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1

by application of Ito's lemma , we have $$d\left(q(t)e^{\Theta\,t}\right)=\Theta \,q(t)e^{\Theta\,t}dt+e^{\Theta\,t}dq(t)+0$$ then $$d\left(q(t)e^{\Theta\,t}\right)=\sigma e^{\Theta\,t}dW_t$$ in other words $$q(t+h)e^{\Theta\,(t+h)}-q(t)e^{\Theta\,t}=\sigma\int_{t}^{t+h}e^{\Theta\,u}dW_u\Rightarrow$$ ...

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In mathematics and statistics, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, if they are present, also do not change over time and do not follow any trends.

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In words the equation says that when the stock price is very high, the value of the call is (approximately) equal to the stock price minus the PV of the exercise. It is fairly intuitive: if K=100 and x is 1000, the stock is so much above K that exercise is for all practical purposes certain; the call today is worth x minus PV(k), since you can set aside ...

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The Laplace transform of the integrated process CIR process is given by, see e.g. Dufresne (2001). you can download it The integrated square-root process

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let $Y_t=(X_t)^\alpha$,then $$dY_t=\alpha Y_tdt+dW_t^P$$ we define $Q$ measure by $$\frac{dQ}{dP}=exp\left(-\alpha\int_{0}^{T}Y_t\,dW_t^p-\frac{1}{2}\alpha^2\int_{0}^{T}Y_t^2 dt\right)$$ this shows that $$W_t^Q=W_t^P+\alpha\,\int_{0}^{t}Y_s\,ds$$ is standard wiener process under $Q$ measure, thus we have $$dW_t^P=dW_t^Q-\alpha\,Y_t dt$$ and dY_t=\alpha ...

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Let $\{P_t \mid t \geq 0\}$ be a compound Poisson process, where \begin{align*} P_t = \sum_{i=1}^{N_t} (V_i -1), \end{align*} and $N_t$ is a Poisson process with intensity $\lambda$ and jump times $\tau_i$, $i = 1, \ldots, \infty$. Let $Y_i=\ln V_i$ and $f(x)$ be the density function. Then \begin{align*} P_t - \lambda t E(V_1) &= P_t - \lambda t ...

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Let's start with the main idea, I hope you can finish the computations yourself. Whenever you want to derive a pricing equation, try the following approach: discounted value of portfolio/option/derivative must be a martingale for non-arbitrage reasons. Since you have a Markovian dynamics in variables $t$ and $S$, you assume that the price is some function ...

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