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.

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