For drifted Brownian motion X(t) = x − μt + Bt (μ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero, τ (x), and the firstpassage area, A(x), swept out by X till the time τ (x). In particular, we establish differential equations with boundary conditions for the joint moments E[τ (x)mA(x)n], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time τ (x) is obtained
Abundo, M. (2017). On the joint distribution of first-passage time and first-passage area of drifted Brownian motion. METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, 19(3), 985-996 [10.1007/s11009-017-9546-7].
On the joint distribution of first-passage time and first-passage area of drifted Brownian motion
abundo mario
2017-01-01
Abstract
For drifted Brownian motion X(t) = x − μt + Bt (μ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero, τ (x), and the firstpassage area, A(x), swept out by X till the time τ (x). In particular, we establish differential equations with boundary conditions for the joint moments E[τ (x)mA(x)n], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time τ (x) is obtained| File | Dimensione | Formato | |
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