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risk management is a main topic in the study of finance . in the present paper , we consider the problem of minimizing the downside risk associated with an investor s total wealth in a certain incomplete market model . more precisely , let @xmath2 be the price of a riskless asset with the dynamics @xmath3 , @xmath4 the prices of the risky assets , and @xmath5 , @xmath6 , the number of shares of @xmath7th security . then the total wealth that the investor possesses is defined as @xmath8 and we assume a self - financing condition , @xmath9 when setting the proportion of the portfolio invested in the @xmath7th security as @xmath10 , we have @xmath11 and the total wealth is denoted by @xmath12 , which is the solution to this stochastic differential equation for a given strategy @xmath13 . let us consider minimizing the probability @xmath14 for a given target growth rate @xmath15 by selecting portfolio choice @xmath16 . let us make clear the meaning of the probability . if we choose strategy @xmath17 , then we have @xmath18 thus , the probability is always @xmath19 for large time @xmath0 and @xmath20 . accordingly , in considering the above minimization , we investigate the extent for which we can improve the probability by selecting a strategy , as compared with the trivial strategy of investing the total wealth in a riskless asset . the latter strategy is considered the benchmark in terms of finance . we shall consider the asymptotic behavior of the probability @xmath21 according to the theory of large deviation , it is natural to relate ( [ eq1.2 ] ) to @xmath22\ ] ] for @xmath23 . namely , as @xmath1 , @xmath24   \biggr )   \to   -\inf_{k\in ( -\infty,\kappa]}\sup_{\gamma<0}\{\gamma k-{\hat\chi}(\gamma)\}\ ] ] is expected to hold since the legendre transform @xmath25 of @xmath26 , @xmath27 is regarded as the rate function of the asymptotics , if @xmath26 is a convex function ; cf . note that we can see from hlder s inequality that@xmath28=\log e[e^{\gamma\log({{v_t(h)}/{s_t^0}})}]$ ] is a convex function of @xmath29 , but this does not always imply the convexity of its infimum @xmath30 = \inf_{h_{\cdot}}\log e\bigl[e^{\gamma\log ( { { v_t(h)}/{s_t^0}})}\bigr].\ ] ] therefore , the convexity of @xmath31 can not be determined immediately and the above idea does not directly apply . in the present paper , we will find the convexity of @xmath32 by identifying the solution of the h - j - b equation of ergodic type with the limit value ( [ eq1.3 ] ) ; cf . proposition  [ prop4.2 ] and corollary  [ cor4.1 ] . then we shall see that the duality relation between ( [ eq1.2 ] ) and ( [ eq1.3 ] ) holds under suitable conditions , as expected ; cf . theorem  [ theo2.4 ] . minimization ( [ eq1.4 ] ) , which is equivalent to power utility maximization , could be regarded as a risk - sensitive control problem . the infinite time horizon counterpart of ( [ eq1.4 ] ) without a benchmark , @xmath33,\ ] ] has been extensively studied as risk - sensitive control ( e.g. , @xcite ) , and a benchmarked case has recently been reported in @xcite . from the viewpoint of stochastic control theory , it may appear more natural , compared with the above relationship between ( [ eq1.2 ] ) and ( [ eq1.3 ] ) , to relate @xmath34\ ] ] to @xmath35 which is considered in the present paper as well ; cf . theorem  [ theo2.5 ] . we note that the problem relating ( [ eq1.2 ] ) to ( [ eq1.3 ] ) is thought to be equivalent to considering @xmath36 and @xmath37\ ] ] without a benchmark . however , the arguments in this article may be simpler than in the case without a benchmark ( cf . remark  [ rem2.2 ] ) .    in previous papers @xcite , we studied similar asymptotic behavior without benchmarks for linear gaussian models in relation to the asymptotics of risk - sensitive portfolio optimization . indeed , we established a duality relation between these problems , and as a result , an explicit expression of the limit value of the probability minimizing downside risk for each case of full and partial information . to obtain those results , the key analysis involved poisson equations derived by taking derivatives with respect to @xmath29 of the h - j - b equations of ergodic type corresponding to risk - sensitive control over an infinite time horizon . since the solutions of the h - j - b equations can be explicitly expressed as quadratic functions by using the solutions of the riccati equations for linear gaussian models , analysis of the differentiability of the solutions of the riccati equations with respect to @xmath29 was essential in those works .    in the present paper , we shall consider general diffusion market models and discuss the above - mentioned duality relation between the asymptotics of the minimization of downside risk and the risk - sensitive stochastic control for large time . since the solutions of h - j - b equations of ergodic type do not always have explicit expressions , we need to consider , in general , the differentiablity with respect to @xmath29 of h - j - b equations of ergodic type . the analysis is presented in sections  [ sec5 ] and  [ sec6 ] based on the results concerning h - j - b equations of ergodic type and related stochastic control problems given in sections  [ sec3 ] and  [ sec4 ] . here , we mention the ongoing work of hata and sheu @xcite , which is closely related to the present paper and examines similar problems under the assumptions that @xmath38 in ( [ eq2.2 ] ) in section  [ sec2 ] is bounded and that @xmath39 for @xmath40 in place of ( [ eq2.20 ] ) . we shall explain more precisely the relationships between these papers in remark  [ rem2.3 ] . we note that maximization of an upside chance probability for the long term was studied by pham @xcite for continuous time models , and then by stettner @xcite for discrete time models , in relation to risk - sensitive portfolio optimization in the risk - seeking case . by regarding the maximization problem as large deviation control , pham established a duality relation between these two types of problems . explicit calculation of the limit value is given in the case of 1-dimensional gaussian models . the problem was later extended to a nonlinear case by hata and sekine @xcite and also to the partial information case by hata and iida @xcite for 1-dimensional gaussian models . see also related works @xcite . however , asymptotic estimates of downside risk probabilities and upside chance probabilities can not be obtained in parallel . indeed , obtaining the estimates of downside risk is rather difficult than those of upside chance and further analysis of h - j - b equations is required to show the estimates as was shown in @xcite . further note that large deviations control ( [ eq1.7 ] ) is an unconventional optimization problem , and thus we need to employ a new approach to study it . consider a market model with @xmath41 securities and @xmath42 factors , where the bond price is governed by the ordinary differential equation @xmath43 the other security prices and factors are assumed to satisfy the stochastic differential equations @xmath44 \\[-8pt ]    s^i(0)&=&s^i , \qquad    i=1,\ldots , m , \nonumber\end{aligned}\ ] ] and @xmath45   \\[-8pt ]    x(0)&=&x , \nonumber\end{aligned}\ ] ] where @xmath46 is a standard @xmath47-dimensional brownian motion process on a probability space @xmath48 . let @xmath5 be the number of the shares of the @xmath7th security . then , the total wealth that the investor possesses is defined as @xmath49 and the proportion of the portfolio invested in the @xmath7th security is @xmath50 @xmath51 [ or , @xmath52 is called self - financing if @xmath53 thus , under the self - financing condition , we have @xmath54 here we note that @xmath13 is defined as an @xmath55-vector consisting of @xmath56 since @xmath57 holds by definition . the filtration that must be satisfied by admissible investment strategies @xmath58 is relevant in the present problem , and we introduce the following definition .    @xmath59 is said to be an investment strategy if @xmath60 is an @xmath61 valued @xmath62-progressively measurable stochastic process such that @xmath63    the set of all investment strategies is denoted by @xmath64 . for a given @xmath65 , the process @xmath12 representing the total wealth of the investor at time @xmath66 is determined by the stochastic differential equation shown above . @xmath67 \\[-8pt ]   v_0&=&v_0 , \nonumber\end{aligned}\ ] ] where @xmath68 . we are interested in the asymptotics of minimization of a downside risk for a given constant @xmath15 in comparison with investing the whole portfolio in a  riskless security as the benchmark @xmath69 we also examine downside risk minimization with the benchmark @xmath70 over an infinite time horizon , @xmath71 where @xmath72 @xmath73 will be shown to be related to the following risk - sensitive asset allocation problem with benchmark @xmath70 . namely , for a given constant @xmath74 , let us consider the asymptotics @xmath75 where @xmath76=\log e\bigl[e^{\gamma \log ( { { v_t(h)}/{s_t^0}})}\bigr ] , \ ] ] and @xmath16 ranges over the set @xmath77 of all admissible investment strategies defined by @xmath78=1\bigr\}.\ ] ] then we shall see that ( [ eq2.6 ] ) could be considered the dual problem to  ( [ eq2.8 ] ) , while ( [ eq2.7 ] ) is the dual problem to risk - sensitive asset allocation over an infinite time horizon , @xmath79 where @xmath80 we shall consider these problems under the assumptions that @xmath81 and @xmath82 hold . in considering these problems , we first introduce the value function @xmath83.\ ] ] note that @xmath84 where @xmath85 . therefore @xmath86 where @xmath87 by introducing a probability measure @xmath88,\ ] ] the dynamics of the factor process can be written as @xmath89 with the new brownian motion process @xmath90 defined by @xmath91 and the value function written as @xmath92.\ ] ] the h - j - b equation for the value function @xmath93 is @xmath94+\frac{1}{2}(dv)^*\lambda\lambda^*dv\vspace*{1pt}\cr\displaystyle   \qquad { } + \inf_h\{[\beta+\gamma \lambda\sigma^*h]^*dv+\gamma\eta(x , h)\}=0 ,   \vspace*{2pt}\cr\displaystyle v(t , x)=\gamma\log v_0 , } \ ] ] which is also written as @xmath95+\beta_{\gamma}^*dv+\frac{1}{2}(dv)^*\lambda n_{\gamma } ^{-1}\lambda^*dv - u_{\gamma}=0 ,   \vspace*{2pt}\cr\displaystyle v(t , x)=\gamma\log v_0 , } \ ] ] where @xmath96 and @xmath97    @xmath98^*dv+\gamma\eta(x , h)\ }   \\ & & \qquad = \inf_{h\in r^m}\biggl\{[\gamma\lambda\sigma^*h]^*dv-\frac{\gamma ( 1-\gamma)}{2}h^*\sigma\sigma^*h+\gamma h^*{\hat\alpha}\biggr\ } \\ & & \qquad = \inf_{h\in r^m}\biggl\{-\frac{\gamma(1-\gamma)}{2}\biggl[h-\frac { 1}{1-\gamma}(\sigma\sigma^*)^{-1}({\hat\alpha}+\sigma\lambda ^*dv)\biggr]^*\\ & & \qquad\hphantom{=\inf_{h\in r^m}\biggl\ { } { } \times\sigma\sigma^*\biggl[h-\frac{1}{1-\gamma}(\sigma\sigma ^*)^{-1}({\hat\alpha}+\sigma\lambda^*dv)\biggr ] \\ & & \qquad\hphantom{=\inf_{h\in r^m}\biggl\ { } { } + \frac{\gamma}{2(1-\gamma)}({\hat\alpha}+\sigma\lambda ^*dv)^*(\sigma\sigma^*)^{-1}({\hat\alpha}+\sigma\lambda^*dv)\biggr\}. \ ] ]    therefore , the function @xmath99 defines the generator of the optimal diffusion @xmath100 for @xmath101 : @xmath102+\biggl[\beta+\frac{\gamma}{1-\gamma}\lambda\sigma^*(\sigma\sigma ^*)^{-1}({\hat\alpha}+\sigma\lambda^*dv)\biggr]^*d\psi,\ ] ] which is seen in proposition  [ prop2.1 ] . set @xmath103 then , @xmath104+\beta_{\gamma}^*d{\bar v}-\frac{1}{2}(d{\bar v})^*\lambda n_{\gamma}^{-1}\lambda^*d{\bar v}+u_{\gamma}=0 , \vspace*{2pt}\cr\displaystyle { \bar v}(t , x)=-\gamma\log v_0 . } \ ] ] since @xmath105 , which is easily seen by taking @xmath106 with @xmath107 orthogonal to the range of @xmath108 , and seeing that @xmath109 , we have @xmath110 as for existence of a solution to ( [ eq2.16 ] ) satisfying sufficient regularities , we have the following results ; cf . @xcite . [ theo2.1 ] assume ( [ eq2.11 ] ) and ( [ eq2.12 ] ) . then h - j - b equation  ( [ eq2.16 ] ) has a solution such that @xmath111 { \bar v } , \frac{\partial{\bar v}}{\partial t } , \frac{\partial { \bar v}}{\partial x_k } , \frac{\partial^2 { \bar v}}{\partial x_k\,\partial x_j}&\in & l^p(0,t;l^p_{\mathrm{loc}}(r^n ) ) , \qquad   1<\forall p<\infty , \\[-2pt ] \frac{\partial^2{\bar v}}{\partial t^2 } , \frac{\partial^2{\bar v}}{\partial x_k\,\partial t } , \frac{\partial^3 { \bar v}}{\partial x_k\,\partial x_j\,\partial x_l } , \frac{\partial^3 { \bar v}}{\partial x_k\,\partial x_j\,\partial t}&\in & l^p(0,t;l^p_{\mathrm{loc}}(r^n ) ) , \\[-2pt ] \frac{\partial{\bar v}}{\partial t}&\leq&0,\end{aligned}\ ] ] and @xmath112 & & \hphantom{c\bigl ( }   { } + |\nabla\beta_{\gamma}|_{2\rho}+|\beta_{\gamma } |_{2\rho}^2+|u_{\gamma}|_{2\rho}+|\nabla u_{\gamma}|_{2\rho}+1\bigr ) \ ] ] for @xmath113 and @xmath114 , where @xmath115 , @xmath116 , @xmath117 , @xmath118 is a universal constant and @xmath119 .    for @xmath120 , we consider the stochastic differential equation @xmath121 and define @xmath122 for the solution @xmath123 of the stochastic differential equation . note that the solution of this stochastic differential equation is obtained by the change of measure from the solution of ( [ eq2.3 ] ) . indeed , we can see that @xmath124 has at most linear growth under assumptions ( [ eq2.11 ] ) and  ( [ eq2.12 ] ) from the above gradient estimates , and therefore , @xmath125=1\ ] ] holds . thus @xmath126\vadjust{\goodbreak}\ ] ] defines a probability measure . under this measure , @xmath123 turns out to be a  solution of the above stochastic differential equation . the following is a verification theorem , the proof of which is almost the same as the proof of proposition 2.1 , @xcite , and thus is omitted here . [ prop2.1 ] assume ( [ eq2.11 ] ) and ( [ eq2.12 ] ) . then @xmath127 and it is optimal @xmath128=\log e\bigl[e^{\gamma(\log v_t({\hat h})-\log s_t^0)}\bigr].\ ] ]    let us consider an h - j - b equation of ergodic type that is thought to be the limit equation of ( [ eq2.15 ] ) . namely , @xmath129+\beta_{\gamma } ^*d w+\tfrac{1}{2}(d w)^*\lambda n_{\gamma}^{-1}\lambda^*d w - u_{\gamma}.\ ] ] set @xmath130 and assume that @xmath131 and @xmath132 under these assumptions , we have a solution to the h - j - b equation of ergodic type , and the proof is given in proposition  [ prop3.1 ] in section  [ sec3 ] . assume ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.20 ] ) and ( [ eq2.21 ] ) . then ( [ eq2.19 ] ) has a solution @xmath133 such that @xmath134 , @xmath135 and the solution satisfying this condition is unique up to additive constants with respect to @xmath136 . we further assume that @xmath137 then we have the following theorem , and the proof is given after proposition  [ prop4.2 ] in section  [ sec4 ] . [ theo2.2 ] under assumptions ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.20 ] ) and ( [ eq2.22 ] ) , we have @xmath138    the following results are important to prove our main results , and the proofs are given in lemma  [ lem6.3 ] , lemma  [ lem7.1 ] and corollary  [ cor4.1 ] . [ theo2.3 ] let @xmath139 be a solution to ( [ eq2.19 ] ) . then under the assumptions of theorem  [ theo2.2 ] , @xmath140 and @xmath141 are differentiable with respect to @xmath29 and @xmath140 is convex . their derivatives satisfy @xmath142+\bigl(\beta_{\gamma}^*+\bigl(d w^{(\gamma)}\bigr)^*\lambda n_{\gamma } ^{-1}\lambda^*\bigr)d w_{\gamma } \nonumber \\[-8pt ] \\[-8pt ] & & { } + \frac{1}{2(1-\gamma)^2}\bigl\{{\hat\alpha}+\sigma \lambda^*dw^{(\gamma)}\bigr\}^*(\sigma\sigma^*)^{-1}\bigl\{{\hat\alpha } + \sigma\lambda^*dw^{(\gamma)}\bigr\ } , \nonumber\end{aligned}\ ] ] where @xmath143 . furthermore , @xmath144    [ rem2.2 ] it is important to know the limit value @xmath145 since it determines the left endpoint of the interval of the target growth rate  @xmath15 , which makes @xmath146 finite . here we compare the results above with those to be expected for the case without a benchmark , considering asymptotics @xmath37.\ ] ] the h - j - b equation of ergodic type of this problem becomes @xmath147+\beta_{\gamma}^*d{\check w}+\tfrac{1}{2}(d { \check w})^*\lambda n_{\gamma}^{-1}\lambda^*d { \check w}-u_{\gamma}+\gamma r(x),\ ] ] and we can obtain its derivative @xmath148+\bigl(\beta_{\gamma}^*+(d { \check w})^*\lambda n_{\gamma}^{-1}\lambda^*\bigr)d { \check w}_{\gamma } \\ & & { } + \frac{1}{2(1-\gamma)^2}\{{\hat\alpha}+\sigma \lambda^*d{\check w}\}^*(\sigma\sigma^*)^{-1}\{{\hat\alpha}+\sigma \lambda^*d{\check w}\}+r \ ] ] through almost the same arguments as the current ones provided to obtain the results in the present article . the difference appears in considering the asymptotics of @xmath149 as @xmath150 . indeed , @xmath151 could be seen as in @xcite , where @xmath152 is the invariant measure of @xmath153-diffusion process and @xmath154 is defined by @xmath155+(\beta_{\gamma}+\lambda n_{\gamma}^{-1}\lambda^*d{\check w})^*d\psi.\ ] ] note that @xmath153 corresponds to @xmath156 defined by ( [ eq4.16 ] ) in the present paper and can be shown to be ergodic under suitable conditions in a manner similar to the proof of proposition  [ prop4.3 ] . now we can state our main theorems . the proofs are given in sections  [ sec7 ] and  [ sec8 ] . [ theo2.4 ] under the assumptions of theorem  [ theo2.2 ] , for @xmath157 , @xmath158}\sup_{\gamma<0}\{\gamma k-{\hat\chi}(\gamma)\}=-\sup_{\gamma<0}\{\gamma\kappa-{\hat\chi } ( \gamma)\}.\ ] ] moreover , for @xmath159 such that @xmath160 , take a strategy @xmath161 defined in proposition  [ prop2.1 ] . then , @xmath162 for @xmath163 , @xmath164    for the solution @xmath165 to h - j - b equation ergodic type ( [ eq2.19 ] ) , let us set @xmath166 further consider the stochastic differential equation @xmath167 and define @xmath168 for the solution @xmath123 of the stochastic differential equation . then we have the following theorem . [ theo2.5 ] under the assumptions of theorem  [ theo2.2 ] , let @xmath157 and @xmath159 be the same as above . we also assume that @xmath169 then @xmath170}\sup _ { \gamma<0}\{\gamma k-{\hat\chi}(\gamma)\}=-\sup_{\gamma<0}\ { \gamma\kappa-{\hat\chi}(\gamma)\}\ ] ] and @xmath171    [ rem2.3 ] in our previous paper @xcite , we studied similar problems without benchmarks in the case of linear gaussian models . specifically , we discussed the case where @xmath172 , @xmath173 , @xmath174 , @xmath175 and @xmath176 , in which @xmath177 , @xmath178 , @xmath179 and @xmath180 ( resp . , @xmath181 and @xmath182 ) are all constant matrices ( resp . , vectors ) , and @xmath183 is a constant . under the main assumption that @xmath184 which corresponds to ( [ eq2.20 ] ) above , we obtained results similar to theorems  [ theo2.4 ] and  [ theo2.5 ] . the present paper is a natural extension to general diffusion incomplete market models . on the other hand , hata and sheu @xcite treat the case where @xmath38 is bounded , and @xmath39 for @xmath185 , in which linear gaussian models are excluded . in that case , @xmath186 becomes bounded and they employ quite different methods from ours to analyze h - j - b equation  ( [ eq2.19 ] ) , while assumption ( [ eq2.22 ] ) is crucial in our settings . for that reason our theorems do not include the case where @xmath38 is bounded . the generator of the optimal diffusion process governed by ( [ eq2.25 ] ) for risk - sensitive control problem ( [ eq2.10 ] ) is defined by @xmath187+\biggl[\beta_{\gamma}^*+\frac{\gamma}{1-\gamma}(dw)^*\lambda\sigma ^*(\sigma\sigma^*)^{-1}\sigma\lambda^*\biggr]d\psi.\ ] ] on the other hand , in proving theorem  [ theo2.2 ] we introduce another type of stochastic control problem ( [ eq4.9 ] ) with ( [ eq4.7 ] ) . the generator of the optimal diffusion process for this problem is defined by ( [ eq4.16 ] ) . @xmath188+[\beta_{\gamma}^*+(dw)^*\lambda n_{\gamma}^{-1}\lambda^*]d\psi,\ ] ] where @xmath136 is a solution to h - j - b equation ( [ eq2.19 ] ) of ergodic type . then we note that @xmath156 is related to @xmath189 through the gauge transform , @xmath190\varphi=\bigl[{\bar l}-\bigl(\gamma\eta-\chi(\gamma ) \bigr)\bigr]\varphi.\ ] ] further , we see that @xmath191 is an eigenfunction of @xmath192 @xmath193 for the principal eigenvalue @xmath140 ; cf . note that @xmath156 is ergodic as is seen in proposition  [ prop4.3 ] , while @xmath189 is not always ergodic . we assume ( [ eq2.11 ] ) and ( [ eq2.12 ] ) and that @xmath194 , @xmath195 with an @xmath196 ( resp . , @xmath197 ) matrix - valued bounded function @xmath177 ( resp . , @xmath178 ) , and an @xmath55 ( resp . , @xmath42)-vector - valued bounded function  @xmath181 ( resp . , @xmath182 ) such that :    @xmath198    the real parts of all eigenvalues of @xmath199 is less than @xmath200 , @xmath201    @xmath202 .    in this case @xmath203 and we see that ( [ eq2.20 ] ) holds . furthermore , ( [ eq2.22 ] ) holds because of ( i ) . instead of ( [ eq2.19 ] ) , we shall study an h - j - b equation of ergodic type for @xmath204 . @xmath205+\beta _ { \gamma}^*d{\bar w}-\tfrac{1}{2}(d{\bar w})^*\lambda n_{\gamma } ^{-1}\lambda^*d{\bar w}+u_{\gamma}.\ ] ]    [ prop3.1 ] assume ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.20 ] ) and ( [ eq2.21 ] ) . then ( [ eq3.1 ] ) has a solution @xmath206 such that @xmath207 , @xmath208 and the solution satisfying this condition is unique up to additive constants with respect to @xmath209 . the following notation is useful for the task at hand . set @xmath210 . then @xmath211 moreover , we see that @xmath212    to prove proposition  [ prop3.1 ] , we first consider the h - j - b equation of discounted type , @xmath213+\beta_{\gamma}^*d v_{{\varepsilon}}-\tfrac{1}{2}(d v_{{\varepsilon}})^*\lambda n_{\gamma}^{-1}\lambda^*d v_{{\varepsilon}}+u_{\gamma}.\ ] ] note that ( [ eq3.2 ] ) can be written as @xmath214+g^*d v_{{\varepsilon}}-\tfrac{1}{2}(\lambda^ * d v_{{\varepsilon}}-\sigma^*{\hat\alpha})^ * n_{\gamma}^{-1}(\lambda^*d v_{{\varepsilon}}-\sigma^*{\hat\alpha})\nonumber\hspace*{-15pt } \\[-9pt ] \\[-9pt ] & &   { } + \tfrac{1}{2}{\hat\alpha}\sigma\sigma ^*{\hat\alpha}. \nonumber\end{aligned}\ ] ] then , we consider the linear equation @xmath215 where @xmath216+g^*d\varphi.\ ] ] under assumptions ( [ eq2.11 ] ) , ( [ eq2.12 ] ) and ( [ eq2.20 ] ) , ( [ eq3.4 ] ) has a solution @xmath217 . indeed , set @xmath218 , @xmath219 . then , by taking @xmath220 to be sufficiently large , we can see that there exists @xmath221 such that for @xmath222 , @xmath223 therefore , when setting @xmath224 @xmath225 turns out to be a supersolution to ( [ eq3.4 ] ) , and we can see that there exists a solution @xmath217 to ( [ eq3.4 ] ) such that @xmath226 since @xmath227 is a subsolution . we note that @xmath228 is a supersolution to ( [ eq3.2 ] ) which is the same equation as ( [ eq3.3 ] ) . [ lem3.1 ] under the assumptions of proposition  [ prop3.1 ] , ( [ eq3.2 ] ) has a solution such that @xmath229 and @xmath230 .    in proving the existence of the solution , we introduce a dirichlet problem on @xmath231 , @xmath232 : @xmath233+\beta_{\gamma}^*d v_{{\varepsilon}}-\tfrac{1}{2}(d v_{{\varepsilon}})^*\lambda n_{\gamma}^{-1}\lambda^*d v_{{\varepsilon}}+u_{\gamma } \quad     \mbox{in }   b_r , \nonumber\hspace*{-15pt } \\[-9pt ] \\[-9pt ]    \qquad v_{{\varepsilon}}(x)&=&\varphi_{{\varepsilon } } , \qquad     x\in   \partial b_r . \nonumber\end{aligned}\ ] ] owing to theorem 8.3 ( @xcite , chapter 4 ) , dirichlet problem ( [ eq3.4 ] ) has a solution @xmath234 . we extend @xmath234 to the whole euclidean space as @xmath235 then we can see that @xmath236 is nonincreasing with respect to @xmath237 . indeed , for @xmath238 , @xmath236 is a supersolution to ( [ eq3.3 ] ) in @xmath239 , and we have @xmath240+\beta _ { \gamma}^*d(v_{{\varepsilon},r}-v_{{\varepsilon},r ' } ) \\ & & \qquad   \quad { } -\tfrac{1}{2}(d v_{{\varepsilon},r})^*\lambda n_{\gamma}^{-1}\lambda^*d v_{{\varepsilon},r}+ \tfrac{1}{2}(d v_{{\varepsilon},r'})^*\lambda n_{\gamma}^{-1}\lambda^*d v_{{\varepsilon},r ' }     \qquad \mbox{in }   b_r ' \\ & & \qquad = \tfrac{1}{2}\operatorname{tr}[\lambda\lambda^*d^2 ( v_{{\varepsilon},r}-v_{{\varepsilon},r'})]+\beta_{\gamma}^*d ( v_{{\varepsilon},r}-v_{{\varepsilon},r ' } ) \\ & & \qquad   \quad { } -\tfrac{1}{2}(d v_{{\varepsilon},r}+d v_{{\varepsilon},r'})^*\lambda n_{\gamma } ^{-1}\lambda^*d ( v_{{\varepsilon},r}-v_{{\varepsilon},r ' } ) . \ ] ] therefore , from the maximum principle ( cf . theorem 3.1 in @xcite ) we see that @xmath241 since @xmath242 . we further note that @xmath243 for each @xmath237 because @xmath244 is a subsolution to ( [ eq3.2 ] ) , and the maximum principle again applies . similar to the proof of lemma  2.6 in @xcite , we have the following gradient estimate : for each @xmath237 and @xmath245 , @xmath246 where @xmath247 is a constant independent of @xmath237 , @xmath248 . thus , when taking a sequence @xmath249 such that @xmath250 , @xmath251 forms a family of uniformly bounded and equicontinuous functions . thus we can choose a subsequence @xmath252 converging to a continuous function @xmath234 . furthermore , since @xmath253 for a positive constant @xmath254 independent of @xmath249 and @xmath248 , it converges weakly in @xmath255 to @xmath234 by taking a subsequence if necessary . by similar arguments to lemma  6.8 in @xcite , the convergence can be strengthened as @xmath256 converges strongly in @xmath257 to @xmath258 . as a result we can see from the regularity theorems that we have a solution @xmath229 to ( [ eq3.2 ] ) . since @xmath259 , for each @xmath232 from the maximum principle as well as ( [ eq3.7 ] ) , we see that @xmath260 . set @xmath261 then , by taking @xmath262 to be sufficiently small , we can see that there exists @xmath263 such that for @xmath264 , @xmath265 therefore , we see that @xmath266 and @xmath267 satisfy assumption ( [ eq9.3 ] ) in the last section . set @xmath268 , @xmath269 and @xmath270 then for @xmath271 there exists a solution @xmath272 to @xmath273 if and only if @xmath274 where @xmath275 is an invariant measure for @xmath266 ; cf . proposition  [ prop9.4 ] in in the . therefore , setting @xmath276 there exists a solution @xmath277 to @xmath278 and it is known that @xmath279 converges to @xmath280 as @xmath281 uniformly on each compact set . now we can prove proposition  [ prop3.1 ] . proof of proposition  [ prop3.1 ] we first note that @xmath282 because of lemma  [ lem3.1 ] . therefore , we have @xmath283 where @xmath284 is a constant independent of @xmath248 . moreover , @xmath285 for a positive constant @xmath286 independent of @xmath248 in view of ( [ eq3.8 ] ) . thus , similarly to the proof of theorem 3.1 in @xcite , we can prove the existence of the solution @xmath287 to ( [ eq3.1 ] ) such that @xmath288 . from regularity theorems we see that @xmath207 . the proof of uniqueness is similar to the proof of lemma 3.2 in @xcite . now we have the following proposition .    under the assumptions of proposition  [ prop3.1 ] , the solution @xmath209 to ( [ eq3.1 ] ) satisfies @xmath289 ) , then , for each @xmath290 , there exists a positive constant @xmath291 such that the nonnegative solution @xmath292 , @xmath293 , satisfies @xmath294    set @xmath295 . then we shall prove for each @xmath296 that @xmath297 \\[-8pt ]   & & \hphantom{k\biggl(}{}+|u_{\gamma } |_r+|\nabla u_{\gamma}|_r+\frac{|\beta_{\gamma}|_r}{r}+|\nabla \beta_{\gamma}|_r+c\biggr ) \nonumber\end{aligned}\ ] ] for positive constants @xmath298 and @xmath220 , where @xmath299 . note that ( [ eq3.13 ] ) implies ( [ eq3.11 ] ) because of our assumptions on the coefficients @xmath300 and  @xmath183 . we have @xmath301 since @xmath302 and @xmath303 as @xmath281 . in the following @xmath304 , @xmath305 and @xmath186 are abbreviated to @xmath306 , @xmath307 and @xmath308 , respectively . @xmath309 is abbreviated to @xmath310 .    by differentiating ( [ eq3.1 ] ) with respect to @xmath311 , we have @xmath312 \\[-8pt ] & & { } -d_iwq^{ij}d_{jk}w-\tfrac{1}{2}d_iwq^{ij}_kd_jw+u_k . \nonumber \ ] ] set @xmath313 then we have @xmath314 here we have used ( [ eq3.14 ] ) and the matrix inequality @xmath315)^2\leq n c\operatorname{tr}[ab^2]\]]for symmetric matrix @xmath178 and nonnegative definite symmetric matrix @xmath177 , where @xmath118 is the maximum eigenvalue of @xmath177 . set @xmath316 then @xmath317\geq-\frac{4n}{r^2}c_2 , $ ] @xmath318 and @xmath319 . let @xmath320 be the maximum point of @xmath321 in @xmath322 . then @xmath323 and @xmath324(x)\leq0 $ ] . therefore , from the maximum principle we have @xmath325 \\ & & { } -f\biggl\{\frac{1}{2}(\lambda\lambda^*)^{ij}d_{ij}\tau-\frac{(\lambda \lambda^*)^{ij}d_i\tau d_j\tau}{\tau } -\beta^id_i\tau + q^{ij}d_jwd_i\tau\biggr\}. \ ] ] since @xmath326 , by taking @xmath262 to be sufficiently small , @xmath327 therefore , it follows that @xmath328 we can assume @xmath329 and @xmath330 ; thus , @xmath331 accordingly , we have @xmath332 therefore , we obtain @xmath333 and universal constant @xmath219 . including the case where @xmath334 , @xmath335 , we obtain @xmath336 and ( [ eq3.13 ] ) has been proved . now let us prove ( [ eq3.12 ] ) . for each @xmath337 take a point @xmath338 such that @xmath339 . set @xmath340 where @xmath341 is a positive constant determined later . then , @xmath342 in @xmath343 and @xmath344 on @xmath345 . set @xmath346 then , @xmath347 note that @xmath348 then we have @xmath349-\beta ^*dz \\ & & \qquad = -\frac{1}{2}(d{\bar w})^*\lambda n_{\gamma}^{-1}\lambda^*d{\bar w}+u_{\gamma}+\frac{1}{2}\operatorname{tr}[\lambda\lambda^*d^2r]+\beta ^*dr \\ & & \qquad = -\frac{1}{2}d({\bar w}+r)^*\lambda n_{\gamma}^{-1}\lambda ^*d({\bar w}-r)-\frac{1}{2}(dr)^*\lambda n_{\gamma}^{-1}\lambda ^*dr+u_{\gamma } \\ & & \qquad   \quad { } + \frac{1}{2}\operatorname{tr}[\lambda\lambda^*d^2r]+\beta^*dr \\ & & \qquad \geq -\frac{1}{2}d({\bar w}+r)^*\lambda n_{\gamma}^{-1}\lambda^*dz-\frac { c_2}{2}|dr|^2+u_{\gamma}+\frac{1}{2}\operatorname{tr}[\lambda\lambda ^*d^2r]+\beta^*dr . \ ] ] noting that @xmath350 , for a positive constant independent of @xmath29 , @xmath349-\beta ^*dz+\frac{1}{2}d({\bar w}+r)^*\lambda n_{\gamma}^{-1}\lambda^*dz   \\ & & \qquad \geq-\frac{c_2}{2}|dr|^2+u(x)-\frac{4c_{\rho}}{\rho^2}\operatorname { tr}[\lambda\lambda^*]-4c c{\rho } \\ & & \qquad \geq -\frac{8c_2c_{\rho}^2}{\rho^2}+\frac{-\gamma}{2(1-\gamma ) } c_0\biggl(\frac{|\rho|^2}{4}+1\biggr)-\frac{4c_2nc_{\rho}}{\rho^2}-4cc_{\rho } \\ & & \qquad \geq-\biggl(8c_2\frac{c_{\rho}^2}{\rho^2}+4c_2n\frac{c_{\rho}}{\rho ^2}+4cc_{\rho}\biggr)+\frac{-\gamma_0c_0\rho^2}{8(1-\gamma_0)}+\frac { -\gamma_0c_0}{2(1-\gamma_0)}. \ ] ] by setting @xmath351 with @xmath352 such that @xmath353 and @xmath354 , we see that @xmath355-\beta^*dz+\tfrac { 1}{2}d({\bar w}+r)^*\lambda n_{\gamma}^{-1}\lambda^*dz\geq m>0\qquad \mbox{in }   d_{\rho}\ ] ] for some positive constant and sufficiently large @xmath356 . then @xmath357 is superharmonic in @xmath343 and @xmath358 . therefore @xmath359 , from which we have @xmath360 hence , @xmath361 . let us come back to h - j - b equation ( [ eq2.16 ] ) . according to assumption ( [ eq2.11 ] ) , we have a  positive constant @xmath362 such that @xmath363 we strengthen condition ( [ eq2.21 ] ) to ( [ eq2.22 ] ) . then we have the following lemma . [ lem4.1]assume ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.22 ] ) and @xmath364 . then for each @xmath365 there exist positive constants @xmath366 and @xmath367 such that @xmath368    choose a positive constant @xmath220 such that @xmath369 and set @xmath370 , where @xmath371 , and set @xmath372 where @xmath373 is a solution to the riccati equation @xmath374 and @xmath375 is a solution to the ordinary equation @xmath376-\frac{c c_{\beta } } { 2}-c_{\gamma}'=0,\qquad q(t)=-\gamma\log v_0,\ ] ] where @xmath377 . set @xmath378 then @xmath379-\beta_{\gamma}^*dz \\ & & \qquad = \frac{1}{2}(d{\bar v})^*\lambda n_{\gamma}^{-1}\lambda^*d{\bar v}+u_{\gamma}+\frac{\partial r}{\partial t}+\frac{1}{2}\operatorname { tr}[\lambda\lambda^*d^2r]+\beta^*dr \\ & & \qquad = -\frac{1}{2}d({\bar v}+r)^*\lambda n_{\gamma}^{-1}\lambda^ * d({\bar v}-r)-\frac{1}{2}dr^*\lambda n_{\gamma}^{-1}\lambda^ * dr+u_{\gamma } \\ & & \qquad   \quad { } + \frac{\partial r}{\partial t}+\frac{1}{2}\operatorname{tr}[\lambda \lambda^*d^2r]+\beta^*dr \\ & & \qquad \geq-\frac{1}{2}d({\bar v}+r)^*\lambda n_{\gamma}^{-1}\lambda^ * d({\bar v}-r)-\frac{c_2}{2(1-\gamma)}(dr)^*i_ndr+c_{\gamma } |x|^2-c_{\gamma } ' \\ & & \qquad   \quad { } + \frac{1}{2}x^*{\dot p}(t)x+{\dot q}(t)+\frac{c_1}{2}\operatorname { tr}[p(t)]-\frac{c}{2}\beta_{\gamma}^*\beta_{\gamma}-\frac{1}{2c}(dr)^*dr . \ ] ] therefore , @xmath380-\beta^*dz+\frac{1}{2}d({\bar v}+r)^*\lambda n_{\gamma}^{-1}\lambda^*dz \\ & & \qquad \geq\frac{1}{2}x^*{\dot p}(t)x-\frac{1}{2}\biggl(\frac{c_2}{1-\gamma } + \frac{1}{c}\biggr)x^*p(t)i_np(t)x+\biggl(c_{\gamma}-\frac{cc_\beta } { 2}\biggr)|x|^2 \\ & & \qquad   \quad { } + { \dot q}(t)+\frac{c_1}{2}\operatorname{tr}[p(t)]-\frac { cc_{\beta}}{2}-c_{\gamma } ' \\ & & \qquad \geq\frac{1}{2}\biggl ( c_{\gamma}-\frac{cc_\beta}{2}\biggr)|x|^2\geq0 . \ ] ] thus we see that @xmath381 is super harmonic in @xmath382 , and @xmath383 . therefore we have @xmath384 , that is , @xmath385 since @xmath373 is positive definite , @xmath386    let us rewrite ( [ eq2.16 ] ) as @xmath387+g^*d{\bar v } \cr\displaystyle \hphantom{0=}{}-\frac{1}{2}(\lambda ^*d{\bar v}-\sigma^*{\hat\alpha})^ * n_{\gamma}^{-1}(\lambda ^*d{\bar v}-\sigma^*{\hat\alpha})+\frac{1}{2}{\hat\alpha}^*\sigma\sigma^*{\hat \alpha } , \vspace*{4pt}\cr\displaystyle   { \bar v}(t , x)=-\gamma\log v_0 . } \ ] ] noting that @xmath388 , \ ] ] we can rewrite it again as @xmath389+g^*d{\bar v}+\inf_{z\in r^{n+m}}\ { ( \lambda z)^*d{\bar v}+\varphi(x , z)\},\cr\displaystyle   { \bar v}(t , x)=-\gamma\log v_0 , } \ ] ] where @xmath390 this h - j - b equation corresponds to the following stochastic control problem , the value of which is defined as @xmath391,\ ] ] where @xmath392 is a controlled process governed by the stochastic differential equation @xmath393 with control @xmath394 . here , @xmath395 is the set of all @xmath396 valued progressively measurable processes such that @xmath397<\infty.\ ] ] to study this problem , we introduce a value function for @xmath398 , @xmath399.\ ] ] by the verification theorem , the solution @xmath400 to ( [ eq4.5 ] ) can be identified with the value function @xmath401 . indeed , set @xmath402 which attains the infimum in ( [ eq4.5 ] ) , and consider the stochastic differential equation @xmath403 from the estimates obtained in theorem  [ theo2.1 ] , we see that ( [ eq4.8 ] ) has a unique solution . it is also seen by using it s theorem that @xmath404\ ] ] holds , where @xmath405 . in a similar way , we can see that @xmath406\ ] ] for each @xmath407 , hence , @xmath408 . let us consider the following stochastic control problem with the averaging cost criterion : @xmath409,\ ] ] where @xmath392 is a controlled process governed by controlled stochastic differential equation ( [ eq4.7 ] ) with control @xmath410 . the solution @xmath392 of ( [ eq4.7 ] ) is sometimes written as @xmath411 to make clear the dependence on control @xmath410 , and the set @xmath412 of all admissible controls is defined as follows : @xmath413=0 ,   e\biggl[\int_0^t|z_s|^2\,ds\biggr]<\infty , \forall t\biggr\}. \ ] ] corresponding to this stochastic control problem , h - j - b equation of ergodic type ( [ eq3.1 ] ) can be written as @xmath414+g^*d{\bar w}+\inf_{z\in r^{n+m}}\{(\lambda z)^*d{\bar w}+\varphi ( x , z)\}.\ ] ] we then set @xmath415 and consider stochastic differential equation @xmath416 we shall prove the following proposition . [ prop4.1 ] @xmath417 and @xmath418,\ ] ] where @xmath419 .    for the proof of this proposition , we prepare the following lemma . [ lem4.2 ] under assumptions ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.20 ] ) and ( [ eq2.22 ] ) the following estimates hold . for each @xmath420 there exist positive constants @xmath421 and @xmath422 independent of @xmath0 and @xmath29 with @xmath423 such that @xmath424\leq c\ ] ] and also @xmath425\leq c.\ ] ]    let us set @xmath426+(g+\lambda{\hat z})^*d\psi \nonumber \\[-8pt ] \\[-8pt ] & = & \tfrac{1}{2}\operatorname{tr}[\lambda\lambda^*d^2\psi]+(\beta_{\gamma } -\lambda n_{\gamma}^{-1}\lambda^*d{\bar w})^*d\psi.\nonumber \ ] ] then we have @xmath427 therefore , by applying it s formula , we have @xmath428 thus , for @xmath429 , @xmath430 taking into account ( [ eq2.22 ] ) and ( [ eq2.17 ] ) , for @xmath421 such that @xmath431 , we have @xmath432 for @xmath433 . thus , we obtain @xmath434 therefore , taking @xmath435 and setting @xmath436 we see that @xmath437&\leq & e^{\delta{\bar w}(x)}+\delta e\biggl[\int_0^{t\wedge\tau } e^{p\delta s+\delta{\bar w}({\bar y}_s)}\{-k_1|{\bar y}_s|^2+k_2\}\,ds\biggr ] \\ & \leq & e^{\delta{\bar w}(x)}+\delta k_2e\biggl[\int_0^{t\wedge \tau}e^{p\delta s+\delta{\bar w}({\bar y}_s)}{\mathbf1}_{\{|{\bar y}_s|^2\leq{{k_2}/{k_1}}\}}\,ds\biggr ] \\ & \leq & e^{\delta{\bar w}(x)}+\delta k_2e^{\delta k_3}e\biggl[\int_0^{t\wedge\tau}e^{p\delta s}\,ds\biggr ] \\ &   = & e^{\delta{\bar w}(x)}+ k_2e^{\delta k_3 } e\biggl[\frac { 1}{p}\bigl(e^{p\delta(t\wedge\tau)}-1\bigr)\biggr ] . \ ] ] by letting @xmath237 tend to @xmath438 , we have @xmath439\leq e^{\delta{\bar w}(x)}+ k_2e^{\delta k_3}\frac{1}{p}(e^{p\delta t}-1).\ ] ] hence , @xmath440&\leq & e^{-p\delta t+\delta{\bar w}(x)}+ k_2e^{\delta k_3}\frac{1}{p}(1-e^{-p\delta t } ) \\ & \leq & e^{\delta{\bar w}(x)}+ k_2e^{\delta k_3}\frac{1}{p}. \ ] ]    finally , we see that ( [ eq4.15 ] ) follows from ( [ eq4.14 ] ) because of ( [ eq3.12 ] ) . proof of proposition  [ prop4.1 ] from ( [ eq4.10 ] ) it follows that @xmath441+g^*d{\bar w}+(\lambda z)^*d{\bar w}+\varphi(x , z)\vadjust{\goodbreak}\ ] ] for each @xmath442 . therefore , for each control @xmath410 we have @xmath443(y_s)\,ds \\ & \geq&\int_0^t(d{\bar w}(y_s))^*\lambda(y_s)\,dw_s-\chi t-\int _ 0^t\varphi(y_s , z_s)\,ds . \ ] ] thus , @xmath444,\ ] ] from which we obtain @xmath445 \\ & = & \limsup_{t\to\infty}\frac{1}{t}e\biggl[\int_0^t\varphi(y_s , z_s)\,ds\biggr ] \ ] ] since @xmath446 . namely , we have @xmath447 . on the other hand , by taking @xmath448 , we have @xmath449 and thus @xmath450.\ ] ] lemma  [ lem4.2 ] implies @xmath451\geq\rho,\ ] ] and we see that @xmath417 .    let us define @xmath452=\limsup_{t\to \infty}\frac{1}{t}{\bar v}(0,x;t).\ ] ] then we can see that @xmath453    [ prop4.2 ] assume ( [ eq2.11 ] ) , ( [ eq2.12 ] ) , ( [ eq2.20 ] ) and ( [ eq2.22 ] ) . then @xmath454    the proof of theorem  [ theo2.2 ] follows directly from this proposition since @xmath455 because of proposition  [ prop2.1 ] .    for the proof of the present proposition , we prepare some lemmas .    for each @xmath456 , we take the controlled process @xmath457 defined by ( [ eq4.8 ] ) and control @xmath458 . taking a sequence @xmath459 such that @xmath460,\ ] ] we have the following lemma .    under the assumptions of proposition  [ prop4.2 ] , for each @xmath461 we have @xmath462=0.\ ] ] set @xmath463+\bigl(g+\lambda{\hat z}(t , x)\bigr)^*d\psi.\ ] ] then @xmath464 therefore , @xmath465.\ ] ] since @xmath466\geq{\bar\chi},\ ] ] we have @xmath467=0.\ ] ] noting that @xmath468 and @xmath469 , because of lemma  [ lem4.1 ] , we obtain @xmath470\leq0,\ ] ] and our lemma has been proved . under the assumptions of proposition  [ prop4.2 ] , there exists a  subsequence @xmath471 such that @xmath472=0.\ ] ]    @xmath473 & & { } + 2\int_{t - t}^t{\hat y}^{(t)}_s\bigl\{g\bigl({\hat y}^{(t)}\bigr)+\lambda\bigl({\hat y}_s^{(t)}\bigr){\hat z}\bigl(s,{\hat y}_s^{(t)}\bigr)\bigr\}\,ds\\[-2pt ] & & { } + \int_{t - t}^t\operatorname { tr}\bigl[\lambda\lambda\bigl({\hat y}_s^{(t)}\bigr)\bigr]\,ds . \ ] ]    therefore , @xmath474&=&e\bigl[\bigl|{\hat y}_{t - t}^{(t)}\bigr|^2\bigr]+2e\biggl[\int _ { t - t}^t{\hat y}^{(t)}_s\bigl\{g\bigl({\hat y}^{(t)}\bigr)+\lambda\bigl({\hat y}_s^{(t)}\bigr){\hat z}\bigl(s,{\hat y}_s^{(t)}\bigr)\bigr\}\,ds   \\[-2pt ] & & \hphantom{e\bigl[\bigl|{\hat y}_{t - t}^{(t)}\bigr|^2\bigr]+2e\biggl[}\hspace*{91pt } { } + \int_{t - t}^t\operatorname{tr}\bigl[\lambda\lambda\bigl({\hat y}^{(t)}_s\bigr)\bigr]\,ds\biggr ] . \ ] ] by using the gradient estimates in theorem  [ theo2.1 ] and ( [ eq2.20 ] ) we obtain @xmath475 for some positive constant @xmath220 and @xmath476 & \leq&e\bigl[\bigl|{\hat y}_{t - t}^{(t)}\bigr|^2\bigr]+ce\biggl[\int _ { t - t}^t\bigl|{\hat y}_s^{(t)}\bigr|^2\,ds\biggr ] \\[-2pt ] & \leq&e\bigl[\bigl|{\hat y}_{t - t}^{(t)}\bigr|^2\bigr]+c'e\biggl[\int_{t - t}^t\varphi\bigl({\hat y}_s^{(t)},{\hat z}\bigl(s,{\hat y}_s^{(t)}\bigr)\bigr)\,ds\biggr ] \ ] ] since ( [ eq2.22 ] ) is assumed and @xmath477 & = & \frac{1}{2}(\lambda^*d{\bar v}-\sigma^*{\hat\alpha})^ * n_{\gamma } ( \lambda^*d{\bar v}-\sigma^*{\hat\alpha } ) \\[-2pt ] & & { } + ( \lambda^*d{\bar v}-\sigma^*{\hat\alpha } ) ^*n_{\gamma}^{-1}\sigma^*{\hat\alpha}+\frac{1}{2}{\hat\alpha } \sigma\sigma^*{\hat\alpha } \\[-2pt ] & = & \frac{1}{2}(d{\bar v})^*\lambda n_{\gamma}^{-1}\lambda^*d{\bar v}-\frac{\gamma}{2(1-\gamma)}{\hat\alpha}\sigma\sigma^*{\hat \alpha}. \ ] ] by it s formula , @xmath478 and we have @xmath479= e[{\bar v}(t - t,{\hat y}_{t - t};t)].\ ] ] take a subsequence @xmath480 such that @xmath481=0\ ] ] and @xmath482=0.\ ] ] then we have @xmath483&\leq&\lim_{t_n'\to\infty}\frac { 1}{t_n'}e\bigl[\bigl|{\hat y}_{t_n'-t}^{(t_n')}\bigr|^2\bigr ] \\ & & { } + c'\lim_{t_n'\to\infty}\frac{1}{t_n'}e\biggl[\int_{t_n'-t}^{t_n'}\varphi \bigl({\hat y}_s^{(t_n')},{\hat z}\bigl(s,{\hat y}_s^{(t_n')}\bigr)\bigr)\,ds\biggr]=0 , \ ] ] and thus the lemma has been proved . proof of proposition  [ prop4.2 ] for each @xmath248 there exists @xmath484 such that @xmath485 < { \varepsilon}t_{{\varepsilon}},\qquad{\bar w}(x ) < { \varepsilon}t_{{\varepsilon } } , & \\ & \displaystyle \biggl|{\bar\chi}-\frac{1}{t_{{\varepsilon}}}e\biggl[\int_0^{t_{{\varepsilon}}}\varphi \bigl({\hat y}_s^{(t_{{\varepsilon}})},{\hat z}_s^{(t_{{\varepsilon}})}\bigr)\,ds\biggr]\biggr|<{\varepsilon}.&\end{aligned}\ ] ] set @xmath486 and consider @xmath487 defined by @xmath488 then , for @xmath489 , @xmath490 and @xmath491\,dt.\end{aligned}\ ] ] therefore , @xmath492\bigr\}\,ds \\ & & \qquad \leq e^{p t_{{\varepsilon}}}\bigl|y_{t_{{\varepsilon}}}^{({\varepsilon})}\bigr|^2 + 2\int _ { t_{{\varepsilon}}}^te^{ps}\bigl(y_s^{({\varepsilon})}\bigr)^*\lambda\bigl(y_s^{({\varepsilon})}\bigr)\,dw_s+\int_{t_{{\varepsilon}}}^te^{ps}\bigl\{-k_1\bigl|y_s^{({\varepsilon})}\bigr|^2+k_2\bigr\}\,ds \ ] ] for some positive constants @xmath433 . by using stopping time arguments , for @xmath489 , we have @xmath493&\leq & e[e^{pt_{{\varepsilon}}}|y_{t_{{\varepsilon}}}|^2]+e\biggl[\int_{t_{{\varepsilon}}}^te^{ps}k_2 \,ds\biggr ] \\ & = & e\bigl[e^{pt_{{\varepsilon}}}\bigl|y_t^{({\varepsilon})}\bigr|^2\bigr]+\frac { k_2}{p}(e^{pt}-e^{pt_{{\varepsilon } } } ) . \ ] ] thus , we see that @xmath494\leq e\bigl[\bigl|y_{t_{{\varepsilon}}}^{({\varepsilon})}\bigr|^2\bigr]+\frac{k_2}{p},\ ] ] from which we obtain @xmath495=0 $ ] . hence , @xmath496 . now , by applying it s formula , we have @xmath497\bigl(y_s^{({\varepsilon})}\bigr)\,ds \\ & \geq&\int_0^{t_{{\varepsilon}}}\bigl(d{\bar w}\bigl(y_s^{({\varepsilon})}\bigr)\bigr)^ * \lambda \bigl(y_s^{({\varepsilon})}\bigr)\,dw_s-\chi t_{{\varepsilon}}-\int_0^{t_{{\varepsilon}}}\varphi\bigl(y_s^{({\varepsilon})},z_s^{({\varepsilon})}\bigr)\,ds . \ ] ] therefore , @xmath498,\ ] ] from which we have @xmath499+\frac{1}{t_{{\varepsilon}}}e\bigl[{\bar w}\bigl(y_{t_{{\varepsilon}}}^{({\varepsilon})}\bigr)\bigr ] \\ & \leq&{\varepsilon}+{\bar\chi}+c{\varepsilon}\ ] ] for some positive constant @xmath219 . therefore , @xmath500 for any @xmath248 , and we have @xmath501 . this completes the proof of the proposition . the following is a direct consequence of proposition  [ prop4.1 ] . [ cor4.1 ] under the assumptions of proposition  [ prop4.2 ] , @xmath502 is a  concave function on @xmath503 , and @xmath31 is a convex function . indeed , @xmath504 is a concave function of @xmath29 , and the infimum of a family of concave functions  @xmath502 is concave . [ prop4.3 ] under the assumptions of proposition  [ prop3.1 ] , @xmath156 is ergodic .    @xmath505 as @xmath506 , and @xmath507 and @xmath219 . moreover , @xmath508 , and the hasminskii conditions ( cf . @xcite ) hold . we are going to consider a poisson equation formally obtained by differentiating h - j - b equation ( [ eq3.1 ] ) of ergodic type with respect to @xmath29 . namely , we consider @xmath509+g^*du-(\lambda^*d{\bar w}-\sigma^*{\hat\alpha})^*n_{\gamma } ^{-1}\lambda^*du \\ & & { }   -\frac{1}{2(1-\gamma)^2}(\lambda^*d{\bar w}-\sigma ^*{\hat\alpha})^*\sigma^*(\sigma\sigma^*)^{-1}\sigma(\lambda ^*d{\bar w}-\sigma^*{\hat\alpha } ) . \ ] ] since @xmath510 we can write @xmath511 note that @xmath156 is ergodic in light of proposition  [ prop4.3 ] , and the pair @xmath512 of a function @xmath513 and a constant @xmath514 is the solution to ( [ eq5.1 ] ) . let us set @xmath515 and let @xmath221 be sufficiently large so that @xmath516 \\[-8pt ] \eqntext{\displaystyle x\in{\mathcal d}^c,}\end{aligned}\ ] ] for @xmath517 , which is possible because of assumption ( [ eq2.22 ] ) . therefore , we see that @xmath156 and @xmath209 satisfy assumption ( [ eq9.3 ] ) in the and also @xmath518 for @xmath519 in the following we always take a solution @xmath209 to ( [ eq3.1 ] ) such that @xmath520 . thus , according to proposition  [ prop9.4 ] we can show the existence of the solution @xmath512 to ( [ eq5.1 ] ) . [ cor5.1 ] equation ( [ eq5.1 ] ) has a solution @xmath521 such that @xmath522 and @xmath523 moreover , this solution @xmath513 is unique up to additive constants . it can be clearly seen that @xmath524 and proposition  [ prop9.4 ] applies . [ lem6.1][lem2.6 ] under the assumptions of proposition  [ prop4.2 ] , @xmath525 where @xmath220 and @xmath262 are positive constants independent of @xmath526 .    inequality ( [ eq6.1 ] ) is a direct consequence of ( [ eq4.15 ] ) in lemma  [ lem4.2 ] since @xmath527 is an ergodic diffusion process with the invariant measure @xmath528 .    in the following , we always work under the assumptions of theorem  [ theo2.2 ] ( proposition  [ prop4.2 ] ) . [ lem6.2 ] let @xmath529 and @xmath530 be solutions to  ( [ eq3.1 ] ) with @xmath29 and @xmath531 , respectively , such that @xmath532 . then @xmath533 converges to @xmath534 in @xmath535 strongly and uniformly on each compact  set . we have @xmath536 and @xmath537 for each @xmath183 in light of ( [ eq3.11 ] ) and ( [ eq3.15 ] ) , where @xmath538 is a positive constant independent of @xmath539 , @xmath540 . therefore it follows that @xmath541 is bounded in @xmath542 and converges to some @xmath543 in @xmath542 weakly for each @xmath183 and also uniformly on each compact set by taking a subsequence if necessary . note that @xmath544 converges to @xmath140 because @xmath140 is convex on @xmath545 and thus continuous . take a function @xmath546 such that @xmath547 , and @xmath548 . with @xmath549 , we test @xmath550+\beta_{\gamma+\delta}^*d{\bar w}^{(\gamma+\delta ) } \\ & & { } -\tfrac{1}{2}\bigl(d{\bar w}^{(\gamma+\delta)}\bigr)^*\lambda n_{\gamma+\delta}^{-1}\lambda^*d{\bar w}^{(\gamma+\delta ) } + u_{\gamma+\delta } \ ] ] and obtain @xmath551 where @xmath552 . therefore , @xmath553 since all terms of the right - hand side converge to @xmath554 , we see that converges strongly to @xmath554 in @xmath555 and @xmath533 to @xmath543 strongly in @xmath556 . thus , we obtain our present lemma because @xmath557 satisfies ( [ eq3.1 ] ) , and the solution is unique up to additive constants with respect to @xmath209 .    [ lem6.8][lem6.3 ] let @xmath558 be a solution to @xmath559 where @xmath560 then , as @xmath561 , @xmath562 converges to @xmath563 and @xmath564 converges to @xmath565 in @xmath535 strongly and uniformly on each compact set , where @xmath566 is a  solution to ( [ eq5.1 ] ) .    note that @xmath567 , and that @xmath568 almost everywhere by taking a subsequence , if necessary , since @xmath569 converges strongly in @xmath535 to @xmath570 by lemma  [ lem6.2 ] . moreover , we note that @xmath571 is tight because of lemma  [ lem6.1 ] . therefore , it converges weakly to some probability measure @xmath572 by taking a subsequence if necessary . the limit can be identified with @xmath573 , and  @xmath574 is the only function satisfying ( [ eq9.26 ] ) for @xmath575 and @xmath576 . thus @xmath577 converges to @xmath578 weakly . therefore , @xmath579 converges to @xmath563 .    on the other hand , since @xmath564 is a solution to ( [ eq6.2 ] ) it satisfies @xmath580 and we have @xmath581 . therefore , we see that @xmath564 is locally bounded by a constant independent of @xmath539 . then , by testing ( [ eq6.2 ] ) with @xmath582 , we can see that @xmath583 is bounded for each @xmath183 . therefore , by taking a subsequence if necessary , @xmath564 converges in @xmath535 weakly to a function @xmath584 , which turns out to be the solution @xmath585 to ( [ eq5.1 ] ) . by similar arguments as in the proof of lemma  [ lem6.2 ] , we can see it converges in @xmath535 strongly . furthermore , by theorem  9.11 in @xcite , we see that @xmath586 for each @xmath587 , where @xmath118 is a constant depending on @xmath588 and the @xmath589 norms of the coefficients of @xmath590 . thus , by the sobolev imbedding theorem , @xmath591 is equicontinuous , and thus @xmath564 converges uniformly to  @xmath585 on each compact set .    [ lem6.4 ] let @xmath529 and @xmath592 be solutions to  ( [ eq3.1 ] ) with @xmath29 and @xmath531 , respectively . set @xmath593 and @xmath594 . then @xmath595 and @xmath596 here , @xmath566 is the solution to ( [ eq5.1 ] ) . here we abbreviate @xmath585 to @xmath513 . from ( [ eq3.1 ] ) it follows that @xmath597 & & \qquad = \tfrac{1}{2}\operatorname{tr}\bigl[\lambda \lambda^*d^2\bigl({\bar w}^{(\gamma+\delta)}-{\bar w}^{(\gamma)}\bigr)\bigr]+\beta _ { \gamma+\delta}^*d{\bar w}^{(\gamma+\delta ) } \\[-2pt ] & & \qquad   \quad { } -\beta_{\gamma}^*d{\bar w}^{(\gamma)}-\tfrac { 1}{2}\bigl(d{\bar w}^{(\gamma+\delta)}\bigr)^*\lambda n_{\gamma+\delta } ^{-1}\lambda^*d{\bar w}^{(\gamma+\delta ) } \\[-2pt ] & & \qquad   \quad { } + \tfrac{1}{2}\bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda n_{\gamma } ^{-1}\lambda^*d{\bar w}^{(\gamma)}+u_{\gamma+\delta}-u_{\gamma } \\[-2pt ] & & \qquad = \tfrac{1}{2}\operatorname{tr}\bigl[\lambda\lambda^*d^2\bigl({\bar w}^{(\gamma + \delta)}-{\bar w}^{(\gamma)}\bigr)\bigr]+\beta_{\gamma}^*d\bigl({\bar w}^{(\gamma + \delta)}-{\bar w}^{(\gamma)}\bigr ) \\[-2pt ] & & \qquad   \quad { } -\bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda n_{\gamma } ^{-1}\lambda^*d\bigl({\bar w}^{(\gamma+\delta)}-{\bar w}^{(\gamma)}\bigr ) -\tfrac{1}{2}\bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda n_{\gamma}^{-1}\lambda ^*d{\bar w}^{(\gamma ) } \\[-2pt ] & & \qquad   \quad { } + \bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda n_{\gamma } ^{-1}\lambda^*d{\bar w}^{(\gamma+\delta)}-\tfrac{1}{2}\bigl(d{\bar w}^{(\gamma+\delta)}\bigr)^*\lambda n_{\gamma+\delta}^{-1}\lambda ^*d{\bar w}^{(\gamma+\delta ) } \\[-2pt ] & & \qquad   \quad { } + ( \beta_{\gamma+\delta}-\beta_{\gamma})^*d{\bar w}^{(\gamma + \delta)}+u_{\gamma+\delta}-u_{\gamma } \\[-2pt ] & & \qquad = { \bar l}(\gamma)\bigl({\bar w}^{(\gamma+\delta)}-{\bar w}^{(\gamma ) } \bigr)\\[-2pt ] & & \qquad   \quad { } -\tfrac{1}{2}d\bigl({\bar w}^{(\gamma+\delta)}-{\bar w}^{(\gamma ) } \bigr)^*\lambda n_{\gamma+\delta}^{-1}\lambda^ * d\bigl({\bar w}^{(\gamma + \delta)}-{\bar w}^{(\gamma)}\bigr ) \\[-2pt ] & & \qquad   \quad { } + ( \beta_{\gamma+\delta}-\beta_{\gamma})^*d{\bar w}^{(\gamma+\delta)}+\tfrac{1}{2}\bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda ( n_{\gamma+\delta}^{-1}-n_{\gamma}^{-1})\lambda^*d{\bar w}^{(\gamma ) } \\[-2pt ] & & \qquad   \quad { } -\bigl(d{\bar w}^{(\gamma)}\bigr)^*\lambda ( n_{\gamma+\delta } ^{-1}-n_{\gamma}^{-1})\lambda^*d{\bar w}^{(\gamma+\delta ) } + u_{\gamma+\delta}-u_{\gamma}. \ ] ] therefore we have @xmath598 where @xmath599 and @xmath600 note that @xmath601 is dominated by @xmath602 with a certain positive constant  @xmath220 and that it converges almost everywhere to @xmath603 by taking a subsequence , if necessary , because @xmath604    therefore , @xmath605 moreover , for @xmath606 , @xmath607 let us consider the equation @xmath608    we shall see that @xmath609 as @xmath610 by specifying suitable ambiguity constants . for that , set @xmath611 then we have @xmath612 by considering constructing the solution to this equation according to the proof of proposition  [ prop9.4 ] in the , we see that @xmath613 as @xmath610 . to begin , let @xmath614 be the solution to ( [ eq9.21 ] ) for @xmath615 , @xmath616 and @xmath617 the solution to ( [ eq9.22 ] ) for @xmath618 , @xmath616 and @xmath619 . the operator  @xmath0 is defined as @xmath620 and the operator @xmath621 is defined in the same way as in ( [ eq9.11 ] ) by replacing @xmath622 with @xmath575 in ( [ eq9.4 ] ) and ( [ eq9.10 ] ) . then starting with @xmath623 , @xmath624 , we define the sequence @xmath625 , @xmath626 successively as the solution to ( [ eq9.10 ] ) with @xmath627 and @xmath628 , and as the solution to ( [ eq9.4 ] ) with @xmath629 and @xmath618 , respectively . then we obtain @xmath630 and the estimate for @xmath631 , @xmath632 to estimate @xmath633 , we set @xmath634 to be the solution to @xmath635 and @xmath636 . then @xmath637 satisfies @xmath638 and we have @xmath639 for some constant @xmath640 . on the other hand , to estimate @xmath641 , we set @xmath642 we can assume that @xmath643 is sufficiently large so that @xmath644 for some @xmath645 . since @xmath646 @xmath647 satisfies @xmath648 noting that @xmath649 we have @xmath650 and thus @xmath651 moreover , @xmath652 uniformly on each compact set as @xmath610 . therefore , @xmath653 uniformly on each compact set and we also obtain the estimates @xmath654 and @xmath655    let @xmath656 be the solution to ( [ eq9.28 ] ) for @xmath618 , @xmath616 and @xmath657 . then @xmath658 uniformly as @xmath610 since @xmath659 is estimated as shown above . let @xmath660 be the solution to ( [ eq9.22 ] ) for @xmath618 , @xmath616 and @xmath661 . then , as above , @xmath662 uniformly on each compact set as @xmath610 . since @xmath663 in @xmath664 and @xmath665 in @xmath666 , we conclude that @xmath667 uniformly on each compact set .    in a similar manner , we have @xmath668 by setting @xmath669 we have @xmath670 since @xmath671 converges to @xmath578 weakly , and @xmath672 converges almost everywhere to @xmath673 by taking a subsequence if necessary , as above , we have @xmath674 \\[-8pt ] \eqntext{\mbox{as }   |\delta    @xmath675 we consider @xmath676 then , in the same manner as above , we see that @xmath677 , as @xmath561 , by specifying ambiguity constants . since @xmath564 converges to @xmath513 , @xmath678 does the same .    from ( [ eq6.4 ] ) , ( [ eq6.5 ] ) , ( [ eq6.8 ] ) and ( [ eq6.9 ] ) , it follows that @xmath679 the converse inequalities of ( [ eq6.5 ] ) and ( [ eq6.9 ] ) hold for @xmath680 , and we have    @xmath681    hence , we see that @xmath682 as @xmath561 . from ( [ eq6.3 ] ) and ( [ eq6.6 ] ) , we have @xmath683 and through arguments similar to those above , we see that @xmath684 since @xmath685 for @xmath606 and @xmath686 as @xmath561 . similarly , from  ( [ eq6.7 ] ) and ( [ eq6.10 ] ) , we have @xmath687 and see that @xmath688 therefore , @xmath689 we likewise have @xmath690 and @xmath691 since @xmath692 for @xmath680 . therefore , we obtain @xmath693 and lemma  [ lem6.4 ] follows . [ rem6.1 ] since @xmath694 is a solution to ( [ eq5.1 ] ) , it has a polynomial growth order . more precisely , we have @xmath695 cf . corollary  [ cor5.1 ] and ( [ eq3.11 ] ) . furthermore , we can see that @xmath696 also has a  polynomial growth order for each @xmath697 . indeed , @xmath698 satisfies @xmath699 where @xmath700 therefore , if we set @xmath701 , and @xmath702 , then @xmath703 satisfies @xmath704 for each @xmath705 , @xmath706 . we note that @xmath707 which can be seen in light of our assumptions since @xmath513 is a solution to ( [ eq5.1 ] ) , and @xmath209 is a solution to ( [ eq3.1 ] ) . equation ( [ eq6.14 ] ) corresponds to ( 13.4 ) in @xcite , chapter 3 , section 13 . therefore , by the same arguments as in that work , @xmath708\ ] ] is seen to hold , where @xmath709 is a cut - off function supported by @xmath710 , @xmath711 , and @xmath712 is a constant dominated by @xmath713 , @xmath422 , @xmath714 . from this inequality we obtain inequality ( 5.12 ) for @xmath715 in @xcite , chapter 2 , section 5 . hence , similarly to the proof of lemma 5.4 in  @xcite , chapter  2 , we see that @xmath703 has a polynomial growth order . we first state the following lemma .    [ lem7.1 ] under the assumptions of theorem  [ theo2.3 ] , @xmath716    note that @xmath717 for @xmath290 and that @xmath149 is nondecreasing . therefore , @xmath718 exists . furthermore , @xmath719 and @xmath720 since @xmath721 . here , @xmath722 is a constant defined by ( [ eq3.10 ] ) . hence , we obtain ( [ eq7.1 ] ) . we next give the proof of theorem  [ theo2.4 ] . for @xmath23 , we have @xmath723e^{-\gamma\kappa t}\biggr\ } \\ & = & \log e\biggl[\biggl(\frac{v_t(h)}{s_t^0}\biggr)^{\gamma}\biggr ] -\gamma\kappa t. \ ] ] therefore , @xmath724 -\gamma\kappa t\\ & \leq & v(0,x;t)-\gamma\kappa t,\end{aligned}\ ] ] from which we obtain @xmath725 for all @xmath23 . hence , we have @xmath726    the converse inequality is more difficult to prove . take a constant @xmath15 and @xmath727 such that @xmath728 . then there exists @xmath729 such that @xmath730 we write @xmath729 as @xmath29 for simplicity in the following . let us introduce a probability measure @xmath731 defined by @xmath732 where @xmath733 then @xmath734 is a martingale under the probability measure @xmath731 and @xmath735 \\[-8pt ] & = & \biggl\{\beta+\frac{\gamma}{1-\gamma}\lambda\sigma^*{\hat\alpha}+ \lambda n^{-1}_{\gamma}\lambda^*dw\biggr\}(x_t)\,dt+\lambda(x_t)\,d{\tilde w}_t . \nonumber \ ] ] note that @xmath736 has the same law as the diffusion process @xmath737 governed by stochastic differential equation ( [ eq4.12 ] ) . we further note h - j - b equation of ergodic type ( [ eq2.19 ] ) can be written as @xmath738 by using @xmath156 defined by ( [ eq4.16 ] ) . on the other hand , ( [ eq5.1 ] ) is written as @xmath739 owing to lemma  [ lem6.3 ] , where @xmath740 . now we have @xmath741 and we set @xmath742 note that @xmath743 . then it follows that @xmath744    taking lemma  [ lem4.2 ] into account , we have @xmath745 & & \qquad   \leq\frac{9}{{\varepsilon}^2 t^2}{\tilde e}\biggl[\int _ 0^tv_1(x_s)\,ds\biggr ] \\[-2pt ] & & \qquad   \leq\frac{c}{{\varepsilon}^2 t } \ ] ] for some positive constant @xmath118 and @xmath746\leq e^{-{{{\varepsilon}t}/{3}}}.\ ] ] thus , by using the following lemma , we can see that @xmath747 for sufficiently large @xmath0 . [ lem7.2 ] for sufficiently large @xmath0 we have @xmath748    by it s formula , @xmath749 & & \qquad   = \int_0^t\{{\bar l}(\gamma)w_{\gamma}(x_s)\}\,ds+\int _ 0^t(\nabla w_{\gamma}(x_s))^*\lambda(x_s)\,d{\tilde w}_s \\[-2pt ] & & \qquad   = -\int_0^tv_1(x_s)\,ds+\chi'(\gamma)t+\int_0^t(\nabla w_{\gamma}(x_s))^*\lambda(x_s)\,d{\tilde w}_s . \ ] ] therefore , @xmath750 & & { } + \frac{1}{t}\int_0^t(\nabla w_{\gamma } ( x_s))^*\lambda(x_s)\,d{\tilde w}_s.\end{aligned}\ ] ] thus , @xmath751 & & \qquad = { \tilde p}\biggl(\frac{1}{t}\log v+\frac{1}{t}\{w_{\gamma } ( x)-w_{\gamma}(x_t)\}+ \frac{1}{t}\int_0^t(\nabla w_{\gamma}(x_s))^*\,d{\tilde w}_s>\frac { { \varepsilon}}{3}\biggr ) \\[-2pt ] & & \qquad \leq{\tilde p}\biggl(\frac{1}{t}\log v>\frac{{\varepsilon}}{9}\biggr)+{\tilde p}\biggl(\frac{1}{t}\{w_{\gamma}(x)-w_{\gamma}(x_t)\}>\frac { { \varepsilon}}{9}\biggr ) \\[-2pt ] & & \qquad   \quad { } + { \tilde p}\biggl(\frac{1}{t}\int_0^t(\nabla w_{\gamma}(x_s))^*\,d{\tilde w}_s>\frac{{\varepsilon}}{9}\biggr ) \\[-2pt ] & & \qquad \leq\frac{81}{{\varepsilon}^2t^2}e[|w_{\gamma}(x)-w_{\gamma}(x_t)|^2 ] + \frac{81}{{\varepsilon}^2t^2}e\biggl[\int_0^t(d w_{\gamma})^*\lambda\lambda ^*dw_{\gamma}(x_s)\,ds\biggr ] . \ ] ] hence , by taking @xmath0 and @xmath237 to be sufficiently large , we obtain our present lemma because of lemma  [ lem4.2 ] ; cf . remark  [ rem6.1 ] . let us complete the proof of theorem  [ theo2.4 ] for @xmath752 . set @xmath753 and @xmath754   a_2&=&\bigl\ { -\tfrac{1}{2}\langle m^{\gamma}\rangle_t\geq\bigl(\chi(\gamma ) -\gamma\chi'(\gamma)-{\varepsilon}\bigr)t\bigr\ } , \\[-2pt ]   a_3&=&\biggl\ { \frac{1}{t}\bigl(\log v_t(h)-\log s_t^0\bigr)\leq\kappa\biggr\}. \ ] ] then @xmath755 & & \qquad = { \tilde e}\biggl[e^{-{\tilde m}^{\gamma}_t-{({1}/{2})}\langle m^{\gamma}\rangle_t};\frac { 1}{t}\bigl(\log v_t(h)-s_t^0\bigr)\leq\kappa\biggr ] \\[-2pt ] & & \qquad \geq{\tilde e}\bigl[e^{-{\tilde m}^{\gamma}_t-{({1}/{2})}\langle m^{\gamma}\rangle_t } ; a_1\cap a_2\cap a_3\bigr ] \\[-2pt ] & & \qquad \geq e^{(\chi(\gamma)-\gamma\chi'(\gamma)-2{\varepsilon})t}{\tilde p}(a_1\cap a_2\cap a_3 ) \\[-2pt ] & & \qquad \geq e^{(\chi(\gamma)-\gamma\chi'(\gamma)-2{\varepsilon})t}\ { 1-{\tilde p}(a_1^c)-{\tilde p}(a_2^c)-{\tilde p}(a_3^c)\}. \ ] ] we have seen that @xmath756 holds for sufficiently large @xmath0 in ( [ eq7.6 ] ) , and it is straightforward to see that likewise @xmath757 for sufficiently large @xmath0 . therefore , taking the following lemma into account as well , we have @xmath758 for sufficiently large @xmath0 , and @xmath759 & = & \chi(\gamma)-\gamma(\kappa-{\varepsilon})-2{\varepsilon}\\[-2pt ] & \geq & \inf_{\gamma<0}\{\chi(\gamma)-\gamma(\kappa-{\varepsilon})\ } -2{\varepsilon}\ ] ] for each @xmath248 . since @xmath140 is smooth and convex , @xmath760 , @xmath20 , is strictly concave , and thus continuous . hence , @xmath761    under the assumptions of theorem  [ theo2.4 ] , @xmath762 holds for sufficiently large @xmath0 . first note that @xmath763 and @xmath764 then , from ( [ eq7.4 ] ) and ( [ eq7.5 ] ) , we have @xmath765 set @xmath766 then @xmath767 by it s formula , we have @xmath768 thus , @xmath769 hence , we obtain the present lemma in the same way as lemma  [ lem7.2 ] .    for @xmath163 , we shall prove theorem  [ theo2.4 ] . by convexity , we have @xmath770 that is , @xmath771 @xmath149 is monotonically nondecreasing , and @xmath772 as @xmath773 . therefore , we see that @xmath774 hence , @xmath775 on the other hand , by taking @xmath776 , we have @xmath777 and @xmath778 for sufficiently large @xmath0 . thus , @xmath779 . for a given constant @xmath752 , take @xmath159 such that @xmath780 , namely , @xmath781 then , since @xmath782-\gamma ( \kappa)\kappa t,\ ] ] we have @xmath783-\gamma(\kappa)\kappa.\ ] ] therefore , if we prove that @xmath784&=&\lim_{t\to\infty}\frac { 1}{t}\log e\biggl[\biggl(\frac{v_t(h^{(\gamma(\kappa))})}{s_t^0}\biggr)^{\gamma ( \kappa)}\biggr]\nonumber \\[-8pt ] \\[-8pt ] \quad & = & \chi(\gamma(\kappa ) ) , \nonumber\end{aligned}\ ] ] then we complete the proof of the present theorem because @xmath785-\gamma(\kappa)\kappa\ ] ] and @xmath760 by theorem  [ theo2.4 ] . ( [ eq8.1 ] ) is proved in the following proposition .    under the assumptions of theorem  [ theo2.5 ] , ( [ eq8.1 ] ) let @xmath786 be a solution to ( [ eq2.19 ] ) for @xmath787 and @xmath788 , where @xmath123 is the solution to ( [ eq2.25 ] ) . noting that @xmath789 \\[-8pt ] & = & \frac{1}{2(1-\gamma)}{\hat\alpha}^*\sigma\sigma^*{\hat\alpha } -\frac{1}{2(1-\gamma)}(dw)^*\lambda\sigma^*(\sigma\sigma ^*)^{-1}\sigma\lambda^*dw , \nonumber \ ] ] we have @xmath790+(\beta+\gamma\lambda\sigma^*{\bar h})^*dw\biggr\}(x_s)\,ds \\ & & \qquad   \quad { } + \int_0^t(dw)^*\lambda(x_s)\,dw_s^{{\bar h } } \\ & & \qquad = \int_0^t\biggl\{\frac{1}{2}\operatorname{tr}[\lambda\lambda^*d^2w]+\beta _ { \gamma}^*dw\\ & & \qquad\hphantom{=\int_0^t\biggl\ { } { } + \frac{\gamma}{1-\gamma}(dw)^*\lambda\sigma^*(\sigma \sigma^*)^{-1}\sigma\lambda^*dw\biggr\}(x_s)\,ds \\ & & \qquad   \quad { } + \int_0^t(dw)^*\lambda(x_s)\,dw_s^{{\bar h } } \\ & & \qquad = \int_0^t\biggl\{\chi+u_{\gamma}+\frac{\gamma}{2(1-\gamma ) } ( dw)^*\lambda\sigma^*(\sigma\sigma^*)^{-1}\sigma\lambda ^*dw\\ & & \hspace*{179.5pt}{}-\frac{1}{2}(dw)^*\lambda\lambda^*dw\biggr\}(x_s)\,ds \\ & & \qquad   \quad { } + \int_0^t(dw)^*\lambda(x_s)\,dw_s^{{\bar h } } \\ & & \qquad = \int_0^t\biggl\{\chi-\gamma\eta\bigl(x_s,{\bar h}_s^{(\gamma)}\bigr)-\frac { 1}{2}(dw)^*\lambda\lambda^*dw(x_s)\biggr\}\,ds\\ & & \qquad   \quad { } + \int_0^t(dw)^*\lambda ( x_s)\,dw_s^{{\bar h}}. \ ] ] thus , @xmath791 \\ & & \qquad = e^{{\bar h}}\bigl[e^{\chi t+w(x)-w(x_t)+\int_0^t(dw)^*\lambda ( x_s)\,dw_s^{\bar h}-{({1}/{2})}\int_0^t(dw)^*\lambda\lambda^*dw(x_s)\,ds}\bigr ] . \ ] ] let us introduce a new measure @xmath792 defined by @xmath793 then @xmath794 is a brownian motion process under @xmath792 and @xmath795 therefore , @xmath796   & & \qquad = -\int_0^te^{-w(x_s)}(dw)^*\lambda ( x_s)\,d{\check w}_s \\[-0.5pt ] & & \qquad   \quad { } + \int_0^te^{-w(x_s)}\biggl\{-\frac{1}{2}\operatorname{tr}[\lambda\lambda ^*d^2w]-(\beta+\gamma\lambda\sigma^*{\bar h})^*dw\\[-0.5pt ] & & \hspace*{183pt}{}-\frac { 1}{2}(dw)^*\lambda\lambda^*dw\biggr\}(x_s)\,ds \\[-0.5pt ] & & \qquad = -\int_0^te^{-w(x_s)}(dw)^*\lambda(x_s)\,d{\check w}_s \\[-0.5pt ] & & \qquad   \quad { } -\int_0^te^{-w(x_s)}\biggl\{\frac{1}{2}\operatorname{tr}[\lambda\lambda ^*d^2w]+\beta_{\gamma}^*dw\\[-0.5pt ] & & \hphantom{\qquad   \quad { } -\int_0^te^{-w(x_s)}\biggl\{}{}+\frac{\gamma}{1-\gamma}(dw)^*\lambda \sigma^*(\sigma\sigma^*)^{-1}\sigma\lambda^*dw \\[-0.5pt ] & & \hspace*{149pt } \qquad   \quad { } + \frac{1}{2}(dw)^*\lambda\lambda^*dw\biggr\}(x_s)\,ds \\[-0.5pt ] & & \qquad = -\int_0^te^{-w(x_s)}(dw)^*\lambda(x_s)\,d{\check w}_s-\int _ 0^te^{-w(x_s)}\bigl\{\chi-\gamma\eta\bigl(x_s,{\bar h}_s^{(\gamma)}\bigr)\bigr\}\,ds . \ ] ] then , by the arguments using the stopping time , we have @xmath797\\[-0.5pt ] & & \qquad = e^{\chi t+w(x)}{\check e}\bigl[e^{-w(x_t)}\bigr ] \\[-0.5pt ] & & \qquad = e^{\chi t+w(x)}{\check e}\biggl[e^{-w(x)}+\int_0^te^{-w(x_s)}\bigl\{\gamma \eta\bigl(x_s,{\bar h}_s^{(\gamma)}\bigr)-\chi\bigr\}\,ds\biggr ] . \ ] ] hence , we obtain @xmath798\leq\chi(\gamma)\ ] ] by taking into account ( [ eq2.26 ] ) and ( [ eq8.2 ] ) . the converse inequality holds since @xmath799 . let @xmath622 be an elliptic operator defined by @xmath800 \\[-8pt]&\hspace*{2.3pt}=&\frac { 1}{2}\sum_{i , j}d_i(a^{ij}(x)d_ju)+\sum_i{\tilde b}^i(x)d_iu , \nonumber\end{aligned}\ ] ] where @xmath801 and @xmath802 are lipschitz continuous functions such that @xmath803 and @xmath804 . we assume that there exists a positive function @xmath805 such that @xmath806 set @xmath807 , @xmath808 and @xmath809 then we consider the following exterior dirichlet problem for a given bounded continuous function @xmath16 on @xmath810 : @xmath811    [ prop9.1 ] exterior dirichlet problem ( [ eq9.4 ] ) has a unique bounded solution @xmath812 , @xmath813 . we first show uniqueness . note that @xmath814 and set @xmath815 . then @xmath816 therefore , @xmath107 satisfies @xmath817 let @xmath818 and @xmath819 be solutions to ( [ eq9.5 ] ) . then @xmath820 satisfies @xmath821 to prove uniqueness , it is sufficient to show that the solution @xmath822 to ( [ eq9.6 ] ) is trivial . for each @xmath727 , there exists @xmath823 such that @xmath824 . take @xmath825 . then we see that @xmath826 since @xmath827 and @xmath828 in @xmath666 . thus , we see that @xmath829 because @xmath248 is arbitrary . let us show the existence of the solution to ( [ eq9.4 ] ) . we can assume @xmath830 . consider the following dirichlet problem for @xmath222 : @xmath831 then we have @xmath832 it is clear that @xmath833 and @xmath238 by the maximum principle . therefore , there exists @xmath834 and @xmath835 when taking @xmath836 we see that @xmath837 thus , @xmath838 converges to @xmath839 weakly in @xmath840 . regularity theorems show that @xmath841 . let us take a bounded domain @xmath664 such that @xmath842 and a bounded borel function @xmath843 on @xmath844 . we consider a dirichlet problem @xmath845 which admits a solution @xmath846 . for this solution , we consider exterior dirichlet problem ( [ eq9.4 ] ) with @xmath847 . then we introduce an operator @xmath848 defined by @xmath849 where @xmath850 is the solution to ( [ eq9.4 ] ) with @xmath847 . in a similar manner to lemma  5.1 in @xcite , chapter ii , we have @xmath851 where @xmath852 moreover , we have the following proposition ; cf . theorem 4.1 , chapter ii in  @xcite . operator @xmath621 defined above satisfies the following properties . @xmath853 and for some @xmath421 , @xmath854 furthermore , there exists a probability measure @xmath855 on @xmath856 such that @xmath857 \\[-8pt ]   \eqntext{\displaystyle   \rho=\log\frac{1}{1-\delta } ,   k=\frac{2}{1-\delta},}\end{aligned}\ ] ] and @xmath858 for any bounded borel function @xmath843 . consider an exterior dirichlet problem for a given function @xmath271 : @xmath859 then we have the following proposition . for a given function @xmath271 , there exists a unique solution @xmath860 , @xmath813 , to ( [ eq9.16 ] ) such that @xmath861    assume that @xmath862 , @xmath271 . for @xmath222 we consider a dirichlet problem on @xmath863 : @xmath864 there exists a unique solution @xmath865 . set @xmath866 then we have @xmath867 to see that , set @xmath868 . then @xmath869 therefore , @xmath870 since @xmath871 is subharmonic in @xmath872 and @xmath873 on @xmath874 . hence , we have @xmath875 .    on the other hand , @xmath876 hence , @xmath877 is superharmonic and @xmath878 on @xmath879 . thus , @xmath880 and ( [ eq9.18 ] ) holds . if @xmath881 and @xmath882 , then , through the same arguments for @xmath883 , we obtain @xmath884 where @xmath885 is the corresponding solution to ( [ eq9.17 ] ) . therefore , for general @xmath886 , we have @xmath887    let @xmath888 be a solution to ( [ eq9.17 ] ) for @xmath889 . then , @xmath888 is nondecreasing with respect to @xmath237 because of the maximum principle . indeed , for @xmath238 , we have @xmath890 since @xmath888 is dominated by @xmath891 , there exists @xmath892 such that @xmath893 set @xmath894 then we have @xmath895 for @xmath896 injection @xmath897 , is compact . therefore , @xmath898 weakly in @xmath840 for each @xmath899 and @xmath892 is a weak solution to @xmath900 by the regularity theorem @xmath901 , @xmath902 . similarly , we have @xmath903 , which is a solution to @xmath904    now let us prove uniqueness . for @xmath540 , we assume that @xmath905 is a solution to ( [ eq9.16 ] ) such that @xmath906 then @xmath907 satisfies @xmath908 we shall prove that @xmath513 satisfying ( [ eq9.19 ] ) is trivial , @xmath909 . for this purpose , we set @xmath910 where @xmath911 is the constant that appears in ( [ eq9.3 ] ) . since @xmath912 we have @xmath913 note that @xmath914 for @xmath915 under assumption ( [ eq9.3 ] ) . moreover , @xmath916 hence , from the maximum principle , we see that @xmath227 as in the proof of proposition  [ prop9.1 ] . let @xmath917 be a function on @xmath918 such that @xmath917 is bounded in @xmath919 and @xmath920 , and @xmath664 a bounded domain such that @xmath921 . we consider @xmath922 and @xmath923 then we set @xmath924 and @xmath925 we further consider @xmath926 then , as in the proof of theorem 5.3 , in @xcite , chapter ii , we obtain the following proposition . here , we only give the proof of the existence of the solution for use in section  [ sec6 ] . [ prop9.4 ] equation ( [ eq9.24 ] ) has a solution unique up to additive constants if and only if @xmath927 . moreover , @xmath928 for @xmath929 , @xmath930 and @xmath931 in distribution sense @xmath932 such that @xmath933 and @xmath934 . furthermore , @xmath935 is the only function in @xmath936 satisfying ( [ eq9.26 ] ) and @xmath937    proof of existence let @xmath938 and @xmath939 , where @xmath940 ( resp . , @xmath839 ) is the solution to ( [ eq9.21 ] ) [ resp . , ( [ eq9.22 ] ) ] . for each @xmath941 define @xmath942 and @xmath943 as follows let @xmath942 be the solution to ( [ eq9.10 ] ) for @xmath944 , and @xmath943 the solution to  ( [ eq9.4 ] ) for @xmath945 . then @xmath946 since @xmath947 , we have @xmath948 by ( [ eq9.14 ] ) . set @xmath949 . then @xmath950 therefore , we see that there exists @xmath951 such that @xmath952 , . moreover , we have @xmath953 note that @xmath954 is the solution to @xmath955 and @xmath956 the solution to ( [ eq9.22 ] ) with @xmath957 . noting that @xmath958 we see that @xmath954 converges in @xmath959 and weakly in @xmath840 since its @xmath960 norm is bounded ; cf . theorem 9.11 in @xcite . by the regularity theorems , the limit @xmath961 and satisfies @xmath962 on the other hand , @xmath963 is the solution to ( [ eq9.4 ] ) with @xmath964 and @xmath965 thus , we see that @xmath966 and @xmath956 converges in @xmath967 and weakly in @xmath968 . the limit @xmath969 and satisfies ( [ eq9.22 ] ) with @xmath970 . setting @xmath971 in @xmath664 and @xmath972 in @xmath666 , we have a solution to ( [ eq9.24 ] ) . the author thanks the anonymous referee for careful reading the manuscript and giving valuable suggestions . the author also thanks dr . y.  watanabe for fruitful discussions .
we consider minimizing the probability of falling below a target growth rate of the wealth process up to a time horizon @xmath0 in an incomplete market model , and then study the asymptotic behavior of minimizing probability as @xmath1 . 
 this problem can be closely related to an ergodic risk - sensitive stochastic control problem in the risk - averse case . indeed , in our main theorem 
 , we relate the former problem concerning the asymptotics for risk minimization to the latter as its dual . as a result 
 , we obtain an expression of the limit value of the probability as the legendre transform of the value of the control problem , which is characterized as the solution to an h - j - b equation of ergodic type , in the case of a markovian incomplete market model .    .