# Tagged Questions

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
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### Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
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### 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 ...
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### Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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### 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}\...
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### Trace term in the Itō formula

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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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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### Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as $e^{m+\frac{1}{2}... 1answer 37 views ### Ito formula when g(t,x) is an integral Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If$Y_t$is defined as a stochastic process as a function of$X_t$, then we can find$dY_t$... 0answers 52 views ### what does this integral stand for? i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ... 1answer 42 views ### Word Problem: Probability of Y books Fitting in Book Case Problem: You have$4600$cm of book case. The thickness of the books are independently distributed with$X \sim N(1.8$cm$,0.7^2)$. Approximately determine what the probability of ... 0answers 19 views ### How should I calculate the MLE based on a random sample from$PAR(\theta,2)$Consider a random sample of size$n$from a Pareto distribution,$X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE,$\hat \theta$, to three decimale places. So I started doing the ... 0answers 52 views ### Radon-Nikodym on a Process wrt to filtration Given a probability space$(\Omega,\mathcal{F},P)$. Let$(X_t)_{t\geq0}$be a stochastic process defined on it with cadlag paths, lets say on$(\mathcal{X},\mathcal{B}(X))$. Let be$\mathcal{F}_{t}$... 1answer 54 views ### Why Are Semimartingales the Largest Possible Class of Stochastic Integrators? I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ... 0answers 47 views ### Convergence of a sequence over supremum Given a cadlag-process$X_{t}$with stationary independent increments (Levy process) for which$E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$for all$t>0$. For$n\in \mathbb{N}$the ... 1answer 26 views ### Do Optional and Progressive Processes Have Counterparts in Discrete Time? We know that predictable$\implies$optional$\implies$progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ... 0answers 28 views ### Distribution of Double Stochastic Integral Assume that$f(s)$is a$C^\infty$univariate function and that$\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$is a two-dimensional, correlated Wiener process. Then, does the random variable$X_T \equiv \int\...
I need to solve the SDE: $$dX_t = (X_t)^3 dt + (X_t)^2 dW_t ; X(0)=1$$ Now what I found is this is an SDE of the form: $$dXt =a(X_t)dt+b(X_t)dW_t$$ where $a(x) = \frac{1}{2} b(x)b′(x)$ Using the ...