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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2
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1answer
242 views

Ito's formula for multivariable Ito integral

I'm having trouble finding something that I think should exist, which is an integral formula of the multivariable Ito lemma. Simply put, suppose I have a function $f$ of two stochastic processes, ...
5
votes
3answers
538 views

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
0
votes
1answer
98 views

An elementary example of Ito's integral

Let $m(t,\omega)=\sum_{j \ge 0}B_{(j+1)2^{-n}}(\omega)I_{[j.2^{-n},(j+1)2^{-n})}(t)$ where $B(t)$ is the Brownian motion and $I_[.]$ is the standard indicator function, Can some body explain me why ...
7
votes
1answer
112 views

Trying to integrate a stochastic RV, $\int_0^t sZ_s \, ds$

I'm not taking an official class (actuarial exams), some fellow "students" created a question (forum discussion), considering the integral in title. This is my attempt at a solution with no real ...
1
vote
0answers
81 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 ...
0
votes
1answer
101 views

Expectation of integral of involving geometric brownian motion

Compute $$\mathbb{E_P} \left( \exp{(\alpha W_t)} \int_0^t \exp{(\gamma W_u)} \,du \right)$$ where $\alpha$ and $\gamma$ are real numbers and $W_t$ is a Brownian Motion.
5
votes
1answer
141 views

existence/uniqueness of solution and Ito's formula

Given the Ito SDE $$ dX_t=a(X_t,t)dt + b(X_t,t) dB_t $$ where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions. Given a function $f(X_t,t) ∈ C^2$ ...
3
votes
1answer
67 views

stochastic integrals and inequalities (boundedness)

We have $X(t)=[X_1(t)\ X_2(t)\ X_3(t)\ \dots\ X_n(t)]$ and $Y(t)=[Y_1(t)\ Y_2(t)\ Y_3(t)\ \dots\ Y_n(t)]$ are two stochastic process such that: $$\sup E[Y_1^2] \leq K,$$ on $[t_0, T]$ with $K$ a ...
3
votes
2answers
138 views

Simple stochastic differential equation

Solve the following stochastic differential equation: $$ dX_t=X_t\,dt+dW_t. $$ Thank you very much for help! I even don't know where to start...
3
votes
1answer
161 views

question about Ito's formula

I'm currently learning about the Ito's lemma / formula In my textbook, a direct application of the formula is to compute quantities like that : (W is a Brownian motion) While trying to prove these ...
1
vote
2answers
211 views

How to solve this stochastic integrals?

how can I solve these two stochastic integrals? $$\int_0^T B_t\,dB_t$$ $$\int_0^T f(B_t)\,dB_t$$ where B_t is the BM. Thank you very very much!
0
votes
1answer
73 views

Local martingale and convergence theorem

I have $E[x(t)^2]\leq A\operatorname{exp}(Bt)+C/Bt$ it's clear that for a finite time less than $T$, x(t)^2 is a "local martingale" because $\lim E[x(t)^2]<\infty$. But one can see that if $t$ ...
2
votes
1answer
164 views

When is the following local martingale strict local martingale?

By Section 5.5 of the book [Karatzas and Shreve 1991], the following 1-d SDE has unique weak solution in the form of \begin{equation} d X_{t} = X_{t}^{\gamma} \cdot I_{\{X_{t}\ge 0\}} dW_{t}, \ ...
2
votes
1answer
145 views

Ito differential equation

Define $$X_t := \left( \begin{matrix} \cos W_t \\ \sin W_t \end{matrix} \right).$$ where $W = \left( W_t,\mathcal F_t \right) _{t\ge0}$ is a standard Wiener process. Find the Ito differential of X ...
2
votes
2answers
85 views

stochastic integral equation

For $0 \leq t \leq T$, define $$Z_t:=\exp {\left\lbrace \int_0^t X_sdW_s - \frac 12 \int_0^t X_s^2ds \right\rbrace }$$ Show that this process satisfies the stochastic integral equation ...
3
votes
1answer
92 views

Basic doubt about stochastic integrals over general local martingales

Consider $M = (M_t)$ is a continuous square integrable local martingale and $$ \mathbb H ^2(M):= \left \{ \psi =(\psi_t)\ \text{is a real previsible process s.t.,} \forall t\geq 0, \ \mathbb E\left ...
2
votes
1answer
159 views

generalized derivative of Wiener process

Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that ...
2
votes
1answer
124 views

Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion

Consider $B=(B_t)_{t\geq 0}$ $\mathcal F_t$ - brownian motion in $\mathbb R ^n, \ (n\geq 2)$ starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
1
vote
2answers
255 views

Moment generating function of a stochastic integral

Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then: $$ \mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
votes
1answer
82 views

Brownian motion and convergence in probability of step functions

For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
1
vote
1answer
117 views

$dX_t=1_{X_t\not=0} dW_t$

Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the stopping time $$ ...
0
votes
0answers
69 views

Product of predictable process and a characteristic function is integrable

Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that $$\int_0^T\theta_u dS_u\ge -a$$ for a $a>0$. Furthermore ...
0
votes
1answer
107 views

Itō's Lemma neglecting terms

In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$ I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small. ...
0
votes
1answer
49 views

Integral: Is there a closed form?

I wonder whether there is a closed form or way to compute explicitly: $$\int_0^t e^{\alpha s} dB_s$$ where $\alpha$ is just a real number and the integral is in the Itô sense. Thank you very much!
0
votes
1answer
90 views

Solve a special non-linear Backward SDE

It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.) How can I solve $dY_t=Z_tdW_t+ ...
1
vote
0answers
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 ...
1
vote
2answers
134 views

Conditional expectation of a functional of an Itô's semimartingale under its equivalent martingale measure

Consider a probability filtered space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfying the habitual conditions and is generated by $1 d $- ...
1
vote
1answer
162 views

Approximation of stochastic integral

Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
1
vote
1answer
109 views

Martingale inequality

Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, ...
7
votes
1answer
509 views

Ito's Lemma and Brownian Motion

Show by using Ito's Lemma, for $k \geq 2$ the following result hold. $$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$ where $W(t) = N(0,t)$ is standard Brownian motion. I think ...
0
votes
0answers
161 views

Intuition: integration of function with respect to stochastic process

Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function. With the above conditions, the following equality ...
3
votes
1answer
117 views

Is this stochastic integral well defined?

Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
4
votes
1answer
127 views

Computation of basic stochastic integral.

I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage, $$\text{Cov }(x_s,x_t)=\sigma^2 ...
6
votes
1answer
201 views

Very basic doubt about Itô's lemma

While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following $$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$ I had some doubt concerning the application of ...
2
votes
1answer
288 views

Ito integral almost sure and $L^2$ limit

why does one define the Ito integral as the $L^2$ limit, although it can be shown by Doob's martingale inequality and Borel-Cantelli lemma that there exists a t continuous version, which is ...
1
vote
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 ...
3
votes
1answer
488 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 ...
0
votes
1answer
210 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 ...
2
votes
1answer
575 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 = ...
2
votes
2answers
919 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 ...
0
votes
1answer
90 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 ...
-1
votes
1answer
218 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 $ $\tau_i$ ...
0
votes
2answers
95 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 ...
1
vote
1answer
915 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 ...
10
votes
1answer
387 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 ...
2
votes
1answer
198 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 ...
11
votes
3answers
356 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
votes
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
votes
1answer
162 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 \, ...
2
votes
1answer
125 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) = ...