# Portfolio optimization with transaction cost with order 1/2

The following question remains unsolved, and post it here. Maybe it’s not that hard, maybe not well-posed.

Let ${(\Omega, \mathcal{F}, P)}$ be a complete probability space, ${W}$ be a standard Brownian Motion, and ${\mathbb{F} =\{\mathcal{F}_t\}_{t\ge 0}}$ be the filtration generated by ${W}$, augmented by the ${P}$-null sets as usual. The financial market consists of two assets, a bank account with interest rate ${r = 0}$ and a stock whose price ${X}$ follows the dynamics

$\displaystyle d X_r = X_r(b dr + \sigma dW_r). \ \ \ \ \ (1)$

We now introduce our optimization problem on a subinterval ${[t,T]}$. Given ${(x,y,z)}$ and an admissible trading strategy ${Z}$, we shall set ${Y_{t-} := y}$ and ${Z_{t-}:=z}$. Then the wealth process satisfies

$\displaystyle Y^{t,x,y,z,Z}_T := Y_T = y + \int_t^T Z_s dX_s - \sum_{t\le s\le T} c(\Delta Z_s). \ \ \ \ \ (2)$

Then, our optimization problem is:

$\displaystyle V(t,x,y,z) := \sup_{Z\in \mathcal{Z}_t} \mathbb{E} \Big[ Y^{t,x,y,z,Z}_T\Big]. \ \ \ \ \ (3)$

Here the set ${\mathcal{Z}_t}$ of the admissible strategies is defined rigorously at below: Given ${t\in [0,T] }$, the set of admissible strategies, denoted by ${\mathcal{Z}_t}$, is the space of ${\mathbb{F}}$-adapted processes ${Z}$ over ${[t,T]}$ such that, for a.s. ${\omega}$,

• ${Z}$ is RCLL and piecewise constant with finitely many jumps;
• ${Z_T=0}$, and ${|Z|\le M}$.

Our main interest is the optimization under

• ${c(x) = |x|^{1/2}}$.

Define

$\displaystyle \mathcal{O}_\varphi (z):= \{(t,x,y): \varphi(t,y,z) > \varphi(t, y-c(\tilde z-z),\tilde z),\forall \tilde z\neq z\}.$

and

$\displaystyle \mathcal{M} \varphi(t,x,y,z) = \sup_{ \tilde z\neq z} \varphi (t, x, y-c(\tilde z-z), \tilde z).$

Then, if ${V}$ is continuous, we have ${V(t,x,y,z) = \mathcal{M} V(t,x,y,z)}$. But it’s not clear ${\mathcal{O}_V(z)}$ is open.

For a smooth function ${\varphi}$, define For a smooth function ${\varphi(t,x,y,z)}$, denote

$\displaystyle L\varphi(t,x,y,z) = {1\over 2} \sigma^2 x^2 [\varphi_{xx} + 2z \varphi_{xy} + z^2\varphi_{yy} ] + bx[\varphi_x + z \varphi_y]. \ \ \ \ \ (4)$

Let ${\mathcal{V}}$ denote the space of functions ${\varphi(t,x,y,z)}$ satisfying:

1. ${\varphi}$ is uniformly Lipschitz continuous in ${(x,y)}$ and continuous in ${t}$;
2. ${\varphi(t,x,y,z)\ge \varphi(t,x,y-c(\tilde z-z),\tilde z), ~~ \forall \tilde z\neq z}$.
3. ${\varphi}$ is continuous in ${z}$ for ${z\neq 0}$, and ${\varphi(t,x,y,0+)}$ and ${\varphi(t,x,y,0-)}$ exist. Moreover, if ${c(0+)=c(0)}$, then ${\varphi(t,x,y,0-)=\varphi(t,x,y,0)}$; if ${c(0-)=c(0)}$, then ${\varphi(t,x,y,0+)=\varphi(t,x,y,0)}$.

Let ${\varphi\in \mathcal{V}}$.

1. ${\varphi}$ is a viscosity subsolution, if for any smooth function ${f\ge \varphi}$ with ${\varphi = f}$ at ${(t, x, y, z)}$ with ${(t, x, y)\in \mathcal{O}_\varphi (z)}$, we have

$\displaystyle (f_t + L f)(t, x, y,z) \ge 0.$

2. ${\varphi}$ is a viscosity supersolution, if for any smooth ${f\le \varphi}$ with ${\varphi = f}$ at ${(t, x, y, z)}$, we have

$\displaystyle (f_t + L f)(t, x, y, z) \le 0.$

3. ${\varphi}$ is a viscosity solution if it is both a viscosity supersolution and a viscosity subsolution.

We may want to characterize ${V}$ by unique viscosity solution with some other conditions.