# 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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### Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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### Show that the only nonnegative superharmonic functions in R are the constants

I am having trouble finding g$^∗$(x) when $$g(x) = \begin{cases} xe^{-x} & \text{for x > 0} \\[2ex] 0 & \text{for x \leq 0}. \end{cases}$$ I would like to use the fact that the only ...
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### Proving that only nonnegative superharmonic functions in R$^2$ are constants

How can I prove that the only nonnegative (B$_t$-) superharmonic functions in R$^2$ are the constants? So far, I know that u is a nonnegative superharmonic function and that there exist x, y ∈ R$^2$ ...
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### 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 ...
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### How to show that $S_k = \inf \{t \geq 0 | \|X(t)\| \geq k \} \to \infty$ as $k \to \infty$ a.s.

stack.exchangers! I am currently working my way through the proof given by Karatzas and Shreve (1988) of the Feynman-Kac Theorem (Theorem 5.7.6). However, I am missing out on the following problem: ...
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### Finding a unique strong solution

I am brushing up on my stochastic approximation. I am having a hard time with the following problem. I have the equation dX$_t$ = ln(1+ X$_t^2$)dt + X$_t$dB$_t$ X$_0$ = x, with x ∈ ℝ I know that ...
I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(... 0answers 37 views ### unbounded variation of$\sin(x)/x$How can I show that the variation of$sin(x)/x$is unbounded? Could you please help me. I know that I have to use but how can I rough estimate that this is bigger than infinity? 0answers 64 views ### Show that$\hat{Y}$is an optimal linear estimator of Y Relevant Information. Let$X(t)$,$t \in T$be a second order process. Let$M_0$be the set of random variables of the form$a + b_1X(s_1)+ \cdots + b_nX(s_n)$for a positive integer$n$and constants ... 0answers 48 views ### Convergence of a sequence over supremum Given a cadlag-process$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}$the ... 0answers 88 views ### stochastic exponential uniformly integrable martingale$N$is a continuous local martingale and$T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$,$c>0$. I need to show that the stochastic exponential$\mathcal{E}(-N)$is a uniformly integrable ... 0answers 53 views ### Question about “Stochastic Analysis on Manifolds” After Definition 2.3.1 Hsu says that if$M$is a closed submanifold of$\mathbb{R}^N$then a semimartingale$X$on$M\subseteq\mathbb{R}^N$should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ dX_s,... 0answers 26 views ### A stochastic volatility model An example of stochastic volatility model:$$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases}$$where$Z_t$and$W_t$are Brownian motions and$...
I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that \$[...