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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1answer
58 views

Optimal Investment Strategy

I am not sure to solve the following investment problem: I have an investor which receives an income $I_n\ge 0$ at the start of year $n$. The investor chooses a proportion $p_n\in[0,1]$ of this in ...
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0answers
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 ...
4
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1answer
153 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
5
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1answer
206 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
0
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1answer
46 views

Cadlag process integration

Let $A,B$ be non-decreasing cadlag processes such that $A_0 = B_0 = 0$ and limits $A_\infty = \lim_{t \to \infty} A_t$ and $B_\infty = \lim_{t \to \infty} B_t$ are finite. I am trying to prove that ...
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1answer
54 views

Question about Ito integral

I was wondering if Ito integral: $\int_0^T B(t)dB(t) $ is Gaussian (in which B(t) is Brownian Motion)?? Thank you so much, I appreciate any help ^^
2
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1answer
267 views

Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $$ V_t = e^{\lambda t} v_o + \int_0^t e^{-\lambda (t-s)} dB_s $$ with $ \lambda > 0$, $v_0 \in \mathbb{R}$, and $B$ a brownian motion. I want to ...
2
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1answer
126 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
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1answer
65 views

Weird equality of expectations involving stochastic integral

First of all, sry for the title. I just couldn't figure out any better description for this weird problem: Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence ...
0
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1answer
77 views

Brownian motion and stochastic integration

How do I compute the following expectation? W(T) is a standard brownian motion (i.e.) W(T)~N(0,T) $E\left[ W(T)\int _{ 0 }^{ T }{ sdW(s) } \right] $ I know that Brownian motion of disjoint time ...
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0answers
53 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
2
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1answer
80 views

Can an Itō integral be $\infty$?

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$? Thanks!
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2answers
124 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
132 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: ...
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0answers
49 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
votes
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
122 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
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1answer
186 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
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5answers
331 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
votes
1answer
94 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
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1answer
149 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
366 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
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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 ...
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1answer
158 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 ...
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1answer
206 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 + ...
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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
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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 ...
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0answers
107 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 $t$. ...
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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
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1answer
166 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
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1answer
136 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 ...
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1answer
107 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
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1answer
116 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 ...
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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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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
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1answer
104 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
234 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
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3answers
506 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: $$\int_0^Tf(t)\;dB(t).$$ ...
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votes
1answer
96 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
111 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 ...
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0answers
77 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
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1answer
98 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
138 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
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1answer
64 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
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2answers
136 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
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1answer
158 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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2answers
204 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!