable to ...
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22 views

Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path. If we equip the space of continuous function starting from $0$, ...
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1answer
42 views

Why is $dX_t=X_t(\mu_t dt+ \sigma_t dW_t)$ an Ito process?

In solving the SDE $dX_t=X_t(\mu_t dt+ \sigma_t dW_t)$ we pick $Y_t=ln X_t$ and then apply Ito's lemma on the twice differential function $f(x)=ln (x)$ .But then why is $X_t$ anIto's prcess given ...
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63 views

Multiple Wiener Integral by Ito

In the context of Wiener-Ito chaos expansion, I had a look at Ito's paper "Multiple Wiener Integral", 1951. I am puzzled by his last result, theorem 5.1, that a multiple Wiener integral $$I_p(f)\...
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21 views

Verify a stochastic integral has normal distribution.

A well-known result is that:if $\sigma$ is a non-random process,then $$\int_0^T\sigma_t\,dW_t\sim N(0,\int_0^T\sigma_t^2\,dt)$$ ( from Shreve's "Stochastic Calculus for Finance" thm 4.4.9) by means of ...
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1answer
35 views

Simple differential equation and Integral Ito

With stochastic differential equation dx(t) = dW (t), and knowing that all integrals occurring are integral Ito. Witch variable changes y = tx. How I can prove? integral between 0 and t[sdW(s)] = tW(...
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1answer
80 views

How can a random variable have random variance?

This seems counter-intuitive to me since variance is a difference of expectations and afaik, unconditional expectation is a real number. Apparently, $X_t$ where $dX_t = Y_t dW_t$, where $Y_t$ is an ...
2
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1answer
49 views

How to find the distribution of the following stochastic integral of a geometric Brownian motion?

$K_{\phi,\lambda}(r)=\int_{0}^{r}\exp\{(r-s)\phi+\lambda(W_r-W_s)\}dB_s$ where $W$ and $B$ are independent standard Brownian motions, and $(\phi,\lambda) \in \mathbb{R} \times \mathbb{R}_+ $ The ...
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0answers
17 views

Stochastic Integrations with respect to non-martingale

Does there exists a theory of stochastic integration with respect to processes which are not local-martingales? For example if I have a general stochastic process $X_t$, can I integrate certain other ...
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1answer
29 views

application of Holder's inequality from Oksendal's book on SDEs

I am following the proof of the existence of solutions of SDE: let $b(t,x)$ and $\sigma(t,x)$ be Lipschitz continuous and consider the following SDE $dX_t=b(t,X_t)dt + \sigma(t,X_t)dB_t$. Define $...
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0answers
43 views

Time integral of Brownian motion's running maximum

Let $\mu \geq 0$ and consider $B_{\mu}(t) := B(t) + \mu t$ a one-dimensional BM with drift $\mu,$ and let $M_t := \max_{0 \leq s \leq t} B_{\mu}(t)$ be its running maximum. My question involves two ...
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0answers
44 views

How to solve a nonlinear SDE analytically

I have a numerical solution for the following equation: x'(t)= x(t) - x^3(t) + n(t) where n(t) is a white gaussian noise with zero mean and unit variance. I am a bit confused on how to go about the ...
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0answers
15 views

Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$ dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k, $$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
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0answers
69 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
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0answers
13 views

How to compute this integral using Ito isometry? [duplicate]

I am trying to evaluate the following integral: $E\Bigg[\Bigg(\int^{t}_{0} \frac{B_s}{e}1\big(B_s\in(-e,e)\big)\Bigg)^2\Bigg]$ I cannot figure out how to apply Ito isometry when the indicator ...
4
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1answer
135 views

Construction of Ito integral

This is in regards to constructing the Ito integral, specifically the first step of approximating bounded, continuous functions by elementary functions. Let $(\Omega, \mathcal{F}, P)$ be a ...
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1answer
66 views

Do we have the following?

Suppose the following integrals \begin{equation} \int_t^T X_s \, ds\ \text{ and }\ \int_t^T Y_s \, ds \end{equation} are well-defined, where $X_s$ and $Y_s$ are continuous stochastic process. Do we ...
2
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0answers
64 views

Integral with respect to Brownian motion, Variance

Good day. Imagene we have a martingale $M(t)=\int_0^t f(s)dB(s)$ which satisfies Dambis-Dubins-Schwarz Theorem. At the same time $M(t)^2 - <M>(t)$ is a Martingale starting in $0$ as well. If i ...