Similar to last assignment. 3 questions, 1 and 3 relatively simple. Please deliver LaTeX code with the solution.Document Preview:1. In this question, you may use the fact that the Fourier transform of the Gaussian probability density function G.r(r):12tro2)-rl2e-n2l?o2), for z ?? IR is given bY d)() –o222/2? forzelR.ando>0. (a) Let f (r,t) satisfy the (log-space) Black-scholes equation o : -rf+ (r -o 12)f, +llo 7,, + S, arrd delirrc ^ fx , 1-Q,t) ??R J*f (r,t)e- dr, z Usc thc R)trlier transf0, and initial condition u(r,a,o) : ug(n,Y), for 0 ( r I I, o I Y 1 I, can be solved using the two-dimensional explicit Euler scheme Ui,[ : Ui,k + p (Ui*r,^ * Uit,* + Uin+t + uint -+Uin), for 1 0. (a) Find those values of p, for which the two-dimensional explicit Euler scheme is spectrally stable. (b) Conjecture the form of the two-dimensional implicit Euler scheme, and prove that it is spectrally stable for all p, > 0. Hint: The underlying matrix is still a Toeplitz matrix, albeit in two dimensions. Its eigenvectors are given by u|,in : sinff ,inff , for 1 ( i,kAttachments: IMG.pdf
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