文章快速检索 高级检索
 浙江大学学报(理学版)  2017, Vol. 44 Issue (3): 296-301  DOI:10.3785/j.issn.1008-9497.2017.03.009 0

### 引用本文 [复制中英文]

[复制中文]
ZHANG Miao, LIU Hui, ZHANG Feilong. The first hitting time of stochastic volatility models[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 296-301. DOI: 10.3785/j.issn.1008-9497.2017.03.009.
[复制英文]

### 文章历史

1. 西安电子科技大学 数学与统计学院，陕西 西安 710126;
2. 北京大学 地球与空间科学学院，北京 100871;
3. 西安电子科技大学 物理与光电工程学院，陕西 西安 710126

The first hitting time of stochastic volatility models
ZHANG Miao1 , LIU Hui2 , ZHANG Feilong3
1. School of Mathematics and Statistics, Xidian University, Xi'an 710126, China;
2. School of Earth and Space Scienecs, Peking University, Beijing 100871, China;
3. School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710126, China
Abstract: This paper explores the first passage times of stochastic volatility CEV model. We mainly solve the joint Laplace transform of the first hitting time and volatility. Firstly, we use the It${\rm{\hat o}}$ formula to construct the martingale which can convert the problem into the process of solving a differential equation. Then, we introduce an appropriate second order variable coefficient ordinary differential equation, after a change of variable, it is turned to the Whittaker's equation. It's not difficult to get the general solution of Whittaker's equation. Thus, the explicit expressions for the joint Laplace transformation of the first passage times of stochastic volatility CEV model can be derived. Finally, selecting the parameters γ be 0, 1/2 and 1, let the asset price process covers the O-U process, geometric Brownian motion and square root process. Under different parameters, we obtain explicit expression of the joint Laplace transformation function, and use Matlab to draw the corresponding diagram and analyze the trend of graph.
Key words: stochastic volatility CEV model    first passage times    martingale method    joint Laplace transforms    Whittaker's equation

1 随机波动CEV模型

 ${\rm{d}}{X_t} = {\mu _X}\left( {{X_t}} \right){\rm{d}}t + {\sigma _X}\left( {{X_t},{Y_t}} \right){\rm{d}}W_t^X,$ (1)

 ${\rm{d}}{Y_t} = {\mu _Y}\left( {{Y_t}} \right){\rm{d}}t + {\sigma _Y}\left( {{Y_t}} \right){\rm{d}}W_t^Y.$ (2)

 ${\rm{d}}{Y_t} = \left( {{c_Y} - {b_Y}{Y_t}} \right){\rm{d}}t + \varphi \left( {{Y_t}} \right){\rm{d}}W_t^Y,$ (3)

 $\begin{array}{l} {\rm{d}}{X_t} = {a_X}X_t^\gamma \sqrt {{Y_t}} {\rm{d}}W_t^Y,\\ {\rm{d}}{Y_t} = \left( {{c_Y} - {b_Y}{Y_t}} \right){\rm{d}}t + {a_Y}\sqrt {{Y_t}} {\rm{d}}W_t^Y, \end{array}$ (4)

2 首中时和随机波动因子的联合拉普拉斯变换

 ${\tau _l} = \inf \left( {t \ge 0;{X_{{\tau _l}}} = l} \right),$ (5)

 $\varphi \left( {l;x,y} \right) = {\mathit{\boldsymbol{E}}_{x,y}}\left[ {\exp \left( { - \alpha \tau l - \beta {T_{{\tau _l}}}} \right)} \right],$ (6)

 ${\mathit{\boldsymbol{E}}_{x,y}}\left[ \cdot \right] = \mathit{\boldsymbol{E}}\left[ { \cdot \left| {{X_0} = x,{Y_0} = y} \right.} \right].$

 $\begin{array}{*{20}{c}} {\frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }} - \rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right) + }\\ {\left( {{b_Y}\beta + \frac{1}{2}a_Y^2{\beta ^2}} \right)f\left( x \right) = 0,} \end{array}$ (7)

 $\varphi \left( {l;x,y} \right) = {\mathit{\boldsymbol{E}}_{x,y}}\left[ {\exp \left( { - \alpha \tau l - \beta {T_{{\tau _l}}}} \right)} \right] = \frac{{{{\rm{e}}^{ - \beta y}}f\left( x \right)}}{{f\left( l \right)}}.$ (8)

 $\begin{array}{*{20}{l}} \begin{array}{l} {{\rm{e}}^{ - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} {\rm{ = }}\\ {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}} {\rm{d}}\left( {f\left( {{X_s}} \right)} \right) + \end{array}\\ {\int_0^t {f\left( {{X_s}} \right){\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}} \right)} + \int_0^t {{\rm{d}}\left( {f\left( {{X_s}} \right)} \right){\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}} \right)} = }\\ {{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + {a_X}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma \sqrt {{Y_s}} {\rm{d}}W_s^X} + }\\ {\frac{1}{2}a_X^2\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f''\left( {{X_s}} \right)X_s^{2\gamma }{Y_s}{\rm{d}}s} - }\\ {\beta \int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\left( {{c_Y} - {b_Y}{T_s}} \right){\rm{d}}s} - }\\ {\beta {a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\sqrt {{Y_s}} {\rm{d}}W_s^X} + }\\ {\frac{1}{2}{\beta ^2}a_Y^2\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right){Y_s}{\rm{d}}s} - }\\ {\beta \rho {a_X}{a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma {Y_s}{\rm{d}}s} .} \end{array}$

 $\begin{array}{l} {M_t} = {a_X}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma \sqrt {{Y_s}} {\rm{d}}W_s^X} - \\ \;\;\;\;\;\;\;\;\beta {a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\sqrt {{Y_s}} {\rm{d}}W_s^X} ,t \ge 0. \end{array}$

 $\begin{array}{l} {{\rm{e}}^{ - \alpha t - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{\rm{d}}\left( {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} + \\ \;\;\;\;\;\;\;\;\;\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right){\rm{d}}\left( {{{\rm{e}}^{ - \alpha s}}} \right)} = \\ \;\;\;\;\;\;\;\;\;{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} - \\ \;\;\;\;\;\;\;\;\;\alpha \int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}a_X^2\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f''\left( {{X_s}} \right)X_s^{2\gamma }{Y_s}{\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\beta \int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right)\left( {{c_Y} - {b_Y}{Y_s}} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}{\beta ^2}a_Y^2\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right){Y_s}{\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\beta \rho {a_X}{a_Y}\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma {Y_s}{\rm{d}}s} . \end{array}$

 $\begin{array}{l} \frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }}y - \beta f\left( x \right)\left( {{c_Y} - {b_Y}y} \right) + \frac{1}{2}a_Y^2{\beta ^2}f\left( x \right)y - \\ \;\;\;\;\;\;\;\rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right)y - \alpha f\left( x \right) = \\ \;\;\;\;\;\;\;\left[ {\frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }} - \rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right) + } \right.\\ \;\;\;\;\;\;\;\left. {\left( {{b_Y}\beta + \frac{1}{2}a_\mathit{Y}^2{\beta ^2}} \right)f\left( x \right)} \right]y - \left( {\beta {c_Y} + \alpha } \right)f\left( x \right) = 0. \end{array}$

 ${{\rm{e}}^{ - \alpha t - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} .$

 ${{\rm{e}}^{ - \alpha \left( {t \wedge {\tau _l}} \right) - \beta {Y_{t \wedge {\tau _l}}}}}f\left( {{X_{t \wedge {\tau _l}}}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^{t \wedge {\tau _l}} {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} .$

 ${\mathit{\boldsymbol{E}}_{x,y}} = \left[ {{{\rm{e}}^{ - \alpha {\tau _l} - \beta {Y_{{\tau _l}}}}}f\left( l \right)} \right] = {{\rm{e}}^{ - \beta y}}f\left( x \right).$

 ${\mathit{\boldsymbol{E}}_{x,y}}\left[ {{{\rm{e}}^{ - \alpha \tau \_\beta {Y_{{\tau _l}}}}}} \right] = \frac{{{{\rm{e}}^{ - \beta y}}f\left( x \right)}}{{f\left( l \right)}}.$

 $\begin{array}{l} z\left( x \right) = \sqrt {4\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]/a_X^2{{\left( {1 - \gamma } \right)}^2}} {x^{1 - \gamma }} = \\ \;\;\;\;\;\;\;\;\;A{x^{1 - \gamma }},x \in {\bf{R}}. \end{array}$

 $\frac{{{\rm{d}}f}}{{{\rm{d}}x}} = \frac{{{\rm{d}}f}}{{{\rm{d}}z}}\frac{{{\rm{d}}z}}{{{\rm{d}}x}},\frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{x^2}}} = \frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{z^2}}}{\left( {\frac{{{\rm{d}}z}}{{{\rm{d}}x}}} \right)^2} + \frac{{{\rm{d}}f}}{{{\rm{d}}z}}\frac{{{{\rm{d}}^2}z}}{{{\rm{d}}{x^2}}},$

 $\begin{array}{*{20}{c}} {z\frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{z^2}}} - \left[ {\frac{\gamma }{{1 - \gamma }} + \frac{{2\rho {a_Y}\beta }}{{{a_X}\left( {1 - \gamma } \right)A}}z} \right]\frac{{{\rm{d}}f}}{{{\rm{d}}z}} + }\\ {\frac{{a_Y^2{\beta ^2} + 2{b_Y}\beta }}{{a_X^2{{\left( {1 - \gamma } \right)}^2}{A^2}}}zf\left( z \right) = 0.} \end{array}$

$f(z) = {z^{\frac{\gamma }{{^2(1 - \gamma )}}}}{\rm{e}^{\frac{{\rho aY\beta }}{{^a{X^{(1 - \gamma )A}}}}z}}g(z)$，得

 $\frac{{{{\rm{d}}^2}g}}{{{\rm{d}}{z^2}}} + \left[ {\frac{{1/4 - {m^2}}}{{{z^2}}} + \frac{k}{z} - \frac{1}{4}} \right]g\left( z \right) = 0,$ (9)

 $\begin{array}{*{20}{c}} {m = \frac{1}{{2\left( {1 - \gamma } \right)}},}\\ {k = - \frac{{\gamma \rho {a_Y}\beta }}{{2\left( {1 - \gamma } \right)}}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - \frac{1}{2}}}.} \end{array}$

 $\begin{array}{l} f\left( x \right) = {C_1}z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{\rm{e}}\left\{ {\frac{{\rho {a_Y}\beta z\left( x \right)}}{{\left[ {2\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } } \right]}}} \right\}{\mathit{\boldsymbol{M}}_{k,m}}\left( {z\left( x \right)} \right) + \\ \;\;\;\;\;\;\;\;\;{C_2}z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{\rm{e}}\left\{ {\frac{{\rho {a_Y}\beta z\left( x \right)}}{{\left[ {2\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } } \right]}}} \right\}{\mathit{\boldsymbol{W}}_{k,m}}\left( {z\left( x \right)} \right) = \\ \;\;\;\;\;\;\;\;\;{C_1}\varphi \left( x \right) + {C_2}\psi \left( x \right). \end{array}$

 $f\left( x \right) = {C_1}{x^{{\lambda _1}}} + {C_2}{x^{{\lambda _2}}},$

 $\varphi \left( {l;x,y} \right) = \frac{{{{\rm{e}}^{ - \beta y}}\left( {{C_1}\varphi \left( x \right) + {C_2}\psi \left( x \right)} \right)}}{{{C_1}\varphi \left( l \right) + {C_2}\psi \left( l \right)}},$ (10)

 $\begin{array}{l} \varphi \left( x \right) = z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}{\mathit{\boldsymbol{M}}_{k,m}}\left( {z\left( x \right)} \right),\\ \psi \left( x \right) = z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}{\mathit{\boldsymbol{W}}_{k,m}}\left( {z\left( x \right)} \right), \end{array}$

 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{k,m}}\left( x \right) = {{\rm{e}}^{ - \frac{x}{2}}}{x^{m + \frac{1}{2}}}M\left( {m - k + \frac{1}{2},1 + 2m;x} \right),\\ {\mathit{\boldsymbol{W}}_{k,m}}\left( x \right) = {{\rm{e}}^{ - \frac{x}{2}}}{x^{m + \frac{1}{2}}}U\left( {m - k + \frac{1}{2},1 + 2m;x} \right). \end{array}$

 $\begin{array}{*{20}{c}} {\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }},\;\;\;\;m = \frac{1}{{2\left( {1 - \gamma } \right)}},}\\ {k = - \frac{{\gamma \rho {a_Y}\beta }}{{2\left( {1 - \gamma } \right)}}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - \frac{1}{2}}}.} \end{array}$
3 推论和结果分析

 $\begin{array}{l} \varphi \left( {l;x,y} \right) = \\ \;\;\;\;\;\;\;\frac{{{{\rm{e}}^{ - \beta \gamma }}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{0,\frac{1}{2}}}\left( {z\left( x \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{0,\frac{1}{2}}}\left( {z\left( x \right)} \right)} \right]}}{{{{\rm{e}}^{\frac{{\varepsilon z\left( l \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{0,\frac{1}{2}}}\left( {z\left( l \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{0,\frac{1}{2}}}\left( {z\left( l \right)} \right)} \right]}}, \end{array}$

 $\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }}.$

 $\begin{array}{l} \varphi \left( {l;x,y} \right) = \\ \;\;\;\;\;\;\;\frac{{{{\rm{e}}^{ - \beta \gamma }}{\rm{z}}{{\left( x \right)}^{\frac{1}{2}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{k,1}}\left( {z\left( x \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{k,1}}\left( {z\left( x \right)} \right)} \right]}}{{{\rm{z}}{{\left( l \right)}^{\frac{1}{2}}}{{\rm{e}}^{\frac{{\varepsilon z\left( l \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{k,1}}\left( {z\left( l \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{k,1}}\left( {z\left( l \right)} \right)} \right]}}, \end{array}$

 $\begin{array}{*{20}{c}} {\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }}.}\\ {k = - \frac{{\rho {a_Y}\beta }}{2}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - 1/2}}.} \end{array}$

 $\varphi \left( {l;x,y} \right) = \frac{{{{\rm{e}}^{ - \beta y}}\left( {{C_1}{x^{{\lambda _1}}} + {C_2}{x^{{\lambda _2}}}} \right)}}{{{C_1}{l^{{\lambda _1}}} + {C_2}{l^{{\lambda _2}}}}},$

 图 1 当γ=0，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 1 The joint Laplace transform function (x, y)→φ(l; x, y) when γ=0

 图 2 当γ= $\frac{1}{2}$，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 2 The joint Laplace transform function (x, y)→φ(l; x, y) when γ= $\frac{1}{2}$
 图 3 当γ=1，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 3 The joint Laplace transform function (x, y)→φ(l; x, y) when γ=1
4 结论

 [1] BLACK F, SCHOLE M. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3): 133–155. [2] HESTON S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. The Review of Financial Studies, 1993, 6(2): 327–343. DOI:10.1093/rfs/6.2.327 [3] HULL J, WHITE A. The pricing of options on assets with Stochastic volatility[J]. Journal of Finance, 1987, 42(2): 281–300. DOI:10.1111/j.1540-6261.1987.tb02568.x [4] STEIN E, STEIN C. Stock price distributions with stochastic volatility: An analytic approach[J]. Review of Financial Studies, 1991(4): 727–752. [5] TALAMANCA G F. Testing volatility autocorrelation in the constant elasticity of variance stochastic volatility model[J]. Computational Statistics and Data Analysis, 2009, 53: 2201–2218. DOI:10.1016/j.csda.2008.08.024 [6] MARIO L. First passage problems for asymmetric wiener processes[J]. Ecolepoly Technique de Montreal, 2006, 43(4): 175–184. [7] CARLSUND A. Cover Times, Sign-Dependent Random Walks, and Maxima[M]. Stockholm: Matematik, 2003. [8] BOROVKOV K, NOVIKOV A. On exit times of Levy-driven Ornstein-Uhlenbeck processes[J]. Statistics & Probability Letters, 2007, 78(12): 1517–1525. [9] PATIE P. On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance[J]. Stoch Process, 2005, 115: 593–607. DOI:10.1016/j.spa.2004.11.003 [10] LOEFFEN R L, PATIE P. Absolute ruin in the Ornstein-Uhlenbeck type risk model[J]. Quantitative Finance, 2010, arXiv:1006.2712. [11] ALILI L, PATIE P, PEDERSEN J L. Representations of the first hitting time density of an Ornstein-Uhlenbeck process[J]. Stock Models, 2005, 21: 967–980. DOI:10.1080/15326340500294702 [12] BO L J, HAO C. First passage times of constant-elasticity-of-variance processes with two-sided reflecting barriers[J]. Journal of Applied Probability, 2012, 49(4): 1119–1133. [13] GERBER H U, YANG H L. Absolute ruin probabilities in a jump diffusion risk model with investment[J]. NAAJ, 2007, 1. [14] BO L J, REN G, WANG Y J. First passage times of reflected generalized Ornstein-Uhlenbeck processes[J]. Stochastics and Dynamics, 2013, 13(1): 1250014. DOI:10.1142/S0219493712500141 [15] BO L J, WANG Y J. Some integral functionals of reflected SDEs and their applications in finance[J]. Quantitative Finance, 2011, 11(11): 343–348. [16] BO L J, TANG D, WANG Y J, et al. On the conditional default probability in a regulated market: A structural approach[J]. Quantitative Finance, 2010, 11(12): 1695–1702. [17] 冯海林, 薄立军. 随机过程——计算与应用[M]. 西安: 西安电子科技大学出版社, 2012. FENG H L, BO L J. Stochastic Process—Calculation and Application[M]. Xi'an: Xidian University Press, 2012. [18] 孙健. 金融衍生品定价模型—数理金融引论[M]. 北京: 中国经济出版社, 2008. SUN J. Financial Derivatives Pricing Model-An Introduction to Mathematical Finance[M]. Beijing: China Economy Press, 2008. [19] 王高雄, 周之铭. 常微分方程[M]. 北京: 高等教育出版社, 2006. WANG G X, ZHOU Z M. Ordinary Differential Equations[M]. Beijing: Higher Education Press, 2006.