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We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation, \begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*} we prove the existence of infinitely many radially symmetric solutions. Also, we perform rigorous asymptotic analysis to identify the blow-up limit when the steady state is close to a constant solution and the blow-down limit when the maximum of the steady state goes to the infinity.

This paper studies a dissipative heat equation with Delta sources of non-linear strength located on a periodic lattice. The model arises from intracellular waves in continuum excitable media with discrete release sites. Due to the presence of Delta sources, the solution of the model has discontinuous spatial derivatives. We focus on the bistable regime of the model, determined by the decay strength parameter a and the separation distance L between release sites, in which the model admits exactly three L-periodic steady states. We establish the existence of pulsating waves spatially connecting them. For the case of waves connecting two stable L-periodic steady states, the uniqueness and global exponential stability of pulsating waves are shown. Also a new technique is introduced to find the fine structure of the tails of pulsating waves.

In this paper, a new model for corporate bond pricing and credit rating is proposed. In this model, credit rating migrations are assumed to depend on the ratio of debt and asset value of the underl...

We study spreading phenomena in a two-component reaction-diffusion system, modeling the Neolithic transition from hunter-gatherer societies to agricultural societies in Europe. When farmer populations were introduced into the region inhabited by hunter-gatherer populations, farmer populations spread as an expanding wave through Europe. The speed of the wave is observed to tend to be a constant. We analytically compute the spreading velocity of the model, and characterize the longtime behavior of solutions of the model. Without comparison principle, we introduce a new technique that can locate the spreading front in reaction-diffusion systems.

In this thesis, we study asymptotic behaviors of a free boundary arised from the evaluation of a corporate bond subject to changes of the credit rating of the underlying company. The credit rating migration is modeled by a free boundary which separates different credit rating regions in a state space. We first formulate the mathematical problem and then we establish the well-posedness of the problem and the long time-to-expiry behavior of the solution. As a result, we describe the location and asymptotic line of the free boundary. Certain numerical simulations are also provided.

Metal−organic frameworks (MOFs) with porous structures, high surface areas, diverse compositions, and functional linkers are promising materials and good carriers for building high-performance devices. In this work, uniform cobalt-doped ZnO nanoparticles (Co-doped ZnO NPs) derived from a MOF mold were synthesized, demonstrating the first example of synthesizing doped semiconductor metal oxide nanostructures using such strategy. The synthesis method produced Co-doped ZnO NPs that had a controllable doping mode, adjustable surface status, good dispensability, ferromagnetism and catalytic activity. The Co-doped ZnO NPs were evaluated as a sensing material for diabetes biomarker detection; the obtained sensors showed a high response to trace acetone (18.2 at 5 ppm), fast response/recovery times, a low detection limit (170 ppb), and long-term stability for 4 months. The enhanced sensing performance can be attributed to the increased number of active sites, additional impurity energy levels, and the catalytic ability of elemental Co. Moreover, the optimized sensor could distinguish between simulated diabetic breath and healthy human breath samples. The MOF-derived Co-doped ZnO NPs are a good candidate for the low-cost and noninvasive diagnosis of diabetes, and the proposed synthesis strategy can be extended to other types of extrinsically doped oxide materials.

In this paper, we propose a new weak solution to an optimal stopping problem in finance and economics. The main advantage of this new definition is that we do not need the Dynamic Programming Principle, which is critical for both classical verification argument and modern viscosity approach. Additionally, the classical methods in differential equations, e.g. penalty method, can be used to derive some useful results.

In this paper, we consider the singular limit of an energy minimizing problem which is a semi-limit of a singular elliptic equation modeling steady states of thin film equation with both Van der Waals force and Born repulsion force. We show that the singular limit of energy minimizers is a Dirac mass located on the boundary point with the maximum curvature.

Long-term stability remains a great challenge for metal halide perovskite solar cells (PSCs). The utilization of ionic liquids (ILs) is a promising strategy to solve the stability problem. However, few studies have focused on controlling the halide anions of ILs, in which different organic cations can modulate the melting point of ILs and film crystal growth. Here, ILs with a 1-ethyl-3-methylimidazolium (EMIM

Consider the inverse first-passage problem: Given a diffusion process $\{\frak{X}_{t}\}_{t\geqslant 0}$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a survival probability function $p$ on $[0,\infty)$, find a boundary, $x=b(t)$, such that $p$ is the survival probability that $\frak{X}$ does not fall below $b$, i.e., for each $t\geqslant 0$, $p(t)= \mathbb{P}(\{\omega\in\Omega\;|\; {\frak{X}}_s(\omega) \geqslant b(s),\ \forall\, s\in(0,t)\})$. In earlier work, we analyzed viscosity solutions of a related variational inequality, and showed that they provided the only upper semi-continuous (usc) solutions of the inverse problem. We furthermore proved weak regularity (continuity) of the boundary $b$ under additional assumptions on $p$. The purpose of this paper is to study higher-order regularity properties of the solution of the inverse first-passage problem. In particular, we show that when $p$ is smooth and has negative slope, the viscosity solution, and therefore also the unique usc solution of the inverse problem, is smooth. Consequently, the viscosity solution furnishes a unique classical solution to the free boundary problem associated with the inverse first-passage problem., Comment: 24 pages

A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox–Ingersoll–Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model; that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

Two mathematical models under so-called intensity and structure frameworks to pricing a double defaultable interest rate swap are established. The default could happen or jump to a high probability in both fixed and floating parties on the predetermined boundaries. The models lead to a new and interesting mathematical problem. As the intensity approaches infinity in designated regions, the solutions of the intensity models converge to a solution of a structure-type model which is an initial value problem of a partial differential equation coupled with two obstacles problem in their restricted regions. According to the value of the fixed rate, three cases are discussed. The free boundary that determines the swap rate and the free boundaries that determine the earlier termination of the contract (due to counterparty’s default) are analysed.

We investigate the asymptotic behavior, as the number of options approaches zero, of the optimal exercise strategy of continuous-exercise perpetual executive stock options. Here the optimal strategy is described in terms of the free boundary of a degenerate variational inequality. To our knowledge, our method based on the convexity of the variational inequality at the initial time is new. Also, the unique formal asymptotic expansions of the free boundary and the solution of the variational inequality are derived.

We solve the optimal portfolio choice problem for an investor who can trade a risk-free asset and a risky asset. The investor faces both Brownian and jump risks and the jump is modeled by a Hawkes process so that occurrence of a jump in the risky asset price triggers more sequent jumps. We obtain the optimal portfolio by maximizing expectation of a constant relative risk aversion (CRRA) utility function of terminal wealth. The existence and uniqueness of a classical solution to the associated partial differential equation are proved, and the corresponding verification theorem is provided as well. Based on the theoretical results, we develop a numerical monotonic iteration algorithm and present an illustrative numerical example.

The reaction-diffusion system $ u_t = u_{xx} - u g(v), v_t = D v_{xx} + u g(v)$, where $ g \in C^{1, \gamma} [0, \infty) \cap C^{2} (0, \infty) \rightarrow [0, \infty) $, $ g(0)= g'(0) =0$, $g(v) >...

In this paper, we study a Boltzmann-type mean field game model proposed in Achdou et al. [Philos. Trans. A, 372 (2014), pp. 1--19] for knowledge diffusion and economic growth, where knowledge diffu...

Genetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright--Fisher model. The diffusion limit of this model is a degen...

This paper investigates a utility maximization problem in a Black--Scholes market, in which trading is subject to a convex cone constraint and the utility function is not necessarily continuous or ...

Background Kindlin Family Members have been reported to be aberrantly expressed in various human cancer types and involved in tumorigenesis, tumor progression, and chemoresistance. However, their roles in non-small cell lung cancer (NSCLC) remain poorly elucidated. Methods We analyzed the prognostic value and immune infiltration of Kindlins in NSCLC through Oncomine, GEPIA, UALCAN, CCLE, Kaplan‑Meier plotter, cBioPortal, TIMER, GeneMANIA, STRING, and DAVID database. Additionally, the mRNA expression levels of Kindlins were verified in 30 paired NSCLC tissues and NSCLC cell lines by real-time PCR. Results The expression level of FERMT1 was remarkably increased in NSCLC tissues and NSCLC cell lines, while FERMT2 and FERMT3 were reduced. Kindlins expressions were associated with individual cancer stages and nodal metastasis. We also found that higher expression level of FERMT1 was obviously correlated with worse overall survival (OS) in patients with NSCLC, while higher FERMT2 was strongly associated with better overall survival (OS) and first progression (FP). Additionally, the expression of FERMT2 and FERMT3 were obviously correlated with the immune infiltration of diverse immune cells. Functional enrichment analysis has shown that Kindlins may be significantly correlated with intracellular signal transduction, ATP binding and the PI3K-Akt signaling pathway in NSCLC. Conclusions The research provides a new perspective on the distinct roles of Kindlins in NSCLC and likely has important implications for future novel biomarkers and therapeutic targets in NSCLC.

We consider a long-term portfolio choice problem with two illiquid and correlated assets and formulate it as an eigenvalue problem in the form of a variational inequality. The eigenvalue is associated with the portfolio’s optimal long-term growth rate, and the free boundaries implied by the variational inequality correspond to the optimal trading strategy. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi-analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one.

We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.

In this paper we investigate a thermoelastic system with Dirichlet or Neumann or natural boundary conditions. For completeness we derive the thermoelastic system model and show the stability results for different boundary conditions which are based on the analysis of the corresponding eigenvalue problems. Further we derive the equilibrium ambient temperature equations that admits a periodic solution modeling the daily heat source from the sun.

Capital gains taxation has important implications for investors’ portfolio choice decisions. To explore these implications, we develop a continuous time investment and consumption model with capital gains tax, Epstein–Zin recursive utility, and regime switching. We find that various factors, such as tax rate, risk aversion, interest rate, stock return, and volatility, jointly affect optimal portfolio allocation, whereas intertemporal substitution does not. In a regime switching market, investors may trade or stop trading purely because of a change in regime, and there is a distinct cross-regime effect on optimal portfolio allocation. In particular, investors tend to raise stock investment in a bear regime so as to reduce potential tax payments upon regime switching. Given reasonable parameter values, regime switching has a greater impact on optimal portfolio allocation in a bear regime than in a bull regime. This paper was accepted by Neng Wang, finance.

Dissolved organic matter (DOM) in wastewater treatment plant (WWTP) effluent threats aquatic environment and challenges engineered aquatic systems. In order to predict its environmental risk and guide the selection of suitable strategy and technology, understanding the removal efficiency and the fate of DOM along the treatment is helpful. Herein, to qualitatively and quantitatively estimate the composition of DOM in a full-scale WWTP in northern China, a combined approach of chromatography and spectroscopy was employed, and the contribution of size-dependent substructure to the global feature of DOM was discussed simultaneously. The results suggest that the WWTP could remove the free proteins of low molecular size by physical treatment with removal ratios of 46.4% and 41.0% for tyrosine and tryptophan, respectively, whereas humic-like and humic-protein complex accumulate in the effluent of treatment with a ratio of 40.6%. Biotreatment is of importance for changing the proportion of different types of DOM, especially for anaerobic tank and the anoxic tank, which results in the increase of humification. Size exclusive chromatography (SEC) coupled with multi emission maps identifies six DOM substructures. Two protein-like DOM were removed by the WWTP, except the composition with the molecular weight ranging from 0.75 to 0.25 kDa, whereas humic-like DOM with whole size range accumulated to varying extent. The standard deviation analysis states that the structure variation of bulk DOM in the WWTP is primarily contributed by the decrease of small-sized proteins and the increase of humic-like DOM. This study presents insight into the fate and behavior of DOM components during the full-scale WWTP treatment and provides guidelines for altering DOM-treating technologies. Besides, innovative strategies suggest using DOM as a resource based on the critical review of previous research.

The reaction–diffusion system a t = a x x − a b n , b t = D b x x + a b n , where n ≥ 1 and D > 0 , arises from many real-world chemical reactions. Whereas n = 1 is the KPP type nonlinearity, which is much studied and very important results obtained in literature not only in one dimensional spatial domains, but also multi-dimensional spaces, but n > 1 proves to be much harder. One of the interesting features of the system is the existence of traveling wave solutions. In particular, for the traveling wave solution a ( x , t ) = a ( x − v t ) , b ( x , t ) = b ( x − v t ) , where v > 0 , if we fix lim x → − ∞ ⁡ ( a , b ) = ( 0 , 1 ) it was proved by many authors with different bounds v ⁎ ( n , D ) > 0 such that a traveling wave solution exists for any v ≥ v ⁎ when n > 1 . For the latest progress, see [7] . That is, the traveling wave problem exhibits the mono-stable phenomenon for traveling wave of scalar equation u t = u x x + f ( u ) with f ( 0 ) = f ( 1 ) = 0 , f ( u ) > 0 in ( 0 , 1 ) and, u = 0 is unstable and u = 1 is stable. A natural and significant question is whether, like the scalar case, there exists a minimum speed. That is, whether there exists a minimum speed v m i n > 0 such that traveling wave solution of speed v exists iff v ≥ v m i n ? This is an open question, in spite of many works on traveling wave of the system in last thirty years. This is duo to the reason, unlike the KPP case, the minimum speed cannot be obtained through linear analysis at equilibrium points ( a , b ) = ( 0 , 1 ) and ( a , b ) = ( 1 , 0 ) . In this work, we give an affirmative answer to this question.

In this paper we study a model which describes the pattern formation of vegetation spots and strips in a semi-arid or arid landscape. The mathematical model consists of a nonlinear cross-diffusion system with evaporation and absorption sources. Global existence and uniqueness in classical sense for the system are established. Some asymptotic behaviors of the solution are derived for a linearized system. It is shown that the solution of the linearized system is asymptotically stable near the steady-state solution. Moreover, we show that the Turing phenomenon occurs for the linearized cross-diffusion system and the cross-diffusion destabilizes the ecosystem.

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m ( $$m > 1$$ ) between two chemical species, a reactant A and an auto-catalyst B, $$ A + m B \rightarrow (m+1) B$$ , and a super-linear decay of order $$ n > 1$$ , $$ B \rightarrow C$$ , where $$ 1< n < m$$ . Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value $$u_0$$ at $$ - \infty $$ . The explicit bound is also derived in terms of $$u_0$$ and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.

We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qinet al.(Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN:http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Laiet al.(2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qinet al.(2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.

A number of techniques, some of which are novel, are introduced to develop a systematic method to study a set of eigenvalue problems arising from the stability analysis of bubble steady states of a Keller–Segel's minimal chemotaxis model. Estimates of the eigenvalue with largest real part of an elliptic system without variational structure and the second eigenvalue of a corresponding subproblem possessing variational structure are obtained. These estimates provide critical information about the stability of the bubble steady state with respect to the time relaxation parameter; in particular, it is shown that the stability decreases to zero as the relaxation parameter goes to infinity.

In this paper we study a corporate bond-pricing model with credit rating migration and a stochastic interest rate. The volatility of bond price in the model strongly depends on potential credit rating migration and stochastic change of the interest rate. This new model improves the previous existing models in which the interest rate is considered to be a constant. The existence, uniqueness and regularity of the solution for the model are established. Moreover, some properties including the smoothness of the free boundary are obtained. Furthermore, some numerical computations are presented to illustrate the theoretical results.

This paper investigates the longtime behavior of solutions of a susceptible-infectious-susceptible (SIS) epidemiological model, proposed to explain an epidemiological phenomenon that pathogen spread does not necessarily keep pace with its host invasion. When the rate of transmission of disease is small, the infected population will vanish. When the rate is of moderate size, a local infection of disease will spread with a speed that is smaller than that of the host population. For large disease transmission rates, the infection can keep pace with the host invasion, provided that the host does not propagate too fast. This paper also modifies a method originated from [F. Hamel et al., Netw. Heterog. Media, 8 (2013), pp. 275--289] to prove the logarithmic lag of the front of the infected population from that of the minimum speed traveling wave of a reduced equation obtained by setting the host population to its capacity. Our analysis can be used to study monostable scalar equations in inhomogeneous media.

We formally derive a simple normal form for the dynamics of a nonlocally coupled neural field model when the local dynamics is near a saddle-node infinite cycle (SNIC) bifurcation. The derivation produces a nonlocally coupled scalar model which does not satisfy the comparison principle (ordered initial data produces ordered dynamical solutions). We prove the existence of unique traveling waves for the corresponding nonlocal evolution problem with a new tool that does not use the comparison principle. We obtain sharp estimates for the speed of fast and slow waves and compare these to numerical results.

The development of a high-performance semiconductor oxide sensor for the accurate detection of trace disease biomarkers in exhaled breath is still a challenge that urgently needs to be addressed. Here, we proposed a self-assembly strategy and spin-coating process to create a graphene quantum dot (GQD)-functionalized three-dimensional ordered macroporous (3DOM) ZnO structure. The strong synergistic effect and the p–n heterojunction between the p-type GQDs and n-type ZnO effectively enlarged the resistance variation due to the change in oxygen adsorption. The specific 3DOM structure induced a hierarchical pore size (286 nm in macroscale and 26 nm in mesoscale) and 3D interconnection, which guaranteed high gas accessibility and fast carrier transportation. As a result, the GQD-modified 3DOM ZnO sensor exhibited a remarkably high response (Rair/Rgas = 15.2 for 1 ppm acetone), rapid response/recovery time (9/16 s), extremely low theoretical detection limit (8.7 ppb), and good selectivity towards acetone against other interfering gases. In particular, the proposed sensor could accurately distinguish trace acetone in the simulated breath of diabetic patients. These results demonstrate a high potential for the feasibility of the GQD-modified 3DOM SMO structure as a new sensing material for the possibility of noninvasive real-time diagnosis of diabetes.

Many finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton‐Jacobi‐Bellman (HJB) equation takes the form of variational inequality, and its penalty approximation equation is linked to a regular control problem. The penalty method, combined with a finite difference scheme, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous‐time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function, which is the minimal (viscosity) solution of the HJB equation problem. The key step is to prove that any admissible singular control can be approximated by a sequence of regular controls related to the corresponding penalized equation problem that admits a unique solution. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.

Sensitive and selective determination of H2S in human exhalation is essential for understanding some physiological and pathological processes. However, effective detection of H2S at ppb level remains a challenge. In this work, three-dimensional inverse opal (3DIO) WO3 structure modified with metal-organic framework (MOF) derived ZnO@Au nanoparticles (NPs) were designed and fabricated for trace H2S detection. The ordered macroporous structure of 3DIO, n-n heterojunctions between WO3 and ZnO, and effective encapsulated of small-sized Au NPs benefited to the sensing performance. Compared to pristine 3DIO WO3 sensor, the response of MOF derived 3DIO WO3/ZnO@Au sensor enhanced 6.5-folds (175 to 10 ppm H2S), the optimal working temperature dramatically decreased to 170 °C, and the actual detection limit extended to 50 ppb with a response of 8.5. The good selectivity and long-term stability of the sensor were also identified. The theoretical simulation indicated that improved conductivity, higher negative charge of O, and increased adsorption energies contributed to the excellent selectivity. Furthermore, the proposed sensor could distinguish the trace H2S concentration change in the exhaled breath environment. This work demonstrates an example of a simple and versatile method for high performance H2S gas sensor design, which is promising for exhaled breath biomarker determination.

Following the idea of Wei and Yan [32] , with new ingredients to take care of the critical dimension n = 5 , we construct infinitely many solutions of the biharmonic equation Δ 2 v = | x | α v n + 4 n − 4 in the unit ball of R n ( n ⩾ 5 , α > 0 ) with the Navier conditions v = Δ v = 0 on the boundary.

This article establishes the existence of traveling waves of a class of reaction–diffusion systems which model the pre-mixed isothermal autocatalytic chemical reaction of order m ( m > 1 ) between two chemical species, a reactant and an auto-catalyst, and a linear decay. Moreover, our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x = − ∞ . This is in strong contrast to either the reaction–diffusion systems of autocatalytic chemical reaction of the order m without decay, or to the systems which have the same order of decay, which were shown by various authors (e.g. [8] , [13] , [17] , [26] ) that the set of traveling wave speeds contains [ c ⁎ , ∞ ) for some c ⁎ > 0 . The same systems also appear in a mathematical model of microbial growth and competition in a flow reactor; see [2] , [24] .

The economic incentive to prepay a mortgage depends on the yield curve of interest rates. If interest rates of various terms are sufficiently lower than the coupon of the mortgage, the borrower has an incentive to replace the existing loan through prepayment. However, if rates are high it is optimal for the borrower to continue with the scheduled payment of the mortgage. This problem can be posed as a parabolic variational inequality. Presented in this paper is an analysis of this problem under the assumption that the term structure is defined by the Cox--Ingersoll--Ross (CIR) model for the short rate. This analysis is significant from both mathematical and financial points of view. The main mathematical difficulty is that the partial differential equation (PDE) in the variational inequality is not uniformly parabolic under the CIR assumption, as opposed to the case in earlier work where the short rate is assumed to follow the Vasicek model. Therefore, standard approaches require modification to address t...