I am given the following identities $$ Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2} $$ $$ \int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(t,x)\right]^2 $$ $$ \int_{t_0}^{t_\ast}dt\left[\int_{t_\ast}^Tdx\alpha(t,x)-\int_t^{t_\ast}dx\sigma(t,x)\int_{t_\ast}^Tdy\sigma(t,y)\right]=\frac{1}{2}\int_{t_0}^{t_\ast}dt\left[\int_{t_\ast}^Tdx\sigma(t,x)\right]^2 $$ and the path integral $$ \int D W e^{-\int_{t_0}^{t_\ast}dt\int_t^{t_\ast}dx\sigma(t,x)W(t)+ip\int_{t_0}^{t_\ast}dt\int_{t_\ast}^Tdx\sigma(t,x)W(t)}e^S. $$ I aim to simplify the path integral above by using the identities. The expression I am supposed to arrive at is $$ \exp\left[-\frac{p^2}{2}\int_{t_0}^{t_\ast}dt\left[\int_{t_\ast}^Tdx\sigma(t,x)\right]^2+\int_{t_0}^{t_\ast}dt\int_{t}^{t_\ast}dx\alpha(t,x)\right]. $$ Now, the first exponential in the path integral's integrated can be written in the form of the first identity: $$ \exp\left[\int_{t_0}^{t_\ast}dtW(t)\left(-\int_t^{t_\ast}dx\sigma+ip\int_{t_\ast}^Tdx\sigma\right)\right], $$ where we left the dependence of $\sigma$ on $t,x$ implicit as simplification. Hence, integrating over $W$, using the first identity $$ \exp\left[\frac{1}{2}\int_{t_0}^{t_\ast}dt\left(-\int_t^{t_\ast}dx\sigma+ip\int_{t_\ast}^Tdx\sigma\right)^2\right]. $$ Working out the square in the exponential gives $$ \left(\int_t^{t_\ast}dx\sigma\right)^2-p^2\left(\int_{t_\ast}^Tdx\sigma\right)^2-2ip\int_t^{t_\ast}dx\sigma(t,x)\int_{t_\ast}^Tdy\sigma(t,y). $$ Using the second identity, this is equal to $$ 2\int_t^{t_\ast}dx\alpha(t,x)-2p^2\int_{t_\ast}^Tdx\alpha(t,x)-2ip\int_t^{t_\ast}dx\sigma(t,x)\int_{t_\ast}^Tdy\sigma(t,y) $$
Plugging this back into the exponential yields $$ e^{\int_{t_0}^{t_\ast}dt\int_{t}^{t_\ast}dx\alpha(t,x)}\exp\left[-\int_{t_0}^{t_\ast}dt\left(p^2\int_{t_\ast}^Tdx\alpha(t,x)+ip\int_t^{t_\ast}dx\sigma(t,x)\int_{t_\ast}^Tdy\sigma(t,y)\right)\right]. $$ This is the furthest I have gotten... Although the second exponential above very much resembles the third identity, I can not figure out how to get rid of the factor "$ip$" in the second term... Any hints/help is much appreciated!
PS: For those interested, this is part of a derivation of the price of a (financial) call option on a zero coupon bond using a path integral approach proposed here: https://arxiv.org/abs/cond-mat/9809199
PPS: This question is purely calculation related, one can assume that all the relevant functions behave properly for all the mathematics to make sense. The functions $\alpha$ and $\sigma$ are deterministic.
