Articles of stochastic integrals

$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X$ 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: $$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X.$$ Is this statement true? How could one prove it? Where can I find the prove?

Expectation of Ito integral

The expectation of an Itô stochastic integral is zero $$ E[\int_0^t X(s)dB(s)\,]=0 $$ if $$ \int_0^t E[X^2(s)]ds\,<\infty $$ It is sometimes possible to check this condition directly if the Itô integrand is simple enough but how would you do it if the integrand is the process itself? For example consider the linear SDE $$ X(t)=X(0) […]

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 $E[\int_0^T \Delta^2_t(\omega)dt]<+\infty$.

Ito Integral surjective?

Let $\Phi\in\mathcal{L}\left(M\right)$ if and only if $\Phi$ is a real predictable process and for every $\left\Vert \Phi\right\Vert_{2,t,M}:=\mathbb{E}\left[\int_{0}^{t}\Phi_{s}^2 d\langle M\rangle_{s}\right]^{\frac{1}{2}}<\infty$. And let $\mathcal{M}_{2}$ be the space of square integrable right continuous martingales. It is well known by Ito Isometry that the mapping $$I_{M}:(\mathcal{L}\left(M\right),\left\Vert .\right\Vert_{2})\rightarrow\left(\mathcal{M}_{2},\left|\left\Vert \right|\right\Vert\right)$$ is injective, where $I_{M}\left(\Phi\right)$ is the Ito Integral of $\Phi$.I would […]

Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and given Λ = λ, the number of storms is Poisson with mean λ. Find the probability there are at least three storms this […]

Kolmogorov Backward Equation for Itô diffusion

Let $(X_t)_{t\ge 0}$ be the solution of the SDE $$ X_t = X_0 + \int_0^t \mu(s,X_s) \,ds + \int_0^t \sigma(s,X_s) \,dB_s, \quad t\ge 0 $$ where $\mu(s,x)$ and $\sigma(s,x) $ are Lipschitz continuous in $x$ uniformly in $t$. My question is related to the last argument in a proof (in the book of Klebaner – […]

Expectation of Ito integral, part 2, and Fubini theorem

I previously asked a question (Expectation of Ito integral). I have additional questions on the same subject. Let’s say that we have an Ito process such as $$ X(t)=X(0) + \int_0^t a ds + \int_0^t b X(s) dW(s) $$ where a and b are constants and W(t) is the standard Brownian motion. Using Itô’s formula […]

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined quadratic variation process? Is there a statement that says what exactly is needed in addition to existence of the quadratic variation in order for […]

How can a random variable have random variance?

This seems counter-intuitive to me since variance is a difference of expectations and afaik, unconditional expectation is a real number. Apparently, $X_t$ where $dX_t = Y_t dW_t$, where $Y_t$ is an independent Brownian motion to $W_t$, has random variance. Solving the SDE gives $X_t = X_0 + \int_0^t Y_s dW_s$ Computing the first and second […]

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ dt \quad \text{in} \quad L^2$$ as $mesh(\pi) \rightarrow 0$ where $\pi: 0 = t_0 < t_1 < \ldots < t_n = T$. Since I have that $$\sum_{i=0}^{n-1}B_{t_i}(t_{i+1}-t_i) \rightarrow \int_0^TB_t\ dt \quad \text{in} \quad […]