I may be completely off, but let me give it a try.

We assume some kind of risk-neutral agent who decides on expected returns, only. As you have specified in the comments, the function $f(t)$ represents the short rate process, i.e. the interest paid in the interval $(t,t+dt)$ is $f(t)dt$. At any point in time $t$, the if we switch from investing at the risk free rate towards buying the lottery ticket with expected (total) return from that point on forward equal to $g(t)$. Both $f(T)$ and $g(T)$ are zero.

The decision maker chooses some optimal time $t^*$ to optimize their expected total future wealth, which we call $W_T$ not as to clutter notation:

$$ W_T=\left( \int_0^{t^*}f(s)ds \right)g(t^*) $$ I.e. they earn at the risk free short rate $f$ until decision time and then use up all their money, invest in the lottery ticket and expect $g(t^*)$ times whatever they invested.

Solving for $f$, we get $f(t)=\alpha+c_1e^{t}$, and the condition $f(T)=0$ fixes the constant $c_1=-\alpha e^{-T}$ so that

$$ f(t)=\alpha\left(1-e^{-(T-t)}\right) $$

Likewise,

$$ h(t)=\beta\left(1-e^{-(T-t)}\right) $$

or

$$ g(t)=h(t)-\gamma = \beta\left(1-e^{-(T-t)}\right) -\gamma $$

You could now solve the integral, write out the product of $\int f ds \times g$ and apply the 'standard' methods to find an optimum.

Does that make sense?

I may be completely off, but let me give it a try.

We assume some kind of risk-neutral agent who decides on expected returns, only. As you have specified in the comments, the function $f(t)$ represents the short rate process, i.e. the interest paid in the interval $(t,t+dt)$ is $f(t)dt$. At any point in time $t$, the if we switch from investing at the risk free rate towards buying the lottery ticket with expected (total) return from that point on forward equal to $g(t)$. Both $f(T)$ and $g(T)$ are zero.

The decision maker chooses some optimal time $t^*$ to optimize their expected total future wealth, which we call $W_T$ not as to clutter notation:

$$ W_T=\left( \int_0^{t^*}f(s)ds \right)g(t^*) $$ I.e. they earn at the risk free short rate $f$ until decision time and then use up all their money, invest in the lottery ticket and expect $g(t^*)$ times whatever they invested.

Solving for $f$, we get $f(t)=\alpha+c_1e^{t}$, and the condition $f(T)=0$ fixes the constant $c_1=-\alpha e^{-T}$ so that

$$ f(t)=\alpha\left(1-e^{-(T-t)}\right) $$

Likewise,

$$ h(t)=\beta\left(1-e^{-(T-t)}\right) $$

or

$$ g(t)=h(t)-\gamma = \beta\left(1-e^{-(T-t)}\right) -\gamma $$

Effectively, $W_T$ is now a function of $t^*$. Given the parameters, we could trace out $t^*$ between $0$ and $T$ and select the value that maximizes $W_T$, or we calculate the first derivative of $W_T$ w.r.t. $t^*$, set it zero and solve for $t^*$ (checking that it's a maximum, of course), thus finding the optimal level.

Does that make sense?