So this is a work based on the previous and ongoing works with Professor Everal do Bonoto

and Professor Marcel Fedder, so both from University of São Paulo.

And I thought, although the title of my talk induced the study of stability, I'm going

to emphasize in the presentation of a new class of equations which we call generalized

stochastic equation and I'm going to talk that this new kind of equation actually

encompasses lots of stochastic equations. So at the end, any theory in this new class

of equations is a unifying theory. In order to do that, I'm going first give a motivation

of our research, so I'm going to talk about modern integration theory and generalize of

these. Then I'm going to start to talk about stochastic calculus, present our main tool

here and stability outcomes. So first, I know most of you are going to be mad at me at this

time because my goal now is to convince you that neither Heumann or Lebesgue integral

are a fully satisfactory theory. So the first one is Heumann, it's the first integral that

we learn, it's the first one that we're thinking about when we are studying calculus, but Heumann

integral is quite limited. It's limited not only by bounded functions but also to functions

who have elementary antiderivatives, this is the fundamental theory of calculus. Also

the Heumann integral does not yield good conversion theorems, for instance a sequence of Heumann

integrable functions may converge, point-wise, to non-Hemann integrable functions, so this

is an example. If I would take Rn, a sequence of irrational numbers and define f1 of x being

1 if x is the first rational number, f2 is 1 if x is the first two rational numbers and

so on, all of these functions here are Heumann integrable but they converge to the direct

functions which we know that's not Heumann integrable. So Lebesgue tried to overcome all

these limitations and in fact he did, but even to define the Lebesgue integral we required

a considerable amount of measured theory, so it's a complex method. Also the Lebesgue

integral adds additional constraints on the fundamental theorem of calculus as we're going

to see and there are still a large number of functions which cannot be integrated and

it does not guarantee that every derivative is integrable. For example, if I take capital

F like this and G the derivative of F, G is not Lebesgue integrable because it's unbounded

of unbounded variations, not absolutely integrable. And in comparing with the improper Heumann

integral and with the Lebesgue integral we find that neither is strictly more general

than the other one. So in the Fivics, Hanestock and Kurzweil separately start to define a

new type of integral which we call Hanestock-Kurzweil integral and in their definition the Heumann

approach is preserved. The only difference is how they take the divisions. So in the

Heumann integral our subintervals are bounded, are limited by a constant delta. In the Hanestock-Kurzweil

integral those subintervals are bounded by a positive function which is called gauge

and also denoted by delta. So let's see. So our tag division of an interval here AB is

a point interval pair so tau E and subintervals where tau E our tag is a point of our interval

here. So given a positive function we say that this tag division here is delta phi and

whenever this inclusion here holds. So this is a geometric representation of delta phi

and fineness. So as we can see our subinterval is contained in a bigger interval whose length

is given by two delta applied or valued in our tag. So Kuzilev ensures that for every

gauge there is always a delta phi tag division. So the definition of Kurzweil integral is

the following. We consider X a Banach space, a function U defined from AB times MB into

X is Hanestock-Kurzweil integrable if there is an element in our Banach space that this

inequality here holds for all delta phi tag division. And we denote the Kurzweil integral

by this capital D here. So if you define U of tau t is equal to F of tau g of t so the

sum that appears in the definition of the Kurzweil integral is like the Perron-Stielt

sum, the Hiemann-Stielt sum. So this means that the Kurzweil integral is equal to the

Perron-Stielt integral and just the notation when g is the identity functions we call this

integral here by Perron. So let's see some advantages of these integrals. So using the

delta phi division instead of the traditional one we have more variation in how we can choose