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Showing bounds of Stochastic Process

Consider the equation $$ \mathrm{d}Y_t = \biggl( 2 + 8 \frac{e^{Y_t} - 1}{e^{Y_t} + 1} \biggr) \, \mathrm{d}t + 4 \, \mathrm{d}B_t. \tag{1}\label{e:1} $$ The coefficients $\mu(y, t) = 2+\operatorname{...
Sangchul Lee's user avatar
1 vote

Expressing a continuous local martingale as an integral against a Brownian motion

Since $X_t$ is constant when $A_t = 0$, we have $\int_0^t 1_{A_s = 0}dX_t = 0$. Since you showed $A^{1/2}_t dB_t = 1_{A_t>0}dX_t$, we have \begin{align*} \int_0^t A^{1/2}_s dB_s &= \int_0^t 1_{...
user6247850's user avatar
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1 vote
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Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

Note that we can rewrite the random variable in the following way: $$\begin{aligned} &e^{-\gamma^2t/2+\gamma W_t}E^P[e^{\gamma^2T/2-\gamma W_T}F(Y_T)|\mathscr{F}_t]\\ &=e^{-\gamma^2t/2+\gamma ...
Snoop's user avatar
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