# 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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### Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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### Derived Point Process

Can anyone give me some hint on the following problem? Thanks a lot! Let $\{T_n:n\ge 0\}$ be a point process and $\{N_t: t\ge 0\}$ be the corresponding counting process which admits a bounded ...
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### Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
I need to show that the Ornstein-Uhlenbeck process, $$dX_t = -\theta X_tdt + dB(t)$$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$X_t=B(t) - \theta \... 0answers 43 views ### Weak stochastic integral I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]:$$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$A ... 1answer 43 views ### Stochastic integral with respect to a stochastic integral [From Bass R.F. Stochastic processes. Exercise 10.4] Let N_t = \int_0^tH_sdM_s where M is a continuous square integrable martingale and H is predictable and integrable and L_t = \int_0^tK_sdN_s ... 1answer 106 views ### dX_t=-\mu X_tdt + \sigma dW_t. Prove that X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u  So the solution says use Ito-s formula, taking Y_t:= e^{\mu t}X_t to obtain dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] . As far as I can see though, Ito's formula says ... 0answers 76 views ### Ito formula for f(X_t, Y_{t-s}) I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}. How do I apply Ito lemma in this case?(is Ito lemma still ... 1answer 261 views ### Use Ito's Lemma to show: I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, f:$$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$Where B(t) ... 1answer 110 views ### Calculate Stochastic Integral I found the following integral \int_{0}^1 B_t t^{-1}dt, where B_t is a standard Brownian motion. Using Ito formula with f(t,x)=x\log(t) I achieved$$0=\log(1)B_1=\int_{0}^1 B_s s^{-1}ds +\...
I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that \lim_{\lambda\to\...