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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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 ...
3
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1answer
392 views

Multidimensional infinitesimal generator of a jump-diffusion

Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE $$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$ where $\mu, \sigma$ and $\beta$ are ...
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1answer
186 views

Stochastic process as an Ito integral with time-dependent integrand

Will the following process $$r(t)=\int_0^ta(s,t)dW(s)$$ be adapted to the Brownian motion $W(s)$? Will $r(t)$ be an Ito process? Edit: Maybe I should rephrase it a bit. The question is: does ...
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1answer
462 views

Stochastic Diff Eq SDE

enter link description here Consider the following SDE $$d\sigma = a(\sigma,t)dt + b(\sigma,t)dW $$ The Forward Equation (FKE) is given by $$\frac{\partial p}{\partial t} = ...
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1answer
474 views

Stochastic Calc

(a) Consider the process $$ \mathrm d\sqrt{v} = (\alpha - \beta\sqrt{v})\mathrm dt + \delta \mathrm dW $$ Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that $$ \mathrm dv = ...
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2answers
661 views

Ito Isometry and quadratic variation

Here is a confusion regarding stochastic integrals. Let $Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
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1answer
78 views

Product rule of stochastic exponents

we know that for standard exponents, $(e^x)(e^y)=e^{(x+y)}$. What is the product rule for stochastic exponents? $E_n(U)E_n(V)=E_n(U+V+[U,V])$ where $U$ and $V$ are stocchastic sequences, $E_n$ is the ...
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1answer
197 views

How can I prove it is a martingale when there is a jump process

Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale. Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $ $\tau_i$ ...
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2answers
89 views

One stochastic integrability problem

On a lecture notes, there is a following arguement: To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
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1answer
684 views

Expectation and variance of this stochastic process

I am trying to compute the expectation and variance of the following stochastic process: $$ Z_t = \exp \left( \frac{1}{2} \int_0^t W_s \, dW_s \right) $$ where $W_t$ is a standard Brownian motion. I ...
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1answer
325 views

Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
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1answer
172 views

About stochastic differential equations

Consider, for all $x \in \mathbb R $, the process $\left( X_t^x\right)_{t\geq 0} $ unique solution of the following SDE: $$ X_t ^x =x + \int _0 ^t \sigma\left( X_s^x\right) ~dB_s + \int _0 ^t ...
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3answers
345 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
3
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1answer
34 views

How to deal with differential in Itô

Suppose I have two Brownian Motion $W$ and $B$ which are connected through Girsanov, i.e. $W_t=B_t-\int_0^t v(u,T)du$. Furthermore I have the following expression $$\exp{(\int_0^tv(u,T)-v(u,S) ...
3
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1answer
149 views

convergence ito integral

It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$ That means I showed that $\int_0^T S_n \, ...
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1answer
103 views

Laplace functional of a Poisson random measure with stochastic intensity

This is one of the problems from Cinlar's 2011 book - "Probability and Stochastics" (Chapter VI, page 262, exercise 2.36) : Let $N$ be a Poisson random measure on $R^{+}$, defined by $N(\omega, B) = ...
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1answer
643 views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
2
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2answers
156 views

How do I derive the Gaussian Mixture distribution of an Ito Integral?

I have a question about the distribution of an Ito Integral. Consider the integral $$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$ where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
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1answer
577 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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2answers
219 views

Basic stochastic integral

I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process? Thanks!
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1answer
71 views

how to derive this form using product formula

Suppose we have the following SDE $$dS(t) = S(t)(\mu(t)dt + \sigma(t)dW(t))=:S(t)dX(t)$$ where $W$ is a Brownian Motion and the processes $\mu,\sigma$ are well defined, such that the expression ...
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1answer
234 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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1answer
251 views

Funny problem about stochastic integrals and Ito' s lemma

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
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2answers
259 views

Explicit solution of a SDE

I'd like an explicit formula as a function of $W_t$ (standard brownien motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
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1answer
117 views

Upper bound for the $\sup$ of a martingale defined as a stochastic integral of a general continuous martingale

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
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1answer
63 views

Darboux versus stochastic integral

I don't know if my question is obscure. I'm astonished why there not mention the Darboux sums in the definition of stochastic integral
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1answer
1k views

Covariance of Brownian Bridge?

I am confused by this question. We all know that Brownian Bridge can also be expressed as: $$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$ Where the Brownian motion will end at b at $t ...
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2answers
658 views

Expectation of Brownian motion Integral

I want to calculate $\mathbb{E} \left[\left(\int_0^tB_s\text{d}B_s\right)^3\right]$ where $B_t$ is a standard Brownian motion. Using Ito's formula for $f:\mathbb{R}\rightarrow\mathbb{R}$ with ...
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1answer
254 views

Ito Isometry for conditional expectations

Is Ito's isometry true for conditional expectations too? I mean, is it true that:$$\mathbb{E}\left[\left(\int_0^tX_sdB_s\right)^2\ |\ \mathcal{F}_t^B\right]=\mathbb{E}\left[\int_0^tX^2_sds\ |\ ...
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2answers
115 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
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2answers
371 views

Conditional Expectation of integral of Wiener process

Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$ where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
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1answer
170 views

Show that $M_t$ is a Standard Brownian Motion

Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$ where $(B_t)_{t\geq0}$ is a Standard Brownian Motion. Show that $M$ is also a Standard Brownian Motion and compute ...
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67 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 ...
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395 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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1answer
59 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
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1answer
178 views

Stochastic representation formula

Consider the following boundary value problem in the domain $[0,T]$ x $R$ for an unknown function F. $\frac{\partial F}{\partial t}(t,x) + \mu(t,x)\frac{\partial F}{\partial x}(t,x) + \frac ...
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1answer
176 views

Expectation of stopping time

Let $X_t$be the solution to the SDE: $dX_t=-X_tdt+dB_t$, $X_0=0$ Then $X_t$ is the Ornstein–Uhlenbeck process $X_t=e^{-t}\int_0^te^sdB_s$. I want to calculate $\mathbb{E}[e^\tau X_\tau]$ when ...
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2answers
195 views

Ito Process $\Longrightarrow$ continuous semimartingale

I know that the Ito integral is defined in general for continuous semimartingales. But it can also be defined only for Ito processes. My question is if every process $X_t$ satisying a SDE of the form ...
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1answer
160 views

Stochastic process integration - Notation

I'm facing with a problem of notation and I hope stack could help me! Let $X(t)$ be a time-continuous stochastic process, with pdf $p_X(x, t)$. Let $g(x, t)$ be a generic function. Now, consider the ...
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1answer
291 views

Table of Ito Integrals

Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of? Did a search but didn't come up with anything and was wondering if anyone knew of ...
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118 views

Can we apply Ito's formula?

Suppose that we are given the following processes: $B=(B_t)_{t\geq0}\ $ a standard Brownian motion starting at zero, $I=I_t=\int_0^t|B_s|^2ds,\ S=S_t=\sup_{0\leq s\leq t} B_s$ for $t\geq0$ and a ...
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2answers
264 views

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
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87 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 ...
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1answer
172 views

expectation of a process of a multidimensional brownian motion

Let $B(t)=(B_{1}(t),B_{2}(t),B_{3}(t))$ be a standard three dimensional Brownian motion (i.e. it has independent components and starts at the origin). Now let $a=(a_{1},a_{2},a_{3})\neq(0,0,0)$ be a ...
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1answer
248 views

Stochastic integrals and new probability measures

Let $B$ be a standard Brownian motion on $(\Omega, \mathcal{F}, P, ({\mathcal{F}_t})_{t\ge0})$, where the filtration is the one generated by $B$. Fix a time interval $[0,T]$. Define the process $X$ as ...
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1answer
295 views

Show that this continuous local martingale is a martingale

We are given the following SDE: $$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$ and $$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$ We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
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1answer
75 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
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1answer
106 views

Bounded variation and continuous local part when using Ito's Formula

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...
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179 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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54 views

Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation

Let $B$ be a standard Brownian motion, and, $$ X_t=e^{\int_0^t f(B_s)ds}, $$ for some function $f$. What are the condition on $f$ for $X_t$ to be of finite variation? Let $Y_t=\int_0^t f(B_s)ds$, if ...