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Ito calculus. ) Let X(t) and Y(t) be two stochastic processes, such that dX = μx(X(t), t)dt +ox(X(t), t)dZx pY(Y(t),t)dt +øy(Y(t),t)dZy dY = with dZx (t), dZy(t) being the increments for two distinct Wiener processes, Zx (t) and Zy (t). Let X (ti) = Xį and Y(ti) = Yį. Show that (Xi+1 − Xi)(Yi+1 − Yi) = Xi+1Yi+1 − XiYi — Xi(Yi+1 − Yi) − Yi(Xi+1 − Xi) . - - Then, using the definition of the Ito integral which is the limit of a discrete sum, show that S X (s)dY(s) = [XY] - [* Y (8)dX(s) – [* dx (s)dy (s). (b) Let Z(t) be a stochastic process satisfying dZ = √dt, and assume Z(0) = 0. Using the result in part (a), show that (assuming Ito calculus) z(s) dz(s) = 2(t)² - 1/2

Ito calculus. ) Let X(t) and Y(t) be two stochastic processes, such that dX = μx(X(t), t)dt +ox(X(t), t)dZx pY(Y(t),t)dt +øy(Y(t),t)dZy dY = with dZx (t), dZy(t) being the increments for two distinct Wiener processes, Zx (t) and Zy (t). Let X (ti) = Xį and Y(ti) = Yį. Show that (Xi+1 − Xi)(Yi+1 − Yi) = Xi+1Yi+1 − XiYi — Xi(Yi+1 − Yi) − Yi(Xi+1 − Xi) . - - Then, using the definition of the Ito integral which is the limit of a discrete sum, show that S X (s)dY(s) = [XY] - [* Y (8)dX(s) – [* dx (s)dy (s). (b) Let Z(t) be a stochastic process satisfying dZ = √dt, and assume Z(0) = 0. Using the result in part (a), show that (assuming Ito calculus) z(s) dz(s) = 2(t)² - 1/2

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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We saw that in Ito calculus, an Ito integral is defined in terms of the limit of a so-called “Ito's sum". For the integral fő Z(s)dZ (s), where Z represents the same stochastic process as in (b), the corresponding Ito's sum is ³¹Z(tj)(Z(tj+1) − Z(tj)). Determine the expected value of this sum.
Transcribed Image Text:We saw that in Ito calculus, an Ito integral is defined in terms of the limit of a so-called “Ito's sum". For the integral fő Z(s)dZ (s), where Z represents the same stochastic process as in (b), the corresponding Ito's sum is ³¹Z(tj)(Z(tj+1) − Z(tj)). Determine the expected value of this sum.
4. (a) Ito calculus. ) Let X(t) and Y(t) be two stochastic processes, such that dX μx (X(t), t)dt +ox(X(t), t)dZx dY pY(Y(t),t)dt+øy(Y(t),t)dZy = = with dZx (t), dzy (t) being the increments for two distinct Wiener processes, Zx (t) and Zy(t). Let X(ti) = Xį and Y(ti) = Yį. Show that (Xi+1 − Xi)(Yi+1 − Yi) = Xi+1Yi+1 − XįYi — Xi(Yi+1 − Yi) − Yi(Xi+1 − Xi) . - - - Then, using the definition of the Ito integral which is the limit of a discrete sum, show that ["X(s)dY (s) = [XY - Y(s)dx (s) – ff ax(s)dY (s). - (b) Let Z(t) be a stochastic process satisfying dz Using the result in part (a), show that (assuming Ito calculus) = [z(s) dz ;) dZ (s) 1/12 (1)² - 1/12 = √dt, and assume Z(0) = 0.
Transcribed Image Text:4. (a) Ito calculus. ) Let X(t) and Y(t) be two stochastic processes, such that dX μx (X(t), t)dt +ox(X(t), t)dZx dY pY(Y(t),t)dt+øy(Y(t),t)dZy = = with dZx (t), dzy (t) being the increments for two distinct Wiener processes, Zx (t) and Zy(t). Let X(ti) = Xį and Y(ti) = Yį. Show that (Xi+1 − Xi)(Yi+1 − Yi) = Xi+1Yi+1 − XįYi — Xi(Yi+1 − Yi) − Yi(Xi+1 − Xi) . - - - Then, using the definition of the Ito integral which is the limit of a discrete sum, show that ["X(s)dY (s) = [XY - Y(s)dx (s) – ff ax(s)dY (s). - (b) Let Z(t) be a stochastic process satisfying dz Using the result in part (a), show that (assuming Ito calculus) = [z(s) dz ;) dZ (s) 1/12 (1)² - 1/12 = √dt, and assume Z(0) = 0.
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