We propose a long term portfolio management method which takes into account a liability. Our approach is based on the LQG (Linear, Quadratic cost, Gaussian) control problem framework and then the optimal portfolio strategy hedges the liability by directly tracking a benchmark process which represents the liability. Two numerical results using empirical data published by Japanese organizations are served: simulations tracking an artificial liability and an estimated liability of Japanese organization. The latter one demonstrates that our optimal portfolio strategy can hedge his or her liability.

Abstract:
We derive an analytic expression of the effective potential at finite temperature (T) and chemical potential (mu) in the strong-coupling lattice QCD for color SU(3) including next-to-next-to-leading order (NNLO) effects in the strong coupling expansion. NNLO effective action terms are systematically evaluated in the leading order of the large dimensional (1/d) expansion, and are found to come from some types of connected two plaquette configurations. We apply the extended Hubbard-Stratonovich transformation and a gluonic dressed fermion technique to the effective action, and obtain the effective potential as a function of T, mu, and two order parameters; chiral condensate and a vector potential field. The next-to-leading order (NLO) and NNLO effects result in modifications of the wave function renormalization factor, quark mass and chemical potential. We find that T_{c,mu=0} and mu_{c,T=0} are similar to the NLO results, whereas the position of the critical point is sensitive to NNLO corrections.

Abstract:
We study the phase diagram of quark matter at finite temperature and density in the strong coupling lattice QCD with one species of unrooted staggered fermions including finite coupling ($1/g^2$) effects for color SU($N_c$). We find that we may have partially chiral restored medium density matter at $N_c=3$, which would correspond to the quarkyonic matter suggested at large $N_c$.

Abstract:
We study the QCD phase diagram in the strong coupling limit with fluctuation effects by using the auxiliary field Monte-Carlo method. We apply the chiral angle fixing technique in order to obtain finite chiral condensate in the chiral limit in finite volume. The behavior of order parameters suggests that chiral phase transition is the second order or crossover at low chemical potential and the first order at high chemical potential. Compared with the mean field results, the hadronic phase is suppressed at low chemical potential, and is extended at high chemical potential as already suggested in the monomer-dimer-polymer simulations. We find that the sign problem originating from the bosonization procedure is weakened by the phase cancellation mechanism; a complex phase from one site tends to be canceled by the nearest neighbor site phase as long as low momentum auxiliary field contributions dominate.

Abstract:
We investigate chiral and deconfinement transitions in the framework of the strong coupling lattice QCD for color SU(3) with one species of unrooted staggered fermion at finite temperature and quark chemical potential. We take account of the leading order Polyakov loop terms as well as the next-to-next-to-leading order (1/g^4) fermionic terms of the strong coupling expansion in the effective action. We investigate the Polyakov loop effects by comparing two approximation schemes, a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The effective potential is obtained in both cases, and we analytically clarify the Polyakov loop contributions to the effective potential. The Polyakov loop is found to suppress the chiral condensate and to reduce the chiral transition temperature at mu=0, and the chiral transition temperature roughly reproduces the Monte Carlo results at beta=2N_c/g^2 \lesssim 4. The deconfinement transition is found to be the crossover and first order for light (am_0 \lesssim 4 at beta=4) and heavy quark masses, respectively.

Abstract:
There is a forbidden region in the parameter space of quasinormal modes of black holes in general relativity. Using both inspiral and ringdown phases of gravitational waves from binary black holes, we propose two methods to test general relativity. We also evaluate how our methods will work when we apply them to Pop III black-hole binaries with typical masses. Adopting the simple mean of the estimated range of the event rate, we have the expected rate of 500 ${\rm yr^{-1}}$. Then, the rates of events with signal-to-noise ratios greater than 20 and greater than 50 are 32 ${\rm yr^{-1}}$ and 2 ${\rm yr^{-1}}$, respectively. Therefore, there is a good chance to confirm (or refute) the Einstein theory in the strong gravity region by observing the expected quasinormal modes.

Abstract:
We investigate chiral and deconfinement transitions in the strong coupling lattice QCD for color SU(3). We combine the leading order Polyakov loop effective action of the strong coupling expansion and the next-to-next-to-leading order (1/g^4) fermionic effective action with one species of unrooted staggered fermion. Two approximation schemes are adopted to evaluate the Polyakov loop effects; a Haar measure method (no fluctuation from the mean field) and a Weiss mean-field method (with fluctuations). The Polyakov loop is found to suppress the chiral condensate and to reduce the chiral transition temperature at mu = 0. The chiral transition temperature roughly reproduces the Monte Carlo results in the region beta = 2N_c /g^2 < 4.

Abstract:
We investigate the QCD phase diagram in the strong coupling limit by using a newly developed auxiliary field Monte-Carlo (AFMC) method. Starting from an effective action in the leading order of the 1/g^2 and 1/d expansion with one species of unrooted staggered fermion, we solve the many-body problem exactly by introducing the auxiliary fields and integrating out the temporal links and quark fields. We have a sign problem in AFMC, which is different from the original one in finite density lattice QCD. For low momentum auxiliary field modes, a complex phase cancellation mechanism exists, and the sign problem is not serious on a small lattice. Compared with the mean field results, the transition temperature is found to be reduced by around 10 % and the hadron phase is found to be extended in the larger chemical potential direction by around 20 %, as observed in the monomer-dimer-polymer (MDP) simulations.

Abstract:
We investigate the phase diagram and the mechanism for the sign problem to appear in finite density QCD at strong coupling in a combined framework of the auxiliary field Monte-Carlo (AFMC) and the chiral angle fixing (CAF) methods. When bosonizing meson hopping terms in the effective action, we need to introduce imaginary number coefficients, which leads to a complex phase in numerical simulations. By using the cut-off technique, we quantitatively show that high momentum modes of auxiliary fields mainly contribute to the weight cancellation; Cutting-off high-momentum auxiliary field modes does not modify order parameters but suppresses statistical weight cancellation, when we choose the cut-off parameter appropriately.