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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96 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
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480 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
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125 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
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0answers
91 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
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47 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
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146 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
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144 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
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163 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
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192 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 ...
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296 views

Ito's Lemma application

$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process. Find $dZ$ ?
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67 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
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41 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
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0answers
691 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
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0answers
80 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, $\partial_BX_t=\sigma X_t$, ...
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0answers
21 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.
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40 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
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0answers
51 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
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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 ...
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106 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 ...
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122 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
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61 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
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0answers
61 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
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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
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0answers
42 views

First-Exit time in 2-dimensional problem

Could someone recommend me some books or papers related to this problem?
2
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0answers
67 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
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0answers
90 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
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0answers
68 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
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0answers
396 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 ...
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137 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 ...
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0answers
30 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 ...
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70 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 ...
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0answers
17 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 ...
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0answers
17 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 $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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0answers
33 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) ...
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29 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 ...
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0answers
44 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 ...
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0answers
68 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 ...
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0answers
38 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 ...
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16 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 ...
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0answers
48 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
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0answers
96 views

Generators of difference of two poisson processes

Assume $X_1(t)$, $X_2(t)$ are two poisson processes with parameters $\rho_1$, $\rho_2$ accordingly. Suppose $Z(t)=X_1(t)=X_1(t)-X_2(t)$. At first I'm interest in knowing generators of the process ...
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75 views

Change of variables for stochastic processus

Let $H$ be a previsible locally bounded process, and let $X$ be a continuous local martingale. If $T$ is a stopping time and $X^T=(X_{t+T}-X_{T},t\geq 0) $ then ...
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46 views

Supermartingale Lemma + related problems

Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
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0answers
64 views

Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$

I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
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0answers
407 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
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0answers
189 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
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0answers
624 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
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0answers
286 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...
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0answers
43 views

a question about covariation in stochastic integration

Let H, K be bounded previsibe process. M, N be two local martingales. How can I prove $d<H.M, K.N>_t = H_tK_td<M,N>_t$ $<M>$ means the quadratic variation of M. Thanks
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0answers
178 views

Constructing Ito integral for adapted process

I am trying to construct Ito integral for adapted process. However, I am stuck at some point. Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...