Let W be a standard Brownian motion, and also a Є R and ẞ Є R. (a) For a given smooth function 9 : R→ R, assume that the function ƒ : R+ × (0, ∞) → R satisfies f(0,x) = g(x) for x = R, as well as af af (T,x) Ξα -(T, x) + Эт მე B² a²ƒ 2 0x2 (T,x), for (T,x) Є R+ × R. Show that f(t, x)=E[g(x+at+BW₁)], for (t, x) = R4 × R.

Let W be a standard Brownian motion, and also a Є R and ẞ Є R. (a) For a given smooth function 9 : R→ R, assume that the function ƒ : R+ × (0, ∞) → R satisfies f(0,x) = g(x) for x = R, as well as af af (T,x) Ξα -(T, x) + Эт მე B² a²ƒ 2 0x2 (T,x), for (T,x) Є R+ × R. Show that f(t, x)=E[g(x+at+BW₁)], for (t, x) = R4 × R.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.1: Introduction To Linear Transformations

Problem 45E: Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a...

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