Choosing When to Switch from Interest to a Lottery Payoff

# To gamble or not to gamble! (solving a system of ODEs maybe?)


# To gamble or not to gamble! (solving a system of ODEs maybe?)












Assume we have some money. At every point in time $0\le t \le T$, we can take either action 1 that is to keep our money until $T$ say in a bank and have an expected return of $f(t)$ or take action 2 that is to buy a lottery ticket, have an expected return of $g(t)$ and the system disappears then!

The problem is that we don't have a closed-form expression for $f(t)$ but we know that $$\frac{df(t)}{dt}-f(t)+\alpha=0.$$ Similarly, we don't know what $g(t)$ is but we know that $$g(t)=h(t)-\gamma,$$ where $$\frac{dh(t)}{dt}-h(t)+\beta=0.$$

$\alpha, \beta, \gamma>0$ and we also know that no matter what action we take, the system disappears at $t=T$ so $f(T)=h(T)=0$, so we would like to take an action that maximizes our return.

If we know that there is a time interval, or even a point in time, say $\tau$ where it's best to buy the lottery ticket, how can we compute the return corresponding to that time interval?

We can solve the two ODEs and we see that both $f(t)$ and $h(t)$ and consequently $g(t)$ are decreasing and concave in $t$, if we use $f(T)=h(T)=0$ as the boundary condition. And that's all I could do!

Can anyone give me a hint?

## Answer by Kermittfrog (score 1, accepted)

https://quant.stackexchange.com/a/69620

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?