zbMATH — the first resource for mathematics

Lower bounds for densities of Asian type stochastic differential equations. (English) Zbl 1196.60105
Consider the following system of stochastic differential equations, $X^1_t = x^1 + \int_0^t \sigma(X_s)dW_s + \int_0^t b_1(X_s)ds, \quad X^2_t = x^2 + \int_0^t b_2(X_s)ds, \quad t \in [0,T].$ They assume that $$\sigma,b_1,b_2$$ are five times differentiable and have bounded derivatives but the functions themselves do not need to be bounded. The main goal of this paper is to give lower bounds for the density $$p_T(x,y)$$ of $$X_T(x)$$. These type of equations are linked to the so-called Asian option set-up as the authors claims.

MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 91G80 Financial applications of other theories 60H07 Stochastic calculus of variations and the Malliavin calculus 91G20 Derivative securities (option pricing, hedging, etc.)
Full Text:
References:
 [1] Aida, S.; Kusuoka, S.; Stroock, D., On the support of Wiener functionals, (), 3-34 · Zbl 0790.60047 [2] Bally, V., Lower bounds for the density of the law of locally elliptic ito processes, Ann. probab., 34, 2406-2440, (2006) · Zbl 1123.60037 [3] Ben-Arous, G.; Leandre, R., Decroissance exponentille du noyau de la chaleur sur la diagonale (II), Probab. theory related fields, 90, 377-402, (1991) · Zbl 0734.60027 [4] Boscain, U.; Polidoro, S., Gaussian estimates for hypoelliptic operators via optimal control, Rend. lincei mat. appol., 18, 333-342, (2007) · Zbl 1146.35026 [5] F. Delarue, S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, 2009, in preparation · Zbl 1223.60037 [6] Fefferman, C.L.; Sanchez-Calle, A., Fundamental solutions of second order subelliptic operators, Ann. of math. (2), 124, 247-272, (1986) · Zbl 0613.35002 [7] Kohatsu-Higa, A., Lower bounds for densities of uniform elliptic random variables on Wiener space, Probab. theory related fields, 126, 421-457, (2003) · Zbl 1022.60056 [8] Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus, part III, J. fac. sci. univ Tokyo sect. 1A math., 34, 391-442, (1987) · Zbl 0633.60078 [9] Malliavin, P.; Nualart, E., Density minoration of a strongly non-degenerated random variable, J. funct. anal., 256, 4197-4214, (2009) · Zbl 1175.60055 [10] Nualart, D., The Malliavin calculus and related topics, (1995), Springer-Verlag Berlin · Zbl 0837.60050 [11] Pascucci, A.; Polidoro, S., Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators, Trans. amer. math. soc., 358, 4873-4893, (2006) · Zbl 1172.35339 [12] Polidoro, S., A global lower bound for the fundamental solution of a kolomgorov – fokker – planck equation, Arch. ration. math. anal., 137, 321-340, (1997) · Zbl 0887.35086 [13] Sanchez-Calle, A., Fundamental solutions and geometry of the sum of square of vector fields, Invent. math., 78, 1, 143-160, (1984) · Zbl 0582.58004 [14] Yor, M., On some exponential functionals of Brownian motion, Adv. in appl. probab., 24, 509-531, (1992) · Zbl 0765.60084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.