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Let (X = (X_t: t \in [0,T])) be a stochastic process satisfying a CIR model.
(dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t,)
where (B_t) is a standard Brownian motion, (\beta) is a negative constant, (\gamma, \sigma) are positive constants. In order for the SDE to make sense, assume that (X_t > 0) for all ( t \in [0,T] ).
Consider following two ways to simulate the model based on Ito-Taylor expansion to disretize the model in (t):
1. the Euler scheme:
(X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t} Z \Delta, )
2. the Milstein scheme:
(X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t}Z\sqrt{\Delta} + \frac{1}{4}\sigma^2 \Delta (Z^2-1))
where (Z) is (N(0, 1)) Gaussian variable. I was wondering why these two schemes have a positive probability of generating negative values of (X_t) and therefore cannot be used without suitable modifications.
Thanks and regards!
(dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t,)
where (B_t) is a standard Brownian motion, (\beta) is a negative constant, (\gamma, \sigma) are positive constants. In order for the SDE to make sense, assume that (X_t > 0) for all ( t \in [0,T] ).
Consider following two ways to simulate the model based on Ito-Taylor expansion to disretize the model in (t):
1. the Euler scheme:
(X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t} Z \Delta, )
2. the Milstein scheme:
(X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t}Z\sqrt{\Delta} + \frac{1}{4}\sigma^2 \Delta (Z^2-1))
where (Z) is (N(0, 1)) Gaussian variable. I was wondering why these two schemes have a positive probability of generating negative values of (X_t) and therefore cannot be used without suitable modifications.
Thanks and regards!