Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on their expectations of the prices they will choose at all future times, as well as of the exogenous random variables determining future events. I can write down the model of their behavior in a few lines, but it is surprisingly hard to solve. But it does not seem technically that difficult, and I am sure a professional mathematician can quickly put me on the right track. Thanks for your help.
Problem description
In this model time is divided into discrete periods: $t$, $t+1$, ... and there is one currency being traded. There is a professional currency trader who sits in the marketplace for this currency, and at each time $t$, she decides the price $P_t$ which she will charge for the currency, given the two pieces of information she knows: how many orders she is getting that period ($x_t$) and what is the "fundamental value" of the currency ($V_t$).
$V_{t}$, the fundamental value of the currency, follows a random walk:
$ V_t = V_{t - 1} + N\left( {0,\sigma _v^2 } \right) \\ $
As for $x_t$: at each $t$ orders to buy or sell the currency come in from ordinary people (tourists, corporations, etc). The sum of these orders (it can be positive or negative) is normally distributed with a mean that depends on the difference of last period's price from last period's fair value:
$ x_t \sim \mu \left( {V_{t - 1} - P_{t - 1} } \right) + N\left( {0,\sigma _x^2 } \right) $
where $P_{t-1}$ was the price the trader chose last period.
Every period, the trader takes all the orders that come in. So, starting at time zero, by time t the sum total of currency she holds is
$ X_t = \sum\limits_{s = 0}^t {x_s } $
Finally, from time $t$ to $t+1$ she earns the interest rate $i_t$ on this currency.
So, the profit $ \Pi \left( {P_t } \right)$ that she will earn from $t$ to $t+1$ is given by the difference of the price she is paid today, versus the price plus interest rate that the currency is worth tomorrow. Because she doesnt like waiting for profits, the parameter $\delta$ (in real-life about 0.99), slightly downweights the future terms, so the expression is:
$ \Pi \left( {P_t } \right) = X_t \left( {P_t - \delta \left( {i_t + P_{t + 1} } \right)} \right) $
For economic reasons, we impose the condition that she sets a price which exactly compensates her for the risk she is taking - that is, her expected profit is proportional to the variance of her profit:
$ E\left[ {X_t \left( {P_t - \delta \left( {i_t + P_{t + 1} } \right)} \right)} \right] = \rho/2* Var\left[ {X_t \left( {P_t - \delta \left( {i_t + P_{t + 1} } \right)} \right)} \right] $
Hence the equation I am trying to solve is as follows:
Solve
$ X_t \left( {P_t - \delta \left( {i_t + EP_{t + 1} } \right)} \right) - \rho/2* \delta ^2 X_t^2 Var\left[ {P_{t + 1} } \right] = 0 $
for $P_t$, in closed-form (ie without terms in the future mean and variance of $P_{t+s}$), given the stochastic processes specified above for the evolution of $V_t$ and $X_t$.
