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
100 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
513 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
144 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
112 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
53 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
157 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
146 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
172 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
195 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
72 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
70 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
45 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
723 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
25 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
104 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
32 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
46 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
54 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
32 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
114 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
161 views

When does almost sure convergence of stochastic integral imply $L^2$ convergence?

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes. ...
2
votes
0answers
85 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
64 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
68 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
70 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
449 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
140 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
42 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
32 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
24 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
28 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
32 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
37 views

Ito's integral from the definition

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
1
vote
0answers
29 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
45 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
37 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
69 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
92 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
18 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 ...
1
vote
0answers
22 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
1
vote
0answers
28 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
1
vote
0answers
38 views

Proving $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$, $f$ a $k$-Lipschitz function, is a continuous martingale

Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function. I would like to show that $(\int_0^t f(X_s) ...
1
vote
0answers
30 views

$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X$ for $C\in\mathcal F_t$?

Given a semi-martingale $X$ on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$, an integrand $A$ and a set $C\in\mathcal F_t$. Show: $$\int_t^T 1_C\cdot ...
1
vote
0answers
65 views

Expected value of stopping time of Stochastic Process.

I am trying to solve the following problem: Let $X$ be the strong solution of the following Stochastic Differential Equation: $\mathrm dX_t = sign(X_t)dt + \mathrm dW_t, X_0 = 0$, where $W_t$ is a ...
1
vote
0answers
72 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
1
vote
0answers
43 views

Clarification on the definition of the îto integral

I have a question regarding the îto integral. In the definition of the integral we basically take the limit in probability of the sum $\Sigma H(t_i)\cdot(B(t_{i+1})-B(t_i))$ for suitable $H$ and a ...
1
vote
0answers
17 views

quantile of Ito integral when integration limit goes to zero.

I woud like to calculate the Value at Risk of an Ito Integral in the following form in the limit! $$\lim_{\Delta t\to 0}\frac{1}{\Delta t}VaR_{q,t}\left[\int_t^{t+\Delta t}b(s,y(s))\pi_y^c(s,y(s))d ...