The aim of this exercise is to compute, for t < T < 1 t < T

The aim of this exercise is to compute, for $t<T<1$, the quantity $E(h\left(W_T\right)1_{\{T<g_1\}}\mid G_t)$, which is the price of the claim $h\left(S_T\right)$ with barrier condition $1_{\{T<g_1\}}$.

Prove that

$$E(h\left(W_T\right)1_{\{T<g_1\}}\mid F_t)=E(h\left(W_T\right)\mid F_t)-E\left(h\left(W_T\right)\Phi \left(\frac{\left|W_T\right|}{\sqrt{1-T}}\right)|F_t\right),$$

where

$$\Phi \left(x\right)=\sqrt{\frac{2}{\pi }}{\int }_0^x\exp (-\frac{u^2}{2})du$$

Define $k\left(w\right)=h\left(w\right)\Phi \left(|w|/\sqrt{1-T}\right)$. Prove that $E(k\left(W_T\right)\mid F_t)=\tilde{k}(t,W_t)$, where

$$\begin{array}{cc}\tilde{k}(t,a)& =E\left(k(W_{T-t}+a)\right)\\ & =\frac{1}{\sqrt{2\pi (T-t)}}{\int }_{R}h\left(u\right)\Phi \left(\frac{|u|}{\sqrt{1-T}}\right)\exp (-\frac{(u-a{)}^2}{2(T-t)}\end{array}$$

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