investors will not pay for risk premiums, and defined by following stochastic differential equations:
Where: S(t) represents the price process of an underlying assets. V(t) is the variance of the corresponding instantaneous returns. And the initial condition S(0), V(0) should be strictly positive. r is the risk-free return rate (non-negative constant). k is the speed of the mean reversion. θ is the mean value of the variance. σ is the volatility of the variance. These three are positive constants. Ws(t) and Wv(t) are Brownian motion variables, and ρ is the correlation coefficient between them. In some probability measure, we assume dWs(t) * dWv(t) = ρdt.
For computational convenient, it is general to employee logarithm to transform the asset price process S(t) into X(t) = Log (S(t)), and apply Itô’s Lemma(3) to equations (1).
We can get following stochastic differential equations:…
To give an another definition of complex number, we have to show z= ew for w, where z is any non-zero complex number. If we assume w = u + iv then
eueiv = |z|e^i arg z and this means that
|z| = e^u , v = arg z .
The equation |z| = e^u is a real equation, so we can write u = ln |z|, where ln |z| is
the ordinary logarithm with positive real numbers. Hence,
w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , ±1 , ±2 , ±3 , . . .6
We profoundly examined the connection between…
1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number?
If the equation is 7th degree then it has 7 roots.
Those roots can be complex or real.
Complex roots always come in pairs, so if it has one, then it has 2, the other one being the conjugate of the first one. This in other words, if one complex root is a + bi, then the other complex root is a – bi.
If at least one root were complex, then we would have a…
Title: Application of Self-Organization Neural Network Technique (SOM) to Optimize Finite- Element Partial Differential Equation (PDEs) Results in Square-Shaped Structures Analysis.
The finite-element method (FEM) is a computationally method for solving partial differential equations (PDEs) with specific boundary conditions over a domain. When we applying the FEM to a domain, it has to divide to a finite number of elements and nodes. The collections of the elements and nodes form the…
Substituting , and into (31), and calculating the derivatives with respect to , we obtain
Since , we have
Hence there is a Hopf bifurcation at . We have the following result:
Theorem 13. Suppose holds. Then the system (29) undergoes a Hopf bifurcation when passes through emanating from the steady state leading to periodic solutions trajectories for either (super-critical bifurcation), or (sub-critical-bifurcation) or at .
The fundamental theorem of Calculus:
The fundamental theorem of calculus asserts the interrelated properties of integration and differentiation. It says that a function when differentiated, can be brought back by integrating (anti-derivative) or a function when integrated, can be brought back by differentiation.
First theorem: Let f be a function that is integrable on [a,x] for each x in [a,b], then let c be such that a≤c≤b and define a new function A as follows,
A(x)=∫_c^x▒f(x)dt, if a≤x≤b.…
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of…
Differential Equations was the course that made me question my decision in taking up Chemical Engineering. I actually had no idea it was this challenging. All I had in mind was that it’s just Calculus 1 and 2 combined. Before we officially started the class, Engr. Cabatingan introduced us the syllabus and made us study it. The topics were “bearable”. However, I really had no idea what was Modeling all about. We even joked about it, more so when we knew that there’s going to be a gallery walk. In…
An alternative modeling approach which remedies the computational aspect is differential equations. In contrast to the discrete event simulations, averaged quantities in case of large quantity production predict the time evolution of parts and include the dynamics inside the different production steps. To derive accurate continuous models the overall modeling goal is to transfer as much of the detailed and complex discrete model to the continuous level. This will be achieved regarding…
If R and r is the Circumradius and Inradius of a non-degenerate triangle then due to Euler we have an Inequality stated as and the equality holds when the triangle is equilateral. This ubiquitous inequality occurs in the literature in many different equivalent forms  and also Many other different simple approaches for proving this inequality are known. (some of them can be found in , , , , and  ). In this article we present a proof for this Inequality based on two basic…