Simulation of jump times of Cox process with CIR intensity

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I want to sample the jump times of a Cox process where the intensity $lambda(t)$ is given by a CIR process, i.e.
$$
mathrmdlambda(t)=kappa(theta-lambda(t))mathrmdt+sigmasqrtlambda (t)mathrmdW(t), quad lambda(0)=lambda_0.
$$
The distribution of the time of the first jump $tau_1:=inflefttBigvert Nleft(int_0^tlambda(s)mathrmdsright)=1right$ can be computed as follows:
$$
mathbbPleft(tau_1geq tright) = mathbbPleft(Nleft(int_0^tlambda(s)mathrmdsright)=0right) \
=mathbbEleft(mathbbEleft(mathbf1_Nleft(int_0^tlambda(s)mathrmdsright)=0Biggvertlambda(s), 0leq sleq tright)right)\
=mathbbEleft(expleft(-int_0^tlambda(s)mathrmdsright)right)\
=mathcalL(1)$$
with the Laplace-transform $mathcalL(mu)$ of $int_0^tlambda(s)mathrmds$ evaluated at 1. This function is known from the bond pricing literature and is given by
$$
mathcalL(mu)= expleft(phi_mu(t)-lambda(0)psi_mu(t)right)
$$
where
$$
phi_mu(t) = frac2kappathetasigma^2logleft(frac2gammamathrme^fract(gamma+kappa)2gamma-kappa+mathrme^gamma t(gamma+kappa)right)
$$
and $$
psi_mu(T)=frac2muleft(mathrme^gamma t-1right)gamma-kappa+mathrme^gamma t(gamma+kappa)
$$
with $gamma=sqrtkappa^2+2sigma^2mu$.
Hence, I can easily derive the density of $tau_1$ and may sample from it by some acceptance-rejection or inversion method. The problem is now the following: The density clearly involves $lambda(0)$ and hence for the simulation of $tau_2$ I would need to know $lambda_tau_1$ to proceed. I haven't found anything about how to sample $lambda_tau_1$ conditional on $tau_1$ on the internet. Is there an algorithm to exactly sample the jump times of the Cox process with CIR intensity?




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    I want to sample the jump times of a Cox process where the intensity $lambda(t)$ is given by a CIR process, i.e.
    $$
    mathrmdlambda(t)=kappa(theta-lambda(t))mathrmdt+sigmasqrtlambda (t)mathrmdW(t), quad lambda(0)=lambda_0.
    $$
    The distribution of the time of the first jump $tau_1:=inflefttBigvert Nleft(int_0^tlambda(s)mathrmdsright)=1right$ can be computed as follows:
    $$
    mathbbPleft(tau_1geq tright) = mathbbPleft(Nleft(int_0^tlambda(s)mathrmdsright)=0right) \
    =mathbbEleft(mathbbEleft(mathbf1_Nleft(int_0^tlambda(s)mathrmdsright)=0Biggvertlambda(s), 0leq sleq tright)right)\
    =mathbbEleft(expleft(-int_0^tlambda(s)mathrmdsright)right)\
    =mathcalL(1)$$
    with the Laplace-transform $mathcalL(mu)$ of $int_0^tlambda(s)mathrmds$ evaluated at 1. This function is known from the bond pricing literature and is given by
    $$
    mathcalL(mu)= expleft(phi_mu(t)-lambda(0)psi_mu(t)right)
    $$
    where
    $$
    phi_mu(t) = frac2kappathetasigma^2logleft(frac2gammamathrme^fract(gamma+kappa)2gamma-kappa+mathrme^gamma t(gamma+kappa)right)
    $$
    and $$
    psi_mu(T)=frac2muleft(mathrme^gamma t-1right)gamma-kappa+mathrme^gamma t(gamma+kappa)
    $$
    with $gamma=sqrtkappa^2+2sigma^2mu$.
    Hence, I can easily derive the density of $tau_1$ and may sample from it by some acceptance-rejection or inversion method. The problem is now the following: The density clearly involves $lambda(0)$ and hence for the simulation of $tau_2$ I would need to know $lambda_tau_1$ to proceed. I haven't found anything about how to sample $lambda_tau_1$ conditional on $tau_1$ on the internet. Is there an algorithm to exactly sample the jump times of the Cox process with CIR intensity?




    probability statistics stochastic-processes stochastic-analysis simulation




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      I want to sample the jump times of a Cox process where the intensity $lambda(t)$ is given by a CIR process, i.e.
      $$
      mathrmdlambda(t)=kappa(theta-lambda(t))mathrmdt+sigmasqrtlambda (t)mathrmdW(t), quad lambda(0)=lambda_0.
      $$
      The distribution of the time of the first jump $tau_1:=inflefttBigvert Nleft(int_0^tlambda(s)mathrmdsright)=1right$ can be computed as follows:
      $$
      mathbbPleft(tau_1geq tright) = mathbbPleft(Nleft(int_0^tlambda(s)mathrmdsright)=0right) \
      =mathbbEleft(mathbbEleft(mathbf1_Nleft(int_0^tlambda(s)mathrmdsright)=0Biggvertlambda(s), 0leq sleq tright)right)\
      =mathbbEleft(expleft(-int_0^tlambda(s)mathrmdsright)right)\
      =mathcalL(1)$$
      with the Laplace-transform $mathcalL(mu)$ of $int_0^tlambda(s)mathrmds$ evaluated at 1. This function is known from the bond pricing literature and is given by
      $$
      mathcalL(mu)= expleft(phi_mu(t)-lambda(0)psi_mu(t)right)
      $$
      where
      $$
      phi_mu(t) = frac2kappathetasigma^2logleft(frac2gammamathrme^fract(gamma+kappa)2gamma-kappa+mathrme^gamma t(gamma+kappa)right)
      $$
      and $$
      psi_mu(T)=frac2muleft(mathrme^gamma t-1right)gamma-kappa+mathrme^gamma t(gamma+kappa)
      $$
      with $gamma=sqrtkappa^2+2sigma^2mu$.
      Hence, I can easily derive the density of $tau_1$ and may sample from it by some acceptance-rejection or inversion method. The problem is now the following: The density clearly involves $lambda(0)$ and hence for the simulation of $tau_2$ I would need to know $lambda_tau_1$ to proceed. I haven't found anything about how to sample $lambda_tau_1$ conditional on $tau_1$ on the internet. Is there an algorithm to exactly sample the jump times of the Cox process with CIR intensity?




      probability statistics stochastic-processes stochastic-analysis simulation




      share|cite|improve this question











      I want to sample the jump times of a Cox process where the intensity $lambda(t)$ is given by a CIR process, i.e.
      $$
      mathrmdlambda(t)=kappa(theta-lambda(t))mathrmdt+sigmasqrtlambda (t)mathrmdW(t), quad lambda(0)=lambda_0.
      $$
      The distribution of the time of the first jump $tau_1:=inflefttBigvert Nleft(int_0^tlambda(s)mathrmdsright)=1right$ can be computed as follows:
      $$
      mathbbPleft(tau_1geq tright) = mathbbPleft(Nleft(int_0^tlambda(s)mathrmdsright)=0right) \
      =mathbbEleft(mathbbEleft(mathbf1_Nleft(int_0^tlambda(s)mathrmdsright)=0Biggvertlambda(s), 0leq sleq tright)right)\
      =mathbbEleft(expleft(-int_0^tlambda(s)mathrmdsright)right)\
      =mathcalL(1)$$
      with the Laplace-transform $mathcalL(mu)$ of $int_0^tlambda(s)mathrmds$ evaluated at 1. This function is known from the bond pricing literature and is given by
      $$
      mathcalL(mu)= expleft(phi_mu(t)-lambda(0)psi_mu(t)right)
      $$
      where
      $$
      phi_mu(t) = frac2kappathetasigma^2logleft(frac2gammamathrme^fract(gamma+kappa)2gamma-kappa+mathrme^gamma t(gamma+kappa)right)
      $$
      and $$
      psi_mu(T)=frac2muleft(mathrme^gamma t-1right)gamma-kappa+mathrme^gamma t(gamma+kappa)
      $$
      with $gamma=sqrtkappa^2+2sigma^2mu$.
      Hence, I can easily derive the density of $tau_1$ and may sample from it by some acceptance-rejection or inversion method. The problem is now the following: The density clearly involves $lambda(0)$ and hence for the simulation of $tau_2$ I would need to know $lambda_tau_1$ to proceed. I haven't found anything about how to sample $lambda_tau_1$ conditional on $tau_1$ on the internet. Is there an algorithm to exactly sample the jump times of the Cox process with CIR intensity?





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