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C.I.R Process belongs the class of affine diffusion processes.For processes within this class, a closed form solution of the characteristic function exists(Duffie,et al). For more details, Suppose we have given a scalar SDEs, i.e., $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ this process ($\{X_t\}_{0\leq t\leq T}$) is said to be of the affine form if ...


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EDIT: My reasoning below seems to be wrong. The process as you write it tends to infinity if $a$ is big enough and positive and if $\lambda_0$ is positive. I would not call this process non-meanreverting OU. It is just an Ito process of a simple form. If we remove the stochastic part then we get $$ d\lambda_t = a \lambda_t dt $$ with the solution (if ...


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The Feynman-Kac theorem can be used in both directions. That is, If we know that $r_t$ follows the Ito process as described by the following stochastic differential equation \begin{align} d{{r}_{t}}=\mu ({{r}_{t}},t)dt+\sigma ...


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1. weighted Milstein Scheme We assume $\{X_t\}_{t\geq0}$ described by the following stochastic differential equation $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ Under the Ito version of this scheme Equation $(1)$ becomes $$dX_{t+\Delta t}=X_t+[\alpha\,\mu(t,X_t)+(1-\alpha)\mu(t+\Delta t,X_{t+\Delta t})]\Delta t+\sigma\sqrt{\Delta t ...


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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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I would also say that the pricing of some exotic products require to compute expectations of functions of the random variable at consideration, and these functions may grow more than linearly : you need finite moments in order for the prices of these exotic derivatives to be bounded.



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