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
79 views

Can an Itō integral be $\infty$?

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$? Thanks!
0
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2answers
120 views

A mean square derivative

I'm doing an exercise where I have to check some properties about these two stochastical processes: $X(t)=At+B\;\;$ and $\;\;Y(t)=\frac{1}{t}\displaystyle\int_{0}^{t}X(\tau)\;d\tau$, $t>0$. ...
0
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1answer
126 views

Riemann integral of a function of the Wiener process

I'm trying to solve this exercise: $\bullet$ Find mean and variance of the next stochastical process, and prove it is a second order stationary process: ...
1
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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} ...
2
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1answer
55 views

Approximation of Stochastic integral with Stieltjes integrals

Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is ...
0
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1answer
109 views

Applying Ito To Geometric Brownian Motion

I'm trying to understand the example problem on the Wikipedia page for Ito's Lemma and need it dumbed down a little bit. $$dS = S(\sigma dB + \mu dt)$$ $$ f(S) = log(S) $$ Given Ito's lemma, ...
4
votes
1answer
182 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
2
votes
5answers
319 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
0
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1answer
92 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
2
votes
1answer
141 views

Why can I exchange the order of integration in a multiple Ito stochastic integral?

Stochastic Processes for Physicists by Jacobs says that we can exchange the order of a multiple Ito stochastic integral, giving the example: I don't see how this works either for a regular integral ...
0
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1answer
28 views

find the soultion $Y(t)$ of the SDE $dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$

find the soultion $Y(t)$ of the SDE $$dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$$ as a function of the inital conditon $Y(0) = y_0$ where $\theta$, $\gamma$ and $\sigma$ are ...
0
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2answers
352 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
0
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1answer
60 views

Questions around the establish of Ito integral

I got a some detailed questions on the Ito integral and hope someone can help. I'm reading Chap 3 of Oksendal's SDE book. There he establishes the Ito integral and the Ito isometry for simple ...
0
votes
1answer
156 views

Preservation of Martingale property

Can someone help me to prove this? If possible I'd like the prove can avoid the use of local martingale. Prove the Ito integral $\int_0^T \Delta_t(\omega) dW_t(\omega)$ is a martingale if $E[\int_0^T ...
0
votes
1answer
203 views

Mean and variance of a brownian bridge

I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$. $$X_t = y + ...
1
vote
1answer
44 views

Not using stochasstic integral how to prove $E\int_0^T W^2(t)dt<+\infty$?

Can anyone help me to prove this? Suppose $W_t$ ~ $N(0,t)$, then not using stochasstic integral (or anything related with Ito) how to prove $E\int_0^T W^2(t)dt<+\infty$? Thanks.
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
votes
1answer
103 views

Characteristics of stochastic integral?

I need to describe a couple of integrals which are supposed to be evaluated in terms of Ito calculus. $$ I_1 = \int_0^t e^{-2\tau}dW(\tau); \\ I_2 = \int_0^t e^{-3 W(\tau)} dW(\tau); $$ Here ...
0
votes
0answers
105 views

Is Brownian motion an adapted process?

In establishing theorems in stochastic calculus, a basic stochastic integral is defined as $\int^T_0 \Delta(t) dW(t)$, where $\Delta(t)$ is an adapted process, i.e. $F(t)$ measurable at time $t$. ...
0
votes
0answers
61 views

Expected value of the product of multiple Stratonovich Integrals

I want to calculate: $\mathbb{E}(J_1* J_{10}* J_{10} *J_{110})$ where $J_1=\int 1 dW$, $J_{10}=\int\int 1 dW dt$, $J_{110}= \int \int \int 1 dW dW dt$ are multiple Stratonovich Integrals over the ...
0
votes
1answer
165 views

how to do such stochastic integration $dS = a S^b dt + c S dW$?

How to do stochastic integration $dS = a S^b dt + c S dW$, where $a$, $b$ and $c$ are constant, $b > 0$, and $W$ is the Wiener process. I know how to do integration for $dS = aS dt + cS dW$, or ...
2
votes
1answer
133 views

Girsanov transformation and preservation of independence

If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original ...
1
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1answer
90 views

Need to Prove Result in Stochastic Calculus using Ito's Lemma

I can't figure out where : \begin{align} \delta^2\,dt\\ \end{align} comes from. Consider the process $$ d\sqrt{v} = = (\alpha - \beta\sqrt{v})\,dt + \delta \,dW $$ Here $\alpha, \beta,$ and $\delta$ ...
2
votes
1answer
112 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
2
votes
1answer
94 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
1
vote
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 ...
2
votes
2answers
67 views

The answer of the following stochastic differential equation

I want to solve the following stochastic differential equation $$dX_t=(a(t)+b(t)X_t)dt+(c(t)+d(t)X_t)dB_t$$ where $a,b,c,d$ are continious functions and we have the initial condition $X_0=x$ .
2
votes
1answer
103 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
2
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1answer
224 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
482 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
92 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
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1answer
110 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
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 ...
0
votes
1answer
97 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
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1answer
129 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
63 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
134 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
156 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 ...
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vote
2answers
195 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
161 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
143 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
84 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
90 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
154 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
122 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
238 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
79 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
108 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 ...