# 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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### Expected value of geometric Brownian motion

So everyone knows that the expected value of GBM is given by: $X_0 \exp(\mu t)$ My question is that what does this say about such stochastic processes? Since $X_0$ and $\mu$ are within "my control" ...
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### Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds}$$ where the ...
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Given some probability space, assume $X_t$ is a square integrable continuous process adapted to the filtration $\mathcal{F}_{t}$ generated by the standard Brownian process $B_t$. I denote by $(X.B)_t=\... 2answers 534 views ### Further Reading on Stochastic Calculus/Analysis I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ... 1answer 319 views ### Application of the Burkholder Davis Gundy inequality The proof of the Feynman-Kac formula uses a lemma which I need to prove, but I can not figure it out. The lemma is the folllowing: Let$X$be a weak solution of $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$... 1answer 185 views ### Conditional expectation brownian motion Somebody has an idea on how to tackle this quantity $$\mathbb{E}\left[ \left. \frac{\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}{\int_{0}^{T}{{{e}^{-\alpha {{W}_{t}}}}}dt+\int_{0}^{T}{{{e}^{\alpha {{W}_{... 0answers 78 views ### Question about a Bessel process Are there any explicit path-wise solutions for a 3 dimensional Bessel process? E.g. the Ito SDE:$$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$where W_t is a standard Wiener process. 2answers 55 views ### brownian noise and stochastic differential equations Consider the SDE$$dx=3x(t)dt+dW(t)$$Where we're dealing with Brownian noise. Now, dW comes from$$dW(t)=\int_0^{dt}ds\ \eta (s)$$As far as I understood, \eta is the noise distribution (... 0answers 18 views ### System of stochastic equations I want to know if this system of SDE:$$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$... 0answers 26 views ### Quadratic covariation I am trying to solve small task from stocastic calculus. It can be shown that for stochastic processes X and Y , the quadratic covariation satisfies the polarisation formula - [X, Y](T) = 1/2 ([X + ... 1answer 53 views ### Expectation value of stochastic process For which k>0 process X=(e^{kW_s^2})_{s \ge 0} belong to \mathcal{L}^2_{\infty }(W) and for which belong to \Lambda ^2_{\infty }(W). Set one localization sequence (\tau_n)_{n \ge 0} for ... 1answer 18 views ### Vasicek equation [closed] Vasicek interest rate stochastic differential equation is$$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$where \alpha , \beta are positive constants. I need to use Ito-Doeblin formula to compute ... 0answers 44 views ### Stochastic calculus with normal distribution For l=1,2...... prove that E[W^{2l+1} (t)]=0 I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very difficult to start with. I am really appreciate ... 0answers 15 views ### Interpretation of the stochastic integral of simple functions A simple function h (or random staircase function) is defined as$$ h(t) = \sum_0^{n} \xi_i I_{(t_i,t_{i+1}]}(t) $$where 0<t_1< \ldots < t_{n+1}<T and the \xi_i are random ... 1answer 36 views ### Using of Ito formula with martingales We have exam test - \alpha,\beta \in \mathbb{R} and N(t)=e^{\beta t}cos(\alpha W(t)). It is necessary to calculate \mathbb{E}[cos(\alpha W(t))]. I know that \beta can be chosen so that N ... 2answers 66 views ### What's the variance of the following stochastic integral? The stochastic integral is defined as$$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$where W(t) is a standard Brownian motion, \kappa and c are both positive. I ... 1answer 27 views ### Ito integral for simple stochastic process I need for l=1,2...... prove that E[W^{2l} (t)]= \frac{(2l)!}{2^l l!} and E[W^{2l+1} (t)]=0 I know that Ito integral for simple stochastic process satisfies E[I^2 (t)]=E\int_0^t\Delta^2(u)... 1answer 2k views ### How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation? I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: X_t=-a\int^t_0 X_s\,ds +B_t. B is the Brownian motion. And its analytic ... 1answer 41 views ### Girsanov theorem calculations help I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let X_t solve the Ornstein-Uhlenbeck equation$$dX_t = X_t\, ... 1answer 45 views ### Stochastic differential equation substitution reasoning? I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ... 1answer 28 views ### Question about Ito integration in SDE in Stochastic optimal control Here is my question statement. I cannot understand the last equality. Let$U=[-1,1]$. \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\... 0answers 33 views ### The integral is the area under the curve. Is there a similar notion for stochastic integrals? As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ... 0answers 52 views ### A Simple Stochastic Integral Asymptotics Let$B(t)$be the standard Brownian motion,$\mu(t,x)$and$\sigma(t,x)$are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$$(\mu,\sigma)$obeys the linear growth condition$...
Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...