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.

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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.
Transcribed Image Text: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.
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