Stochastic differential equations involving the two- parameter fractional brownian motion

1.2.4 Fractional Generalized Two-parameter LebesgueStieltjes Integrals

Let 0 = ái = 1, i = 1, 2 be fixed. In what follows, we assume that all functions considered belong to the space D(T), i.e., at every point (x1, x2) ? T, they have limits in the four quadrants

Q⁺⁺(x1, x2) = {(s1, s2) ? T : s1 = x1, s2 = x2} , Q^+-(x1, x2) = {(s1, s2) ? T : s1 = x1, s2 < x2} , Q^-+(x1, x2) = {(s1, s2) ? T : s1 < x1, s2 = x2} , Q^--(x1, x2) = {(s1, s2) ? T : s1 < x1, s2 < x2} ;

furthermore,

f(x1, x2) lim f (s1, s2)

=

(s1,s2)?Q++?(x1,x2)

and, on the sides of the rectangle, the limits that can be defined are supposed to exist and denoted as f(x1, b^-₂ ), f(b^-₁ , x2), f(b^-).Denote f_a+(x) = Ä_af(x), x ? T, and fb-(x) := f(x) - f(x1, b^-₂ ) - f(b-1 , x²) + f(b-),

a := (a1, a2), b := (b1, b2).

Definition 1.2.5. Let f, g : T -? R. The generalized two-parameter LebesgueStieltjes integral of the function f w.r.t to the function g is defined by

_{Z Z}

f(x, y)dg(x, y) :=

K(D:1+ á2 _a+)(x, y)(D¹T^á11?á2 gb-)(x, y)dxdy

+ I (D^á+ f_a+ ) (x, a2) (D- ^á1) (g_b- (x, b₂ ) - g_b- (x, a2)) dx

a₁

^a b2

+ (DZ f_a+₂ ) (a1, y)(D1bá2) (gb-(b1 ,y) - g_b-(a1, y)) dy + f (a)Ä_ag(b)

a2

Where

fa⁺₁ (x, a2) = f(x, a2) - f(a), f_a+₂ (a1, y) = f(a1, y) - f(a),

gb-1 (x, b^-₂ ) = g(x, b^-₂ ) - g(b-), gb^-₁ (x, a2) = g(x, a2) - g(b^-₁ , a2),

gb- _2(b-1 , y) = g(b^-₁ , y) - g(b^-), gb- _2(a1, y) = g(a1, y) - g(a1, b- 2 ).

1.2.4 Fractional Generalized Two-parameter Lebesgue-Stieltjes 18 Integrals