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I'm currently reading the article written by David X.Li "On Default Correlation: A copula Function Approach". I'm deepening my interest in subprime mortgage crisis. In the introduction of the paper the author talks about survival probability and hazard rate function. They are linked by the following formula: $$S(t)=e^{-\int_0^th(s)ds},$$ where $S$ denotes the survival probability and $h$ the hazard rate function. The author also defines the following quantity: $$p_{t,x}=\mathbb P\left\{T>t+x|T>x\right\},$$ where "T" is a random variable denoting the default event. Then, he states that: $$p_{t,x}=e^{-\int_0^th(x+s)ds}.$$ According to my calculations: \begin{align} e^{-\int_0^th(s)ds}&=S(t+x)\\ &=\mathbb P\left\{T>t+x\right\}\\ &=\mathbb P\left\{T-x>t\right\}, \end{align} whereas \begin{align} p_{t,x}&=\mathbb P\left\{T>x+t|T>x\right\}\\ &=\frac{\mathbb P\left\{T>x+t,T>x\right\}}{\mathbb P\left\{T>x\right\}}\\ &=\frac{\mathbb P\left\{T>x+t\right\}}{\mathbb P\left\{T>x\right\}}\\ &=\frac{\mathbb P\left\{T-x>t\right\}}{\mathbb P\left\{T>x\right\}}. \end{align} So in the first part of my calculations, I'm missing the denominator since the two expressions must match. Can I kindly ask you where I am wrong?

Thanks in advance.

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  • $\begingroup$ I suppose when you write $\exp \left\{ -\int_0^t h(s) \mathrm{d} s \right\} = S(t + x)$ you meant $\exp \left\{ -\int_0^t h(x + s) \mathrm{d} s \right\} = S(t + x)$? Even if yes, this line is exactly where your mistake is. The (corrected) l.h.s. is equal to $\exp \left\{ -\int_x^{t + x} h(s) \mathrm{d} s \right\}$ as opposed to the r.h.s. $\exp \left\{ -\int_0^{t + x} h(s) \mathrm{d} s \right\}$. $\endgroup$ – LocalVolatility Apr 10 '17 at 11:57
  • $\begingroup$ Thank you very much! Now I see that I got confused with the variables. $\endgroup$ – Giano Rugge Apr 10 '17 at 13:46

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