# Tagged Questions

Questions on the calculus of stochastic processes, or processes that have a random component.

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### Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
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### Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
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### 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 ...
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### Can a Brownian motion be defined for negative time?

I was just looking at fractional brownian motions on this page. The definition of $B_H(t)$ requires integrating on a negative time domain on $dB(t)$ where $B(t)$ is a Brownian motion! Could you please ...
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### Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \rho_n^2(t)=2 \int_0^...
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### $dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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### An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
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### Manipulating a log normal variable

I am wondering given: and is it possible to state: $$\text{Jdq}_t s_t-\text{dq}_t s_t=\text{dq}_t \log (J) s_t$$ And if it is the case can we show how this argument is done?
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### Brownian Motion Hitting Time?

So my problem is the following. Take a 2D Brownian motion $(W_{1t}, W_{2t})$ such that it starts at $(1,1)$. With probability 1 it will hit the x-axis. What is the probability that it will hit the ...
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### Proving bounded expectation and switching order of stochastic integrals

I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-...
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### Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
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### Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$\int_0^T W(t) dt = \int_0^T (T-t) dW(t)$$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ (...
We have $Y_i = \beta_0 +\beta_1(X_i -\bar X )+\epsilon_i$ for i=1,...,n $$\epsilon_i \sim N(0,\sigma^2)$$ We know that $SSR= Y^T P_xY - n\bar Y^2=Y^T (P_x -n^{-1} J_nJ_n^T)Y$ J_n=\big[\begin{...
### Why does $(W_t)^2$ have mean $t$?
This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...