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tion of this barrier function will be given below. The default time is, by definition, the first time that X(t) hits this barrier, ie
 

Summary: tion of this barrier function will be given below. The default time is, by
definition, the first time that X(t) hits this barrier, ie:
(3)
Let f(x, t) be the survival probability density function of X(t), given by:
(4)
for x b(t).
From standard results in probability theory, the function f(x, t) satisfies
the forward Fokker-Planck equation:
(5)
with initial and boundary conditions:
(6)
(7)
We note that the integral:
(8)
represents the survival probability up to time t, 1 P(t). Therefore, the de-
fault probability P(t) is related to the survival probability density f(x, t) and
the barrier b(t) by the equation:
(9)
We see from this equation that the barrier function b(t) must be chosen
appropriately so that the pair {f(x, t), b(t)} is consistent with the given de-

  

Source: Avellaneda, Marco - Department of Mathematics, Courant Institute of Mathematical Sciences, New York University

 

Collections: Mathematics