The Azéma-Yor Embedding in Brownian Motion with Drift
The Azéma-Yor Embedding in Brownian Motion with Drift
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带有漂移的布朗运动中的 Azéma-Yor 嵌入
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发表时间：
2000
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通讯作者：
G. Peskir
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作者：
G. Peskir
设B = (B t) t≥0是起点为0的标准布朗运动，设μ > 0是给定且固定的，设v是密度f为严格正的R上的一个概率测度，则存在一个停止时间t *。B如此
$$left( {{B_{{T_*}}} + mu {T_*}}
ight) sim
u$$ 当
且仅当满足以下条件： $${D_mu }: = int {_R} {e^{ - 2mu x}}vleft( {dx}
ight) leqslant 1$$ 在这种情况下
设置({C＿mu = - }left(2mu{ ight)^ - 1log left{(}D＿mu ight){)，下面的显
式公式是有效的： }}{{}}$${ au _*} = inf left{ {t >left. 0
ight|left( {{B_t} + mu t}
ight) leqslant {h_mu }left( {mathop {max }limits_{0 leqslant r leqslant t} left( {{B_r} + mu r}
ight)}
ight)}
ight}$$
其中s
> C μ的映射（s mapsto {h＿mu} left（s右））通过它的逆表示为 $$h_mu ^{ - 1}left( x
ight) = - frac{1}{{2mu }}log left( {frac{1}{{1 - Fleft( x
ight)}}int {_x^infty } {e^{ - 2mu t}}dFleft( t
ight)}
ight)left( {x in R}
ight)$$ 当s≤C μ
时，设（{h＿mu} left(
s ight) = - inty）。这就解决了b[6]中提出的问题。此外，还证明了r是满足（left(b＿ au＿ * + μ au＿ * {{ight) simv）的逐点最小可能停止时间，它随机地产生了截止到停止时间的过程（{{b＿}}} t + {μ}}t） t≥0的最大可能最大值。极大极小性质表征了T*。独一无二。结果通过在μ↓0时传递到极限，恢复了Skorokhod嵌入问题[1]的Azema-Yor解。为简单起见，我们给出了严格正密度存在的条件，更一般的情况也可以类似地处理。在证明中使用的论据线可以扩展到处理更一般的非循环扩散的情况。
Let B = (B t ) t≥0 be standard Brownian motion starting at zero, let μ > 0 be given and fixed, and let v be a probability measure on R having a strictly positive density F. Then there exists a stopping time T*. of B such that
$$left( {{B_{{T_*}}} + mu {T_*}}
ight) sim
u$$
if and only if the following condition is satisfied:
$${D_mu }: = int {_R} {e^{ - 2mu x}}vleft( {dx}
ight) leqslant 1$$
Setting in this case ({C_mu } = - {left( {2mu }
ight)^{ - 1}}log left( {{D_mu }}
ight)) the following explicit formula is valid:
$${ au _*} = inf left{ {t >left. 0
ight|left( {{B_t} + mu t}
ight) leqslant {h_mu }left( {mathop {max }limits_{0 leqslant r leqslant t} left( {{B_r} + mu r}
ight)}
ight)}
ight}$$
where the map (s mapsto {h_mu }left( s
ight)) for s > C μ is expressed through its inverse by
$$h_mu ^{ - 1}left( x
ight) = - frac{1}{{2mu }}log left( {frac{1}{{1 - Fleft( x
ight)}}int {_x^infty } {e^{ - 2mu t}}dFleft( t
ight)}
ight)left( {x in R}
ight)$$
and we set ({h_mu }left( s
ight) = - infty) for s ≤ C μ . This settles the question raised in [6]. In addition, it is proved that r. is pointwise the smallest possible stopping time satisfying (left( {{B_{{ au _*}}} + mu { au _*}}
ight) sim v) which generates stochastically the largest possible maximum of the process (B t + μt) t≥0 up to the time of stopping. This minimax property characterizes T*. uniquely. The result recovers the Azema-Yor solution of the Skorokhod embedding problem [1] by passing to the limit when μ ↓ 0. The condition on the existence of a strictly positive density is imposed for simplicity, and more general cases can be treated similarly. The line of arguments used in the proof can be extended to treat the case of more general nonrecurrent diffusions.