SOFIA-M (17) -- Science -- 2010

Brownian motion with returns to zero

continuation

We have also the following theorem:
Let U(t;u_0)=f(t)-w(t)+p(t) be HGWMM. Let f(t) is monotone deterministic function of the income, defined for t>=0, and for which
f(0)=u_0,
where u_0 is a number between zero and one. Let also f(t) tends to infinity when t tends to infinity. We suppose that w(t) is a Winner process with identical parts and filtrations and returns to zero at the times t_i, for which i=f(t_i), where i are nonnegative integers for which i>=1. We assume also that p(t)=N(t) is a homogenuous Poisson process and the processes w(t) and p(t) are independent. By B(t) we denote the standard Brownian motion. The probability a the process U not to achieve the zero, i. e. (i. e., probability of no default) in the interval [0,t_k) has the following boundary:
a>=c_k(1-2P({B(t_1)>u_0})),
if [0,t_1)is the longest between intervals
[0,t_1), [t_1, t_2),...[t_(k-1),t_k),
and a>=c_k(P({max_{[0,t_1)} (w(t))<=u_0} and {max_{I_max} (w(t))<=1})),
if I_max is the longest between intervals,
where
c_k=P({N(t_1)=0} and {N(t_2)=0} and {N(t_3)<=1} and {N(t_(k-1))<=k-3} and {N(t_k)<=k-2}). /to be continued/

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