# 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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### proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z$~$N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula ...
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### How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

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, ...
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### Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
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I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in \... 0answers 130 views ### Representation theorem for continuous process of finite variation There is a martingale representation theorem If$M$is a continuous$L^2$-martingale, there is a Brownian motion$B$and a cadlag adapted function$\sigma$such that $$M_t = M_0 + \int_0^t \... 0answers 583 views ### Ito's lemma and application Can someone help me apply Ito's lemma to the function f(t,x,k) where t is the time and x,k dimensions where x and k refer to dynamics dX(t)=\mu(t)dt+\sigma(t)dB(t) dK(t)=\nu(t)dt+\theta(t)dW(t)... 0answers 71 views ### Is the integral of an Ito process still an Ito process? Assume r(t) is an Ito diffusion:$$dr_t = \mu_tdt + \sigma_tdW_t$$Then, define the following process:$$X_t = \int_0^tr(s)ds$$Is X_t still an Ito diffusion? 0answers 81 views ### Relationship of SDE and Feynman-Kac PDE I am struggling with this problem: Given a stochastic differential equation$$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$where W is a Brownian motion and the functions b and \sigma are ... 0answers 204 views ### An exercise from Revuz, Yor; equality in distribution of 2 integrals. Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let B and ... 0answers 225 views ### Determine if this is a Martingale I am trying to check if the process S_t is a martingale, where \mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t, S_0 = 1. We know that S_t is a local martingale because if we stop it ... 0answers 68 views ### 2-D exponential functional brownian motion I'm looking for the distribution of X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt and Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt} (where W_t is a standard brownian motion) On most ... 0answers 212 views ### Integrating the inverse of a squared bessel process - integrability Let X_t be a 4-dimension Squared Bessel Process (BESQ-4). Let M_t be a continuous true martingale. Question: Does \int_0^t \frac{1}{X_s}dH_s exist? If so, is it only a local or a true ... 0answers 159 views ### Calculating \mathbb{E}[\int_0^T N_{t-} dS_t] - an expectation of a simple stochastic integral. I came across some nasty stochastic integral of which I'd like to calculate the expected value" \mathbb{E}[\int_0^T N_{t-} dS_t] where N_t is a Poisson process and S_t is, say, a geometric ... 0answers 215 views ### Observable and unobservable parameters of stochastic processes Consider the following diffusion process$$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$where X,W are 1-dimensional and. Is it true that given a history (X_s,s\leq t) for each s< t one can find \... 0answers 211 views ### stochastic differential equation Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion. need to find the PDE for the transition density of X at ... 0answers 31 views ### Are martingales progressively measurable? (Application to square integrable martingales) This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ... 0answers 33 views ### The integral is the area under the curve. Is there a similar notion for stochastic integrals? As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ... 0answers 34 views ### Find (a,b) such that aX+bY is a Brownian motion Let$$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$Where (W,V) is a two-dimensional Brownian motion and (X,Y) be a strong ... 0answers 27 views ### Finding the mean of X_t = \int_0^t sW_sdW_s For the stochastic integral, where W_t is a Wiener process, I am trying to find the mean of X_t = \int_0^t sW_sdW_s. I have read before that any stochastic integral with dWt has mean zero, but I ... 0answers 44 views ### The limit of the ratio of two stochastic integrals I am just wondering how to calculate the limit of stochastic integrals. Here is one example:$$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$where B(s) is ... 0answers 18 views ### Calculate expectation of stochastic integrals I am trying to calculate$$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$where (B_t)_{t\geq 0} is a brownian motion, h>0 and \lambda > 0 is some ... 0answers 46 views ### Ito's formula and Infinitesmal generator Consider an Ito process$$ dX_t = \sigma_t dB_t $$where \sigma_t is a two-state continuous-time Markov chain with state space \{ \sigma_1, \sigma_2 \} that switches state with Poisson ... 0answers 59 views ### Does Ito's Isometry hold if the integrand has a brownian motion in it? I am wondering what is the distribution of:$$ \int_0^tW_sdW_s $$Solution: (Thanks to @muaddib) Applying Ito's Formula to W_t^2 gives d(W_t^2) = 2W_tdW_t +dt, and so:$$ \int_0^tW_sdW_s= W_t^2 ... 0answers 133 views ### Integral of Brownian Motion with respect to an independent Brownian motion I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions,$B$and$W$, on the same probability space and under the same filtration (I am not so ... 0answers 50 views ### Quadratic Variation of Increasing Process? I am looking through my notes and I came across the following statement: Let$X_s$be a positive local martingale and let$M_t = max_{0 \le s \le t} X_s$. Then since$M_t$is an increasing process,$...
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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 ...
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### Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?