2010年12月25日星期六
2010年11月13日星期六
Things I'm interested
counterparty credit risk
contagion model?
correlated default
Calypso system
RiskMetrics system
Copulas
contagion model?
correlated default
Calypso system
RiskMetrics system
Copulas
2010年11月12日星期五
damn it! the stopping time is independent on the control
The stochastic control is not working, since the stopping time is dependent on the proportion.
I am thinking of the superreplication of the lookback option now.
I am thinking of the superreplication of the lookback option now.
2010年11月8日星期一
We have solved M. problem?
The problem becomes quite interesting once I apply the viscosity solution theory.
Actually the second derivative Y_{xx} might not be always positive. (I made a terrible assumption last time by assuming Y to be strictly concave.) Hence, the value function Y(x) might be convex for small x and concave for larger x. It is very nice, since when it's convex, the value function satisfies the Black-Scholes-Merton equation. We should be fascinated about our result.
Cannot wait to discuss it with "Big Boss".
Actually the second derivative Y_{xx} might not be always positive. (I made a terrible assumption last time by assuming Y to be strictly concave.) Hence, the value function Y(x) might be convex for small x and concave for larger x. It is very nice, since when it's convex, the value function satisfies the Black-Scholes-Merton equation. We should be fascinated about our result.
Cannot wait to discuss it with "Big Boss".
2010年11月1日星期一
replication, superreplication & Stochastic Singular Control
After turning the random duration to finite during, I have discovered some interesting facts:
1. With the discountinous boundary conditions, the value function V(t,x) is not C^{1,2}. c.f. Pham's book (example on singular control)
2. Need to study viscosity solution (totally forgotten subject)
3. Now that it's a finite boundary problem, it becomes how to replicate/superreplicate the digital option with short constraint on portfolios
4. c.f. Cvitanic or Uwe Wystup's dissertation from CMU.
1. With the discountinous boundary conditions, the value function V(t,x) is not C^{1,2}. c.f. Pham's book (example on singular control)
2. Need to study viscosity solution (totally forgotten subject)
3. Now that it's a finite boundary problem, it becomes how to replicate/superreplicate the digital option with short constraint on portfolios
4. c.f. Cvitanic or Uwe Wystup's dissertation from CMU.
2010年10月31日星期日
Sam's Choice @ blogger.com: My First Blog:: On Stochastic Control on Random Ho...
http://samuelyusunwang.blogspot.com/2010/10/my-first-blog-on-stochastic-control-on.html?spref=bl: "The main reference paper is El Karoui's paper: 'Optimal Investment and consumption with random time'. Now the value function..."
My First Blog:: On Stochastic Control with Random Horizon & Terminal T
The main reference paper is El Karoui's paper: "Optimal Investment Decisions When Time Horizon is Uncertain".
Now the value function is associated with a terminal (utility) function g(u)=Id_{u=D}.
I don't know whether the HJB equation has a C^{1,2} solution.
Very difficult to find the close form solution of PDE.
I'm also finding that the recent paper http://www.maths.univ-evry.fr/prepubli/308.pdf could be a useful tool to link the problem to BSDE or FBSDE. Maybe I shall call my advisor?
Now the value function is associated with a terminal (utility) function g(u)=Id_{u=D}.
I don't know whether the HJB equation has a C^{1,2} solution.
Very difficult to find the close form solution of PDE.
I'm also finding that the recent paper http://www.maths.univ-evry.fr/prepubli/308.pdf could be a useful tool to link the problem to BSDE or FBSDE. Maybe I shall call my advisor?
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