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.

learn more… | top users | synonyms

5
votes
0answers
126 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
5
votes
0answers
104 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
5
votes
0answers
527 views

Ito's lemma and application

Can someone help me apply Ito's lemma to the function $f(t,x,k)$ where t is the time and x,k dimensions where x and k refer to dynamics $dX(t)=\mu(t)dt+\sigma(t)dB(t)$ ...
4
votes
0answers
151 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
4
votes
0answers
125 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
votes
0answers
55 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
4
votes
0answers
161 views

Integrating the inverse of a squared bessel process - integrability

Let $X_t$ be a 4-dimension Squared Bessel Process (BESQ-4). Let $M_t$ be a continuous true martingale. Question: Does $\int_0^t \frac{1}{X_s}dH_s$ exist? If so, is it only a local or a true ...
4
votes
0answers
147 views

Calculating $\mathbb{E}[\int_0^T N_{t-} dS_t]$ - an expectation of a simple stochastic integral.

I came across some nasty stochastic integral of which I'd like to calculate the expected value" $\mathbb{E}[\int_0^T N_{t-} dS_t]$ where $N_t$ is a Poisson process and $S_t$ is, say, a geometric ...
4
votes
0answers
177 views

Observable and unobservable parameters of stochastic processes

Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...
4
votes
0answers
198 views

stochastic differential equation

Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion. need to find the PDE for the transition density of X at ...
3
votes
0answers
49 views

Characterize the limit of an O-U process: $dX_t = -\tfrac{\mu}{\theta} X_t dt + \tfrac{\sigma}{\theta^{1/2}} dW_t$ as $\lim_{\theta \to 0}$.

Standard O-U Formulas: Take the Ornstein–Uhlenbeck process defined by the SDE $$ dX_t = -\frac{\mu}{\theta} X_t dt + \frac{\sigma}{\theta^{1/2}} dW_t $$ where $\mu > 0, \theta > 0, $ and ...
3
votes
0answers
82 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
3
votes
0answers
71 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
3
votes
0answers
47 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
3
votes
0answers
742 views

Doleans-Dade exponential formula

How do I apply the Doleans-Dade exponential formula for the following levy stochastic differential equation: $$dZ_t=Z_t\left(\theta_1(t)dW_t^{(1)} +\theta_2(t)dW_t^{(2)}+\int_\mathbb R ...
2
votes
0answers
42 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...
2
votes
0answers
16 views

Numerical integration scheme for stochastic system driven by colored noise (filtered white noise)

I have given quite a few hours to this problem, but I seem to be getting nowhere. Can anyone just give a hint or point towards a text on where to go looking for the concept and solution.
2
votes
0answers
30 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
2
votes
0answers
116 views

How to solve this SDE ? stuck half way

Problem: $dX_t = \sigma X_tdB_t$, $X_0=x$. $dY_t=X_tdt-Z_tdt$ find $Y_t$, where $Z_t$ is a control and $B_t$ is standard Brownian motion. My attempt: From Ito's lemma, $\partial_BX_t=\sigma X_t$, ...
2
votes
0answers
34 views

Question about a Bessel process

Are there any explicit path solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.
2
votes
0answers
55 views

Mean-value like result for stochastic integrals

I'm working through Protter's book on stochastic integration; this is problem 16 from chapter 2. I can't seem to crack it--maybe someone here can give me a hint? Let B be standard Brownian and H be a ...
2
votes
0answers
59 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
2
votes
0answers
33 views

markov spectral radius independent of states?

Let $\Pi$ be a stochastic matrix of an irreducible markov chain. We define the spectral radius of $\Pi$ as: $\rho(\Pi) := \limsup_{n \to \infty} \left( \pi^{(n)}_{(a,b)} \right)^{\frac{1}{n}}$ Why ...
2
votes
0answers
117 views

Clarification in stochastic integration

In the book "Stochastic Processes" by Bass R.F. when he constructs the Stochastic Integral, at some point he defines for $Y$ predictable $$||Y||_2= \left(\mathbb E \int_0^{\infty}Y_t^2\text{d} \langle ...
2
votes
0answers
94 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
2
votes
0answers
67 views

an exetension of Doob's inequality

Doob's inequality gives an estimation of $$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$ where $X$ is a martingale. Now I wonder how to estimate $$\mathbb{P}(\sup_{0\leq t,s\leq 1, ...
2
votes
0answers
69 views

How to calculate the following expectation

I have a problem to find the expectation of the following expression, $$E\left[W_T e^{\int_0^T(W_s)ds}\right].$$ Here, $W_T$ is a Brownian motion. Any suggestions as to how to proceed with it? Many ...
2
votes
0answers
43 views

First-Exit time in 2-dimensional problem

Could someone recommend me some books or papers related to this problem?
2
votes
0answers
73 views

an example indicating the relation between Brownian motion and PDE

I have a question: Let $(B_t)_{t\geq 0}$ be a brownian motion. Consider the following function $u(x)$ defined by ...
2
votes
0answers
94 views

a pair of Stochastic Differential Equations

I'm trying to complete a course on SDEs and I need to solve two stochastic differential equations. They are supposed to be easy, but I'm still a beginner and to be honest I'm quite stuck. The pair of ...
2
votes
0answers
70 views

Are affine SDEs invertible?

If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves ...
2
votes
0answers
468 views

Why is it true that the continuous local martingale with quadratic variation “t” is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in ...
2
votes
0answers
142 views

Stochastic differential equation

Using stochastic(!) methods find explicit solution to each of the two ($i = 1, 2$) initial value problems $$\partial_t u(t, x) = \frac{1}{2} \beta^2 \partial_x^ 2 u(t, x) + (−\alpha x + \gamma ...
1
vote
0answers
20 views

Profit Maximization

I have listed a homework problem below that I have been working on. How do I get the expected number sold/expected number unsold/expected number lost if I do not have the pdf for the demand? Am I ...
1
vote
0answers
44 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
1
vote
0answers
71 views

Expected value of a brownian motion times the deterministic integral of a brownian motion

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, $E (B_t \int_0^tB_s^3ds)$ = ? I try to turn the expected value into a double integral by rewriting the $B_t$ term as 1) $E(\int_0^t ...
1
vote
0answers
25 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
1
vote
0answers
70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
1
vote
0answers
114 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
1
vote
0answers
70 views

Measurability of solution of diffusion equation in sub-sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
1
vote
0answers
48 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf ...
1
vote
0answers
32 views

Stochastic integral density of simple functions no1

I am trying to understand proposition 2.6 page p.134 from Karatza's book Brownian motion and stochastic calculus. If $M$ is continuous square integrable martingale on $(\Omega, \mathcal{F}, P)$ and ...
1
vote
0answers
29 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
1
vote
0answers
36 views

Stochastic integral in Tanaka formula

Tanaka's formula is the following result $$|B_t| = \int_0^t \text{sgn}(B_s)\, dB_s + L_t$$ I can see how to show that the stochastic integral $$M_t = \int_0^t \text{sgn}(B_s)\, dB_s$$ is a martingale ...
1
vote
0answers
34 views

Conditional Expected Value of Occurrence Time in Stochastic Process

I have a stochastic process defined by the intensity function $\lambda(t:F_t)$ where $t$ is time and $F_t$ is the filtration process. The stochastic process is self-exciting and models the occurrence ...
1
vote
0answers
50 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
1
vote
0answers
42 views

stochastic integration with respect to quadratic variation

I have been studying stochastic integral with respect to Brownian motion. At some point my professor generalized our approach such that we are able to integrate with respect to general Martingales. ...
1
vote
0answers
78 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
1
vote
0answers
99 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
1
vote
0answers
21 views

Differential of $ \int_{0}^{t} e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau} c(s)ds $

I think -- using the chain rule -- it's $$ e^{\int_{t}^{t}\cdots d\tau} c(t)dt \cdot e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau}\cdot ...