# Tagged Questions

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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### Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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### Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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### Calculate $E[\exp(iu\int_0^ts \, dB_s)]$ for a Brownian motion $(B_t)_{t \geq 0}$

Since $X_t:=\int_0^ts \, dB_s$ is a process with independent increments, its distribution is infinitely divisible and its variance is $c_t=\frac{1}{3}t^3$. I think, its characteristic function ...
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### Itos formula on a transformation of bessel Processes

Let $W$ be a Brownian motion and $z,\kappa>0$. Let $X_t(z)$ be a solution to the SDE $$dX_t(z)=dW_t+2/(\kappa X_t(z))dt.\quad X_0(z)=z.$$ The solution is well-defined on $t<\tau(z)$ where ...
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### The stochastic integral $\int W_t dW_t$

I'm reading an introduction to Stochastic Calculus. I'm at the point where Ito integrals are developed and constrasted with the Stratonovich integral. Below is a calculation of $\int_0^T W_t d W_t$. ...
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### Itô integral probability distribution

I know in general this must not have an analytical expression in terms of common functions, but how do you (at least in theory) get the probability distribution of $X_t$ for a given $t$ in the ...
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### Conserved quantity for system of Stochastic Differential Equations

I'm considering the set of SDEs (in the sense of Ito) \begin{align*} \mathrm d x &= -yx \mathrm d t+ x^2 \mathrm d B_t \\ \mathrm d y &= -y^2 \mathrm d t + xy \mathrm d B_t\end{align*} ...
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### Wick renormalization of stochastic integral

I am trying to understand a paper that summarizes some results concerning Wick renormalization of some stochastic integral. In the last few lines of the paper the authors say: In Euclidean ...
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### The limit of the ratio of two stochastic integrals

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$\lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is ...
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### I want to simplify the stochastic integral by change variable

Let $f:[0,t]\rightarrow \mathbb{R^+}$ be a deterministic and integrable and $(B_t)_{t\geq 0}$ is a standard Brownian motion. If $X_t=\int_o^tf(s)dB_s$, we know that $X_t$ has normal distribution with ...
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### Why $d\langle X \rangle_t = d X_t dX_t$ if $X_t$ is a semimartingale?

Following this question, proving the equivalence between equation $(1)$ and $(2)$, I deduced that $$d\langle X \rangle_t = d X_t dX_t$$ (where $X_t$ was an Ito's process, hence a semimartingale). I ...
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### How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T]$ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, ...
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### $dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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### Hilbert-Schmidt operator - converging norm series - Cylindrical brownian motion

I am reading about cylindrical brownian motion in the monograph of Prato and Zabczyk. For this construction a Hilbert-Schmidt operator is used, between to separable Hilbert spaces $U$ and $U_1.$ Let ...
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