ABSTRACT
In this chapter and the next two we illustrate some applications to …- nance, risk management and economics which are drawn mainly from our research interests.
The following notation is used throughout this chapter and the next two. Various classes of utility functions and stochastic orders are de…ned as follows. For n = 1; 2; 3;
UAn = fu : R! R : (1)iu(i) 0; i = 1; :::; ng USAn = fu : R! R : (1)iu(i) < 0; i = 1; :::; ng UDn = fu : R! R : u(i) 0; i = 1; :::; ng USDn = fu : R! R : u(i) > 0; i = 1; :::; ng
where u(i) denotes the ith derivative of u. Also,
UEA1 (UESA1 ) = fu : R! R : u is (strictly) increasingg UEA2 (UESA2 ) = fu : R! R : u is increasing, and (strictly) concaveg UED2 (UESD2 ) = fu : R! R : u is increasing, and (strictly) convexg UEA3 (UESA2 ) = fu : R! R : u 2 UEA2 , u0 is (strictly) convexg UED3 (UESD2 ) = fu : R! R : u 2 UED2 , u0 is (strictly) convexg
Let R be the set of extended real numbers and = [a; b] be a subset of
be a measure on ( ; B). For random variables Y; Z with corresponding distribution functions F;G, respectively,
FA1 (x) = PX [a; x] and F D 1 (x) = PX [x; b] for all x 2 ; (8.1)
and
F = Y = E(Y ) =
Z b a x dF (x)
G = Z = E(Z) =
Z b a x dG(x)
HAj (x) =
HDj (x) =
Z b x HDj1(y) dy; j = 2; 3, (8.2)
where H = F or G: We assume
FA1 (a) = 0 and F D 1 (b) = 0: (8.3)
For H = F or G, we de…ne the following functions for Markowitz Stochastic Dominance (MSD) and Prospect Stochastic Dominance (PSD):
Ha1 (x) = H(x) = H A 1 (x)
Hd1 (x) = 1H(x) = HD1 (x) Hdj (y) =
Z 0 y Hdj1(t)dt; y 0
Haj (x) =
Z x 0 Haj1(t)dt; x 0; j = 2; 3: (8.4)
See De…nition 8.5 (and Section 8.3). In order to make the computation easier, we further de…ne
HMj (x) =
HAj (x) for x 0 HDj (x) for x > 0
HPj (x) =
Hdj (x) for x 0 Haj (x) for x > 0
(8.5)
De…nition 8.1 Y dominates Z and F dominates G in the sense of FASD (SASD, TASD), denoted by
Y A1 Z or F A1 G (Y A2 Z or F A2 G; Y A3 Z or F A3 G);
if and only if
for each x in [a; b], where FASD, SASD and TASD stand for …rst, second and third-order ascending stochastic dominance, respectively.
