Mathematical Finance

Problem-Solving
Name
Institution

Problem Solving
Part 1
V(t,x) =f(t)log(x+gty)+h(t) =
=Vt-12(u-rδ)2VX2VXX+rxVX-βV+yVx+maxc-cVX+u1c=0=Vt-logu-rδVX2VXX +rxVX-βV+yVx+maxc-cVX+u1c=0Focusing on the time limit

Such that

Rearranginging

Deviding by h

Taking limits as h0

s.t

The economic interpretation of g(t)y is labor income at the time, t.
Part 2
2. 1. Optimal Trading Strategy
Wealth process
dXtπ=πtTμtdt+σtdWt+Xtπ-1nTπtX0=x0y=EβTI(βTST, where I is the optimal trading.

where ,
and is uniquely determined by
2.2. Optimal Consumption in terms of X(t)+g(t)y=
Such that,left952500

=

Reference
Bensoussan, A., & Zhang, Q. (2009). Mathematical modeling and numerical methods in finance. Amsterdam: Elsevier/North-Holland.