Optimal Betting Strategy question

It's not a full solution, but for a fixed bet amount $k$, so the states/fortune increments are multiples of $k$ the mean first hitting time of the next state (e.g. you start with $X$ coins, $s_0 = X, s_1 = X+k$),

$$\mathbf{E}T_{s_0, s_1}= \sum_{j=1}^{2k+1}\mathbf{E}[T_{s_0, s_1}|D_j]P(D_j)$$

where $D_j$ is the number of days it took to reach the new state. Since you bet every day, the number has to be odd, e.g. $P(D_1)=p_1, P(D_3) = (1-p_1)p_2p_3$, and so on.

Again, it's not a full solution, perhaps there are better approaches (m.b. using Geometric probability/Coupon Collector's Problem), but this will get you started hopefully.,

I don't think there is a general strategy for problems like this unless there is a lot of symmetry (constant probability of heads) or a lot of throws. In your example, you need to increase your capital by a factor $10$, so three wins do not suffice. One approach is to count on winning four, decline to bet the $0.2$, and bet enough that four wins gets you the increase you need. That has a probability of success of $0.105$. You might as well bet your whole bankroll each time, because one loss guarantees you fail anyway.

The other approach is to try and succeed if you get four wins and one loss. If you bet a fraction $f$ of your capital you will have $(1+f)^4(1-f)$ times your capital at the end. We would need that to be $10$, so we solve $(1+f)^4(1-f)=(1+f)^3(1-f^2)=10$. The $1+f$ terms are at most $2$, and the $1-f^2$ term is at most $1$, so there is no solution. Follow the first approach.

Partial solution:

Let $(x_1,...,x_n)\in [-1,1]$ be our sequence of bets on the $n$ days, with $x_i$ meaning that we bet $100|x_i|\%$ of our current capital, with the bet being on head if $x_i>0$, and on tails if $x_i<0$.

Let $(r_1,...,r_n)\in \{0,1\}$ be the sequence of realizations of the coin, with $r_i=1$ meaning head.

Then the probability that the sequence $(x_1,..,x_n)$ of bets leaves us with a capital of at least $m$ is the sum over all probabilities of realizations $(r_1,...,r_n)$ for which holds $$k \cdot\prod_{i=1}^n (1+x_i\cdot(-1)^{r_i})\ge m$$

So in other words, we obtain the following optimization problem:

$$ \max_{(x_1,...,x_n)\in[-1,1]} \sum_{(r_1,...,r_n)\in \{0,1\}^n} \left(\sum_{i=1}^n \left(1-r_i + p_i·(2·r_i - 1)\right) \delta_{\prod_{i=1}^n (1+x_i\cdot(-1)^{r_i})\ge\frac mk}\right) $$

Where $\delta$ is the Dirac-Delta.

Since the inner sum has $2^n$ summands, this maximization problem is only really solvable for small $n$.

It is probably possible to rewrite this optimization problem to fit better within the existing optimization frameworks, though this goes beyond my knowledge.