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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.

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: DOI
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