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The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ implies a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize you process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor you process to $0$ or to reflect you process, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 bu Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps which is independent of the Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ implies a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize your process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor the process to $0$ or to reflect it, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 by Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps which are independents of the Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ imply a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize you process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor you process to $0$ or to reflect you process, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 bu Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps with independent correlation with Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ implies a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize you process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor you process to $0$ or to reflect you process, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 bu Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps which is independent of the Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ imply a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=kv_t$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize you process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor you process to $0$ or to reflect you process, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 bu Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps with independent correlation with Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.

The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ imply a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $\xi$, but you cannot increase it freely as it is bounded by $\xi^2< 2k\theta $. So in general this condition is violated. However this is not really an issue, at the moment the volatility reaches $0$ you'll have $dv_t=k\theta dt$ and the volatility will instantly go away from $0$. The only problem you can get is in discrete time. So when you discretize you process via Euler scheme for example your process can go negative, one solution of this fact which is widely used is to floor you process to $0$ or to reflect you process, which means you take its absolute value. However, for the CIR process there is an algorithm which was published on 2007 bu Leif Andersen which simulate the CIR process without using discretization, it is now the standard algorithm used in the whole market for the Heston model. The article is "Efficient Simulation of the Heston Stochastic Volatility Model". I believe in the final section of the article the author add jumps with independent correlation with Brownian of the CIR process. In this case the simulation is straightforward, you can simulate the CIR using the algorithm and simulate the jump independently. Hope this helps.