Optimization of a Dynamic Profit Function using Euclidean Path Integral
aa r X i v : . [ ec on . T H ] F e b Optimization of a Dynamic ProfitFunction using Euclidean Path Integral
Paramahansa Pramanik and Alan M. Polansky
Department of Statistics and Actuarial ScienceDeKalb, IL 60115 USA
Abstract:
A Euclidean path integral is used to find an optimal strategy fora firm under a Walrasian system, Pareto optimality and a non-cooperativefeedback Nash Equilibrium. We define dynamic optimal strategies and de-velop a Feynman type path integration method to capture all non-additiveconvex strategies. We also show that the method can solve the non-linearcase, for example Merton-Garman-Hamiltonian system, which the tradi-tional Pontryagin maximum principle cannot solve in closed form. Further-more, under Walrasian system we are able to solve for the optimal strategyunder a linear constraint with a linear objective function with respect tostrategy.
MSC 2010 subject classifications:
Primary 93E20; Secondary 49N90.
Keywords and phrases:
Dynamic profit, Euclidean path integral, Wal-rasian system, Pareto Optimality, Non-cooperative feedback Nash equilib-rium, Stochastic differential games.
1. Introduction
In this paper we consider dynamic profit maximization over a time intervalwith finite horizon t >
0. The objective is to find an optimal strategy for afirm in a system whose state dynamics are specified by a stochastic differentialequation. The instantaneous profit function we consider depends on the time s , a real-valued measure of the market share of the firm x ( s ), and the real-valued dynamic strategy of the firm u ( s ). The profit function is represented by π [ s, x ( s ) , u ( s )] ∈ R . Here x ∈ X and u ∈ U , where X is a functional spacecorresponding to the set of all market share trajectories and U is a functionalspace corresponding to the set of all possible strategies available to the firm. Weassume the functional spaces X and U are bounded and complete. The profitover the time interval [0 , t ] is measured by the stochastic integral Z t π [ s, x ( s ) , u ( s )] ds, using the Itˆo representation of the integral (Øksendal, 2003). The dynamics ofthe market share are given by dx ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] dB ( s ) , (1)where B ( s ) is Brownian motion process. imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral In this paper we are interested in calculating three types of equilibria: Wal-rasian, Pareto and Nash. The Walrasian system is a fundamental market struc-ture in economics, and is the basis of many other market systems (Walras, 1900).The main assumption under this system is that each firm is small when com-pared to the entire industry and therefore does not influence the industry price.The industry consists of all the firms, and its price is determined by the entiresystem. In this system, a single firm can earn at least zero profit in the longrun. Therefore, if a single firm wants to survive it has to achieve its average cost(Walras, 1900).
Definition 1.
The continuous path of market share x ∗ ( s ) ∈ X and a continuousset of optimal strategies u ∗ ( s ) ∈ U constitute a Walrasian Equilibrium if forevery time point s ∈ [0 , t ] , E Z t π [ s, x ∗ ( s ) , u ∗ ( s )] ds ≥ E Z t π [ s, x ( s ) , u ( s )] ds, (2) with market dynamics defined in Equation (1). Definition 1 implies that each firm under the Walrasian system faces identicalmarket dynamics. In this case, finding the optimal strategy of a firm correspondsto solving the optimization problemmax u ∈ U Π( u, t ) = max u ∈ U E Z t π [ s, x ( s ) , u ( s )] ds, (3)under the constraint given in Equation (1), and initial condition x (0) = x .Determining Pareto and Nash equilibria requires us to consider the otherfirms in the industry. Suppose that there are k firms in an economy, where thestrategy function of firm ρ is given by u ρ ( s ) for ρ = 1 , ..., k , u ρ ∈ U ρ ⊂ U ,where U ρ is the set of all available strategies of firm ρ , and U is the set of allavailable strategies in the market. Let x ρ ( s ) be the measure of market sharefor firm ρ . Let x ( s ) and u ( s ) be the vectors containing the elements x ρ ( s ) and u ρ ( s ) for ρ = 1 , ..., k , respectively. Each firm has a dynamic profit function π ρ [ s, x ( s ) , u ( s )] , with market dynamics specified by d x ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] d B ( s ) , (4)where µ [ s, x ( s ) , u ( s )] is an k -dimensional drift function, σ [ s, x ( s ) , u ( s )] is an k × m -dimensional diffusion function, and B ( s ) is an m -dimensional Brownianmotion process. The initial condition is x (0) = x ∈ R k .Pareto optimality is an economic environment where each player benefits atthe expanse of the other players (Greenwald and Stiglitz, 1986; Mas-Colell et al.,1995). Therefore, Pareto optimality insures the greatest mutual benefit for allof the players simultaneously. Mathematically this is equivalent to maximizingthe total dynamic profit,Π P ( u, t ) = E Z t k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] ds, imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where α ρ is the profit weight corresponding to ρ th firm such that P kρ =1 α ρ = 1. Definition 2.
The strategies u ∗ ∈ U k , constitute a cooperative Pareto Equilib-rium for the ρ th firm if E Z t k X ρ =1 α ρ π ρ [ s, x ( s ) , u ∗ ( s )] ds ≥ E Z t k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] ds, (5) for ρ = 1 , ..., k subject to the Equation (4) with initial condition x (0) = x ,where α ρ is the profit weight of ρ th firm such that k X ρ =1 α ρ = 1 . Assuming π ρ [ s, x ( s ) , u ( s )] is non-negative and differentiable, Fubini’s Theo-rem implies that the cooperative Pareto equilibrium the optimization problemfor the ρ th firm ismax u ρ ∈U Π( u , t ) = max u ρ ∈ U Z t ( E k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] ) ds, (6)subject to Equation (4), with initial condition x (0) = x . In other words, Equa-tion (6) implies that ρ th firm performs its optimization in light of the optimalstrategies of the other firms. Definition 3.
In the non-Cooperative feedback Nash framework a set of optimalstrategies u ∗ ( s ) form a non-cooperative feedback Nash equilibrium if E (cid:26)Z t π ρ [ s, x ∗ ( s ) , u ∗ ( s )] ds (cid:27) ≥ E (cid:26)Z t π ρ [ s, x ( s ) , ˆ u ρ ( s )] ds (cid:27) , for all ρ ∈ { , ..., k } where t ∈ (0 , ∞ ) , subject to the constraints, dx ∗ ( s ) = µ [ s, x ∗ ( s ) , u ∗ ( s )] ds + σ [ s, x ∗ ( s ) , u ∗ ( s )] dB ( s ) , (7) dx ρ ( s ) = µ [ s, x ρ ( s ) , ˜ u ∗ ρ ( s )] ds + σ [ s, x ρ ( s ) , ˜ u ∗ ρ ( s )] dB ( s ) , (8) and x (0) = x , for ρ = 1 , ..., k , where ˜ u ∗ ρ ( s ) = [ u ( s ) , ..., u ρ − ( s ) , u ∗ ρ ( s ) , u ρ +1 ( s ) , . . . , u k ( s )] ′ , and ˆ u ∗ ρ ( s ) = [ u ∗ ( s ) , ..., u ∗ ρ − ( s ) , u ρ ( s ) , u ∗ ρ +1 ( s ) , . . . , u ∗ k ( s )] ′ . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Hence, firm ρ has the optimization problemmax u ρ ∈ U ˜Π( u ρ , t ) = max u ρ ∈ U E Z t π ρ [ s, x ( s ) , ˆ u ρ ( s )] ds, (9)subject to the constraints in Equations (7) and (8) and initial conditions x (0) = x ∗ (0) = x .Traditionally, these optimization problems are solved by using the Pontrya-gin principle Pontryagin (1987) after solving the Hamilton-Jacobi-Bellman equation. See Bellman (1952, 2013); Bellman and Dreyfus (2015); Ljungqvist and Sargent(2012); Pontryagin (1966); Stokey (1989) and Yeung and Petrosjan (2006). Themain problem with this method is that finding a solution often requires ob-taining a complicated value function. An alternative method for solving opti-mal control problems is based on principles from quantum mechanics and pathintegrals. These methods have previously been used in motor control theory(Kappen, 2005; Theodorou, Buchli and Schaal, 2010; Theodorou, 2011), and fi-nance (Baaquie, 2007). There are three mathematical representations of thisapproach based on partial differential equations, path integrals, and stochasticdifferential equations (Theodorou, 2011). Partial differential equations give amacroscopic view of an underlying physical process, while path integrals andstochastic differential equations give a more microscopic view. Furthermore, theFeynman-Kac formula yields a special set of Hamiltonian-Jacobi-Bellman equa-tions which are backward parabolic partial differential equations (Kac, 1949).Only a few problems in finance are directly tractable by Pontryagin maximumprinciple and solving the Hamiltonian-Jacobi-Bellman equation usually involvessolving a system of differential equations which is often a difficult task. Thepotential advantage of the quantum approach is that a general non-linear sys-tem, such as Merton-Garman Hamiltonian, can be impossible to solve analyti-cally. The quantum method allows a different approach to attack these problemsand sometimes can give simplified solutions (Baaquie, 2007). Path integrals arewidely used in physics as a method of studying stochastic systems. In finance,path integrals have been used to study the theory of options and interest rates(Linetsky, 1997; Lyasoff, 2004). A rigorous discussion of the application of dif-ferent types of quantum path integrals in finance is given in Baaquie (2007).The idea is that, in quantum mechanics a particle’s evolution is random. Thisis analogous to the evolution of a stock price having non-zero volatility.Motivated by Baaquie (2007) we consider a firm’s real-valued measure ofmarket share as a stochastic process and use the principles of path integral asthe basis for our mathematical model. The assumption is that since a firm is avery small part of an industry and an economy, and is subject to many smallstochastic perturbations, the movement of its share will behave like a quan-tum particle in physics. Although these methods have been used in quantumapproaches to financial problems we are not aware of their use in stochasticoptimization problems for the economic systems studied here. imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral
2. Main results
Define a non-negative measurable discounted profit function for a single firm as π [ s, x ( s ) , u ( s )] = exp( − ζs )˜ π [ s, x ( s ) , u ( s )] . Assume that π is a finite C ∞ function with respect to x ( s ) and u ( s ) where ζ ∈ [0 ,
1] is a constant discount rate of profit over s ∈ [0 , t ]. The function˜ π [ s, x ( s ) , u ( s )] is the actual profit at time s , and is assumed to be quadratic interms of change in time, non-decreasing in output price, non-increasing in inputprice, homogeneous of degree one in output and input prices, convex in outputand input prices, continuous in output and input prices, and is continuous withrespect to s . We assume that x ( s ) is a time dependent measure of a stochasticmarket dynamic and the strategy u ( s ) is a deterministic function of x . Furthertechnical assumptions are given in the Appendix.To optimize the dynamic profit function Π defined in Definition 1 with respectto the strategy u we need to specify a function g : [0 , t ] × X → R to favorstrategies that respect the dynamics specified by Equation (1). In the standardLagrangian framework this function is specified as g ( s, x ) = λ [ dx ( s ) − h ( s, x )],where h is a function that specifies the dynamics of the system and λ is theLagrange multiplier. Proposition 1 (Walrasian Equilibrium) . An optimal strategy for maximizingthe dynamic profit function Π( u, t ) with respect to the control u and constraint dx ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] dB ( s ) , with initial condition x (0) = x is the solution of the equation (cid:20) ∂∂u f ( s, x, u ) (cid:21) (cid:20) ∂ ∂x f ( s, x, u ) (cid:21) = 2 (cid:20) ∂∂x f ( s, x, u ) (cid:21) (cid:20) ∂ ∂x∂u f ( s, x, u ) (cid:21) , (10) with respect to u as a function of x and s evaluated at x = x ( s ) , where f ( s, x, u ) = π ( s, x, u ) + g ( s, x ) + ∂∂s g ( s, x )+ µ ( s, x, u ) ∂∂x g ( s, x ) + σ ( s, x, u ) ∂ ∂x g ( s, x ) . (11) Example 1.
Suppose that a firm under a Walrasian system has the objectivefunction E (cid:26)Z t exp( − ζs )[ px ( s ) − cx ( s ) u ( s )] ds (cid:27) , where ζ ∈ (0 , is a constant discount rate over time interval [0 , t ] , p > isconstant price, x ( s ) is the total output, a twice differentiable function of s , c is imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral a positive constant marginal cost, and u is the total expenditure on advertising.Consider market dynamics given by dx ( s ) = [ ax ( s ) − u ( s )] ds + p σx ( s ) u ( s ) dB ( s ) , (12) where a and σ are two positive and finite constants. The negative terms inthe drift part of Equation (12) and the objective function reflect the firm’s costof advertising its product as its strategy. The diffusion component of Equation(12) reflects the amount of variation in the system. To apply Proposition 1 wespecify g ( s, x ) to represent the market dynamics. For a fixed positive Lagrangianmultiplier λ ∗ let g ( s, x ) = λ ∗ s [ ax − b ] where b is a positive number such that a < b . Equation (11) yields f ( s, x, u ) = x exp( − ζs )( p − cu ) + sλ ∗ ( ax − b ) + 2 λ ∗ sax ( ax − u ) + λ ∗ σasux. Therefore ∂∂x f ( s, x, u ) = exp( − ζs )( p − cu ) + 2 λ ∗ sax + 2 λ ∗ as ( ax − u ) + 2 λ ∗ sa x + λ ∗ σasu,∂∂u f ( s, x, u ) = − cx exp( − ζs ) − λ ∗ asx + σasxλ ∗ = A ( s, x ) ,∂ ∂x f ( s, x, u ) = 2 λ ∗ as (1 + 2 a ) = B ( s ) , and ∂ ∂x∂u f ( s, x, u ) = − c exp( − ζs ) − λ ∗ as + σλ ∗ as = D ( s ) . Equation (10) then implies that an optimal Walrasian strategy for this systemis given by φ ∗ w ( s, x ) = 1 σλ ∗ as − λ ∗ as − c exp( − ζs ) (cid:20) A ( s, x ) B ( s )2 D ( s ) − E ( s, x ) (cid:21) , (13) where E ( s, x ) = p exp( − ρs ) + 2 λ ∗ asx + 4 λ ∗ a s x , D ( s ) = 0 , σλ ∗ as = 2 λ ∗ as + c exp( − ρs ) and A ( s, x ) B ( s )2 D ( s ) − E ( s, x ) = 0 . In Example 1 both the objective function and the market dynamics are linearcontinuous mappings from strategy space to the real line. According to theGeneralized Weierstrass Theorem there exists an optimal strategy. One suchstrategy is given in Equation (13). The Pontryagin maximum principle cannotbe used to find a closed-form optimal strategy for this system.
Example 2.
Suppose that a firm under the Walrasian system produces con-sumer goods with objective function E (cid:26)Z t exp( − ζs ) (cid:2) R ( x ) − cu (cid:3) ds (cid:27) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where ζ ∈ (0 , is a constant discount rate over time interval [0 , t ] , R ( x ) is thetotal revenue function such that it can be multiplicatively separable by d /ds asdiscussed in the Appendix, c is the constant cost multiplied by squared strategyfunction u ( s ) . The main difference between this example with Example 1 is that,the strategy u ( s ) is a C function and hence, we can calculate optimal strategyusing Pontryagin’s maximum principle. Assume the market dynamics of the firmfollow dx ( s ) = [ bx ( s ) − u ( s )] ds + p bx ( s ) dB ( s ) , (14) where b is a positive constant. We will use our method and the traditional Pon-tryagin maximum principle to find the optimal strategy of this Walrasian firmunder a consumer good industry.As the consumption of consumer goods increases exponentially, a Walrasianfirm under this sector should face the market dynamics which shows the behaviorin Equation (14) (Cohen, 2004; Remus, 2019). Assume for a fixed Lagrangianmultiplier λ ∗ the g ( s, x ) function is an exponential function with the trend ofEquation (14). That is g ( s, x ) = λ ∗ exp( sbx − d ) . Equation (11) yields f ( s, x, u ) = exp( − ζs ) (cid:2) R ( x ) − cu (cid:3) + g ( s, x )[1 + bx + sb x (1 − b ) − sbu ] . Therefore ∂∂x f ( s, x, u ) = exp( − ζs ) ∂∂x R ( x ) + g ( s, x )[ b + sb (1 − b )]+ ∂∂x g ( s, x ) { bx [1 + sb (1 − b )] } − sbu ∂∂x g ( s, x )= A ( s, x ) − sbu ∂∂x g ( s, x ) ,∂∂u f ( s, x, u ) = − [2 cu exp( − ζs ) + sbg ( s, x )] ,∂ ∂x f ( s, x, u ) = exp( − ζs ) ∂ ∂x R ( x ) + ∂ ∂x g ( s, x ) { bx [1 + sb (1 − b )] } +2 ∂∂x g ( s, x )[ b + sb (1 − b )] − sbu ∂ ∂x g ( s, x )= A ( s, x ) − sbu ∂ ∂x g ( s, x ) , and ∂ ∂x∂u f ( s, x, u ) = − sb ∂∂x g ( s, x ) , where A ( s, x ) = exp( − ζs ) ∂∂x R ( x ) + g ( s, x ) (cid:2) b + sb (1 − b ) (cid:3) + ∂∂x g ( s, x ) { bx [1 + sb (1 − b )] } imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral and A ( s, x ) = exp( − ζs ) ∂ ∂x R ( x ) + ∂ ∂x g ( s, x ) { bx [1 + sb (1 − b )] } + 2 ∂∂x g ( s, x ) (cid:2) b + sb (1 − b ) (cid:3) Equation (10) yields a cubic strategy function u such that, B ( s, x ) u + B ( s, x ) u + B ( s, x ) u + B ( s, x ) = 0 , with B ( s, x ) = 2 c ( sb ) exp( − ζs ) g ( s, x ) ,B ( s, x ) = s b g ( s, x )[ s b g ( s, x ) − c exp( − ζs ) A ( s, x )] ,B ( s, x ) = 2 (cid:8) c exp( − ζs ) A ( s, x ) − s b g ( s, x ) [ A ( s, x ) − (cid:9) ,B ( s, x ) = sbg ( s, x ) (cid:2) A ( s, x ) − sbA ( s, x ) (cid:3) , and the optimal Walrasian strategy becomes, φ ∗ w ( s, x ) = D ( s, x ) + (cid:26) D ( s, x ) + h D ( s, x ) + (cid:0) D ( s, x ) − D ( s, x ) (cid:1) i (cid:27) + (cid:26) D ( s, x ) − h D ( s, x ) + (cid:0) D ( s, x ) − D ( s, x ) (cid:1) i (cid:27) , such that D ( s, x ) = − B ( s, x )3 B ( s, x ) ,D ( s, x ) = D ( s, x ) + B ( s, x ) B ( s, x ) − B ( s, x ) B ( s, x )6 B ( s, x ) ,D ( s, x ) = B ( s, x )3 B ( s, x ) and B ( s, x ) = 0 . The important part of this result is that we start with a g ( s, x ) function such that it is a C function within [0 , t ] and we get the optimal strategyby solving a cubic equation.For comparison, Walrasian optimal strategy under Pontryagin maximum prin-ciple is found by Yeung and Petrosjan (2006) as φ ∗ w ( s, x ) = 0 or φ ∗ w ( s, x ) = bx exp( − ζs )exp( − ζs ) (cid:2) exp( ζs ) (cid:3) . Example 3.
Suppose that a pure Walrasian firm in the consumer goods industryhas the objective function E (cid:26)Z t exp( − ζs )[ R ( x ) − cu ] ds (cid:27) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where ζ ∈ [0 , is a constant discount rate over [0 , t ] , R ( x ) is the total revenuefunction, and c is the constant cost multiplied by squared strategy function u ( s ) .As we assume the the firm is pure Walrasian, the market dynamics it faces doesnot depend on the strategy and has the form dx ( s ) = bx ( s ) ds + p σx ( s ) dB ( s ) , where b and σ are two positive constants.For the Quantum approach assume g ( s, x ) = λ ∗ exp( sbx ) for a fixed Lagrangemultiplier λ ∗ (Cohen, 2004; Remus, 2019). Equation (11) yields f ( s, x, u ) = exp( − ρs ) (cid:2) R ( x ) − cu (cid:3) + g ( s, x ) (cid:2) sb x (cid:0) sσ (cid:1)(cid:3) . Therefore, ∂∂x f ( s, x, u ) = exp( − ζs ) ∂∂x R ( x ) + ∂∂x g ( s, x ) × [1 + sb x (cid:0) sσ (cid:1) ] + sb (1 + sσ ) g ( s, x ) ,∂∂u f ( s, x, u ) = − cu exp( − ζs ) ,∂ ∂x f ( s, x, u ) = exp( − ζs ) ∂ ∂x R ( x ) + ∂ ∂x g ( s, x )[1 + sb x (cid:0) sσ (cid:1) ]+ sb (1 + sσ ) (cid:20) g ( s, x ) + ∂∂x g ( s, x ) (cid:21) , and ∂ ∂x∂u f ( s, x, u ) = 0 . The right hand side of Equation (10) becomes zero and the Walrasian optimalstrategy is φ ∗ w ( s, x ) = 0 .The corresponding Hamiltonian-Jacobi-Bellman Equation is − ∂∂s V ( s, x ) − σx ∂ ∂x V ( s, x )= max u ∈ U (cid:26) exp( − ζs )[ R ( x ) − cu ] + bx ∂∂x V ( s, x ) (cid:27) . (15) After solving for the right hand side of Equation (15) we get φ ∗ w ( s, x ) = 0 .In this example we conclude that if the trend of the market dynamics does notdepend on u ( s ) , there is no optimal strategy under both of quantum approachand Pontryagin maximum principle. Another important example considers problems involving European call op-tions, which have been well studied in finance, and provide the basis for theBlack-Scholes formula and further generalizations by Merton-Garman. In the imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral generalized approach the stock volatility is stochastic and is derived by a parabolicpartial differential equation (Baaquie, 1997; Merton, 1973). As constructing aHamiltonian-Jacobi-Bellman equation becomes impossible in this case, methodsof theoretical physics have been applied to get an optimal solution (Bouchaud and Sornette,1994). For example, the Feynman-Kac lemma has been used in Baaquie (1997)and Baaquie (2007) to find a solution of a Merton-Garman-Hamiltonian typeequations using the Dirac bracket method Bergmann and Goldberg (1955). InProposition 2 we use a path integral approach to a situation where the firm’sobjective is to maximize its portfolio subject to a Merton-Garman-Hamiltoniantype stochastic volatility in an European call option with controls. Using thefunction g as defined for Proposition 1, the result given below provides an op-timal investment strategy for this framework.For this type of problem suppose that the firm has the objective of maximizingΠ MG ( u, t ) = E Z t π [ x, H ( s, K, V ) , V ( s ) , u ( s )] ds, where u ( s ) is the strategy, and H is the European call option price which is afunction of the time s , the stock price of the security at time s is representedby K ( s ), and the volatility at time s is represented by V ( s ). It is assumed thatthe stock price and the volatility follow Langevin dynamics of the form dK ( s ) = µ [ s, u ( s )] K ( s ) ds + σ [ s, u ( s )] K ( s ) dB ( s ) , and dV ( s ) = µ [ s, u ( s )] V ( s ) ds + σ [ s, u ( s )] V ( s ) dB ( s ) , where µ [ s, u ( s )] is the expected return of the security, µ [ s, u ( s )] is the expectedrate of increase in V ( s ), and B ( s ) and B ( s ) are standard Brownian motionprocesses such that the correlation between dB ( r ) and dB ( s ) is zero unless s = r for which case it equals a value γ ∈ [ − , Proposition 2 (Merton-Garman Hamiltonian Type Equation) . Suppose thata firm’s objective portfolio is given by maximizing Π MG ( u, t ) with respect to thestrategy u ∈ U . Let f ( s, K, V, u ) = π [ s, H ( s, K, V ) , V, u ] + g ( s, K, V ) + ∂∂s g ( s, K, V )+ Kµ ( s, u ) ∂∂K g ( s, K, V ) + V µ ( s, u ) ∂∂V g ( s, K, V )+ K σ ( s, u ) ∂ ∂K g ( s, K, V ) + Kρσ ( s, u ) × ∂ ∂K∂V g ( s, K, V ) + V σ ( s, u ) ∂ ∂V g ( s, K, V ) . (16) An optimal Walrasian strategy is the functional solution of − (cid:20) ∂∂u f ( s, K, V, u ) (cid:21) Ψ s ( K, V ) = 0 imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where Ψ s ( K, V ) = exp {− sf ( s, K, V, u ) } I ( K, V ) is the transition wave functionat time s and states K ( s ) and V ( s ) with initial condition Ψ ( K, V ) = I ( K, V ) . Proposition 2 is the extension of the framework of Baaquie (1997) that ac-counts for the firm’s portfolio and has drift and diffusion components that arefunctions of the feedback control system and considers an optimal Walrasianstrategy.Proposition 3 considers the case of the cooperative environment outlined inDefinition 2.
Proposition 3 (Cooperative Pareto Optimality) . A cooperative Pareto optimalsolution for firm ρ where all the firms maximize the total dynamic profit Π P ( u, t ) subject to d x ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] d B ( s ) , with initial condition x (0) = x is obtained by solving − ∂f [ s, x ( s ) , u ( s )] ∂u ρ Ψ s ( x ) = 0 , (17) with respect to ρ th firm’s strategy, where Ψ s is the transition wave functiondefined as Ψ s ( x ) = exp[ − f ( s, x , u )]Ψ ( x ) with initial condition Ψ ( x ) and f is defined as f ( s, x , u ) = k X ρ =1 α ρ π ρ ( s, x , u ) + g ( s, x ) + ∂∂s g ( s, x )+ µ ′ ( s, x , u ) D x g ( s, x ) + σ ′ ( s, x ) H x g ( s, x ) σ ( s, x ) , where D x is the gradient vector and H x is the Hessian matrix. Example 4.
Suppose that a firm under a Cooperative Pareto system has theobjective function E (Z t exp( − rs ) k X ρ =1 α ρ (cid:0) px ρ − cx ρ u ρ (cid:1) ds ) , where r ∈ (0 , is a constant discount rate over time interval [0 , t ] , p > isconstant price, α ρ is the weight corresponding to ρ th firm such that P kρ =1 α ρ = 1 , x ρ is ρ th firm’s total output, c is a positive constant marginal cost for each firm,and u ρ is the total expenditure on advertising of the ρ th firm. Consider marketdynamics d x ( s ) = [ x ′ ( s ) ax ( s ) − u ( s )] ds + x ( s ) σ ′ d B ( s ) , where x and u both are k -dimensional vectors such that x ρ ∈ X and u ρ ∈ U ρ ∈ U , a is a k × k -dimensional constant symmetric matrix, σ is an m -dimensional imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral constant vector and B is an m -dimensional Brownian motion process. For agiven Lagrangian multiplier λ ∗ assume g ( s, x ) = sλ ∗ [ x ′ ax − b ] . Therefore, f ( s, x , u ) = exp( − rs ) k X ρ =1 α ρ (cid:0) px ρ − cx ρ u ρ (cid:1) + (1 + s ) λ ∗ [ x ′ ax − b ]+ 2 sλ ∗ [ x ′ a ′ x − u ′ ] x ′ a + sλ ∗ σ x ′ ax σ ′ . Equation (17) implies, φ ∗ pρ ( s, x ) = sλ ∗ x ′ a ′ cα ρ x ρ exp( − rs ) , such that cα ρ x ρ exp( − rs ) = 0 . Example 5.
Consider a resource extraction problem of two players as discussedin the Section . . of Yeung and Petrosjan (2006). Suppose, there are two play-ers with objective function max u ,u E Z t exp( − rs ) (cid:26)(cid:20) ( k u ( s )) / − c u ( s ) x / ( s ) (cid:21) + α (cid:20) ( k u ( s )) / − c u ( s ) x / ( s ) (cid:21)(cid:27) ds, subject to d x ( s ) = h a x ( s ) − b x ( s ) − u ( s ) − u ( s ) i ds + σ x ′ ( s ) d B ( s ) . In the above problem u ρ ∈ U ρ ∈ U is the control strategy vector of player ρ for ρ ∈ { , } , a and b are positive constant scalar, σ is an m -dimensionalconstant, α ∈ [0 , ∞ ) is the optimal cooperative weight corresponding to player and B ( s ) is am m -dimensional Brownian motion. Here [ k ρ u ρ ( s )] is player ρ ’s level of satisfaction from the consumption of the resource extracted at time s and c − ρu ρ ( s ) x − ( s ) is the dissatisfaction level brought about by the costextraction. Finally, k , k , c , c are positive constant scalars.(i) Quantum approach: For a given fixed Lagrange multiplier λ ∗ and a positiveconstant scalar d assume g ( s, x ) = sλ ∗ (cid:2) a x ( s ) − b x ( s ) − d (cid:3) , where d takescare of the variability coming from P kq =1 u ∗ q ( s ) + u ρ ( s ) . Hence, ∂∂s g ( s, x ) = λ ∗ (cid:2) a x ( s ) − b x ( s ) − d (cid:3) , D x g ( s, x ) = sλ ∗ h a x − − b i and H x g ( s, x )= − sλ ∗ a x − . Therefore, f ( s, x , u , u ) = exp( − rs ) (cid:26)(cid:20) ( k u ) − c u x (cid:21) + α (cid:20) ( k u ) − c u x ( s ) (cid:21)(cid:27) + (1 + s ) λ ∗ h a x − b x − d i + sλ ∗ h a x ′ − b x ′ − u − u i (cid:16) a x − − b (cid:17) − sλ ∗ a x σ ′ x − σ x ′ . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Equation (17) gives us the cooperative Pareto optimal strategy of two players as φ ∗ p ( s, x ) = k " exp( − rs ) c x − exp( − rs ) + sλ ∗ ( a x − b ) ,φ ∗ p ( s, x ) = k " α exp( − rs ) α c x − exp( − rs ) + sλ ∗ ( a x − b ) , where c x − exp( − rs ) + sλ ∗ ( a x − b ) = 0 and α c x − exp( − rs ) + sλ ∗ ( a x − b ) = 0 .(ii) Pontryagin maximum principle: From Example . . in Yeung and Petrosjan(2006) we get cooperative Pareto optimal strategies of two players as, φ ∗ p ( s, x ) = k x h c + exp( − rs ) x D x W α ( s, x ) i ,φ ∗ p ( s, x ) = k x h c + α exp( − rs ) x D x W α ( s, x ) i , where for s ∈ [0 , t ] the value function is W α ( s, x ) = exp( − rs ) h A α ( s ) x + B α ( s ) i , such that, A α ( s ) and B α ( s ) satisfy: ∂∂s A α ( s ) = (cid:2) r + σ ′ σ + b (cid:3) A α ( s ) − k h c + A α ( s ) i − α k (cid:20) c + α A α ( s ) (cid:21) and, ∂∂s B α ( s ) = rB α ( s ) − aA α ( s ) . Finally, we find optimal strategy of the ρ th firm using a non-cooperativefeedback Nash equilibrium. We assume that a firm is rational in decision makingand earns more profit at the cost of the profit of the other firms in the market.Hence, Firm ρ seeks to maximizeΠ N ( u, t ) = E Z t π ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] ds with respect to the strategy u ρ where u ∗− ρ ( s ) is the optimized strategies for firmsother than the ρ th firm. Proposition 4.
A non-cooperative Nash optimal solution for maximizing Π N ( u, t ) subject to d x ( s ) = µ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] ds + σ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] d B ( s ) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral with initial condition x (0) = x is the solution of − ∂f ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] ∂u ρ Ψ s ( x ) = 0 , (18) where Ψ s is the transition wave function defined as Ψ s ( x ) = exp[ − f ( s, x , u ρ ( s ) , u ∗− ρ ( s ))]Ψ ( x ) with initial condition Ψ ( x ) and f ρ [ s, x , u ρ ( s ) , u ∗− ρ ( s )] = π ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )]+ g ρ [ s, x ( s )] + ∂∂s g ρ [ s, x ( s )]+ µ ′ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] D x g ρ [ s, x ( s )]+ σ ′ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] H x g ρ [ s, x ( s )] × σ ′ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] . Example 6.
Consider an economy endowed with a renewable resource with k ≥ firms such as in section . in Yeung and Petrosjan (2006). We can com-pare our Nash equilibrium strategy through quantum approach with traditionalPontryagin maximum principle in Yeung and Petrosjan (2006). Suppose, ρ th firm’s resource extraction in time s ∈ [0 , t ] is u ρ ( s ) for all ρ = { , , ..., k } . De-fine u ∗− ρ = P kq =1 u ∗ q ( s ) where ρ = q and k -dimensional vector x ( s ) is the sizeof the resource stock at time s such that x ( s ) > . Under this construction ρ th firm’s objective function is E Z t exp( − rs ) k X q =1 u ∗ q ( s ) + u ρ ! − u ρ ( s ) − c x ( s ) u ρ ( s ) ds , subject to the resource dynamics d x ( s ) = " a x ( s ) − b x ( s ) − k X q =1 u ∗ q ( s ) − u ρ ( s ) ds + σ x ′ ( s ) d B ( s ) , where cu ρ ( s ) / [ x ( s )] is ρ th firm’s cost of resource extraction at time s , σ is am m -dimensional constant diffusion vector component and, vector B ( s ) is an m -dimensional Brownian motion. In this model assume a, b and c are the scalars.For a given fixed Lagrange multiplier λ ∗ assume g ρ ( s, x ) = sλ ∗ h a x ( s ) − b x ( s ) − d i ,where d takes care of the variability coming from P kq =1 u ∗ q ( s ) + u ρ ( s ) . Hence, ∂∂s g ρ ( s, x ) = λ ∗ h a x ( s ) − b x ( s ) − d i , D x g ρ ( s, x ) imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral = sλ ∗ h a x − − b i and H x g ρ ( s, x ) = − sλ ∗ a x − . Therefore, f ρ ( s, x , u ρ , u ∗− ρ ) = exp( − rs ) k X q =1 u ∗ q + u ρ ! − u ρ − c x u ρ + (1 + s ) λ ∗ h a x − b x − d i + sλ ∗ " a x ′ − b x ′ − k X q =1 u ∗ q − u ρ a x − − b (cid:17) − sλ ∗ a x σ ′ x − σ x ′ . Finally, Equation (18) implies the feedback Nash Equilibrium as φ ρ ∗ NQ ( s, x ) = 2 k X q =1 u ∗ q ! c x + sλ ∗ exp( rs ) (cid:16) a x − − b (cid:17) − k X q =1 u ∗ q ! − . From section . of Yeung and Petrosjan (2006) we know, the feedback Nashequilibrium from Pontryagin maximum principle is φ ρ ∗ NP ( s, x ) = x (2 k − hP kq =1 (cid:16) c + exp( rs ) D x V q x (cid:17)i × ( k X q =1 " c + D x V q x exp( − rs ) − (cid:18) k − (cid:19) " c + D x V ρ x exp( − rs ) , where V ρ and V q are the value function of firms ρ and q with their gradients D x V ρ and D x V q respectively. By Corollary . . in Yeung and Petrosjan (2006)Hamiltonian-Jacobi-Bellman system has a solution V ρ ( s, x ) = exp( − rs ) h A ( s ) x + B ( s ) i , where A ( s ) and B ( s ) satisfies, ∂∂s A ( s ) = (cid:2) r + σ ′ σ − b (cid:3) A ( s ) − k − k (cid:2) c + A ( s ) (cid:3) − + c (2 k − k (cid:2) c + A ( s ) (cid:3) − + (2 k − A ( s )8 k (cid:2) c + A ( s ) (cid:3) , ∂∂s B ( s ) = rB ( s ) − aA ( s ) .
3. Proofs
The arguments here are based on the use of the quantum Lagrangian ac-tion function. Further details are given in the Appendix. Equation (1) implies imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral ∆ x ( s ) = x ( s + ds ) − x ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] dB ( s ). FollowingChow (1996) from Equation (40), the Euclidean action function is, A ,t ( x ) = Z t E s { π [ s, x ( s ) , u ( s )] ds + λ [∆ x ( s ) − µ [ s, x ( s ) , u ( s )] ds − σ [ s, x ( s ) , u ( s )] dB ( s )] } . Let ε >
0, and for a normalizing constant L ε > s,s + ε ( x ) = 1 L ε Z R exp[ − ε A s,s + ε ( x )]Ψ s ( x ) dx ( s ) , (19)where Ψ s ( x ) is the value of the transition function at time s and state x ( s ) withthe initial condition Ψ ( x ) = Ψ .Fubini’s Theorem implies that the action function on time interval [ s, s + ε ]is A s,s + ε ( x ) = E s Z s + εs π [ ν, x ( ν ) , u ( ν )] dν + g [ ν + ∆ ν, x ( ν ) + ∆ x ( ν )] . where g [ ν + ∆ ν, x ( ν ) + ∆ x ( ν )] = λ [∆ x ( ν ) − µ [ ν, x ( ν ) , u ( ν )] dν − σ [ ν, x ( ν ) , u ( ν )] dB ( ν ) + o (1) . This conditional expectation is valid when the strategy u ( ν ) is determined attime ν and the measure of firm’s share x ( ν ) is known (Chow, 1996). The evolu-tion of a process takes place as if the action function is stationary. Therefore, theconditional expectation with respect to time only depends on the expectationof initial time point of this time interval.Itˆo’s Lemma implies, ε A s,s + ε ( x ) = E s { επ [ s, x ( s ) , u ( s )] + εg [ s, x ( s )]+ ε ∂∂s g [ s, x ( s )] + εµ [ s, x ( s ) , u ( s )] ∂∂x g [ s, x ( s )]+ εσ [ s, x ( s ) , u ( s )] ∂∂x g [ s, x ( s )] dB ( s )+ εσ [ s, x ( s ) , u ( s )] ∂ ∂x g [ s, x ( s )] + o ( ε ) (cid:27) , and [∆ x ( s )] ∼ ε as ε →
0. Feynman (1948) uses an interpolation method tofind an approximation of the area under the path in [ s, s + ε ]. Using a similarapproximation, A s,s + ε ( x ) = π [ s, x ( s ) , u ( s )] + g [ s, x ( s )] + ∂∂s g [ s, x ( s )]+ µ [ s, x ( s ) , u ( s )] ∂∂x g [ s, x ( s )]+ σ [ s, x ( s ) , u ( s )] ∂ ∂x g [ s, x ( s )] + o (1) , (20) imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where E s [ dB ( s )] = 0 and E s [ o ( ε )] /ε → ε →
0. Combining Equations (19)and (20) yieldΨ s,s + ε ( x ) = 1 L ε Z R exp (cid:20) − ε (cid:26) π [ s, x ( s ) , u ( s )] + g [ s, x ( s )] + ∂∂s g [ s, x ( s )]+ µ [ s, x ( s ) , u ( s )] ∂∂x g [ s, x ( s )] + σ [ s, x ( s ) , u ( s )] ∂ ∂x g [ s, x ( s )] (cid:27)(cid:21) × Ψ s [ x ( s )] dx ( s ) + o ( ε / ) , (21)as ε →
0. Taking a first order Taylor series expansion on the left hand side ofEquation (21) yieldsΨ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε Z R exp (cid:26) − ε (cid:2) π [ s, x ( s ) , u ( s )]+ g [ s, x ( s )] + ∂∂s g [ s, x ( s )] + ∂∂x g [ s, x ( s )] µ [ s, x ( s ) , u ( s )]+ σ [ s, x ( s ) , u ( s )] ∂ ∂x g [ s, x ( s )] (cid:3)(cid:27) Ψ s [ x ( s )] dx ( s ) + o ( ε / ) . (22)For fixed s and τ let x ( s ) = x ( τ ) + ξ and assume that for some 0 < η < ∞ we have | ξ | ≤ q ηεx ( s ) so that 0 < x ( s ) ≤ ηε/ξ . Furthermore, as our stochasticisoperimetric non-holonomic constraint follows Theorem 1 along with Assump-tions 1, 2 in the Appendix, and dξ is a cylindrical measure where ξ contributessignificantly, Ψ τ [ x ( ξ )] of Equation (22) can be expanded using a Taylor seriesof ξ around 0. Therefore,Ψ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε Z R (cid:20) Ψ τs ( x ) + ξ ∂ Ψ τs ( x ) ∂x + o ( ε ) (cid:21) exp (cid:26) − ε (cid:2) π [ s, x ( τ ) + ξ, u ( s )]+ g [ s, x ( τ ) + ξ ] + ∂∂s g [ s, x ( τ ) + ξ ] + ∂∂x g [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ( s )]+ σ [ s, x ( τ ) + ξ, u ( s )] ∂ ∂x g [ s, x ( τ ) + ξ ] (cid:3)(cid:27) dξ + o ( ε / ) . Let f [ s, ξ, u ( s )] = π [ s, x ( τ ) + ξ, u ( s )] + g [ s, x ( τ ) + ξ ] + ∂∂s g [ s, x ( τ ) + ξ ]+ ∂∂x g [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ( s )]+ σ [ s, x ( τ ) + ξ, u ( s )] ∂ ∂x g [ s, x ( τ ) + ξ ] + o (1) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral so thatΨ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε ) = Ψ τs ( x ) 1 L ε Z R exp (cid:8) − εf [ s, ξ, u ( s )] } dξ + ∂ Ψ τs ( x ) ∂x L ε Z R ξ exp (cid:8) − εf [ s, ξ, u ( s )] (cid:9) dξ + o ( ε / ) . where f [ s, ξ, u ( s )] = f [ s, x ( τ ) , u ( s )] + ∂∂x f [ s, x ( τ ) , u ( s )][ ξ − x ( τ )]+ ∂ ∂x f [ s, x ( τ ) , u ( s )][ ξ − x ( τ )] + o ( ε ) , where ε → x → m = ξ − x ( τ ) so that dξ = dm , then standard integration techniquescan be used to show that Z R exp {− εf [ s, ξ, u ( s )] } dξ = exp {− εf [ s, x ( τ ) , u ( s )] }× Z R exp (cid:26) − ε (cid:20) ∂∂x f [ s, x ( τ ) , u ( s )] m + ∂ ∂x f [ s, x ( τ ) , u ( s )] m (cid:21)(cid:27) dm. ThenΨ τs ( x ) 1 L ε Z R exp (cid:8) − εf [ s, ξ, u ( s )] } dξ = Ψ τs ( x ) 1 L ε r πεa exp (cid:26) ε (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21)(cid:27) , where a = ∂ ∂x f [ s, x ( τ ) , u ( s )] and b = ∂∂x f [ s, x ( τ ) , u ( s )].Similarly, it can be shown that ∂ Ψ τs ( x ) ∂x L ε Z R ξ exp [ − εf [ s, ξ, u ( s )]] dξ = ∂ Ψ τs ( x ) ∂x L ε exp (cid:26) ε (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21)(cid:27)(cid:20) x ( τ ) − b a (cid:21)r πεa . ThereforeΨ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε ) = Ψ τs ( x ) 1 L ε r πεa exp (cid:26) ε (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21)(cid:27) + ∂ Ψ τs ( x ) ∂x L ε r πεa exp (cid:26) ε (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21)(cid:27)(cid:20) x ( τ ) − b a (cid:21) + o ( ε / ) . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Assuming L ε = p πεa > τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε ) = (cid:26) ε (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21)(cid:27) × (cid:26) Ψ τs ( x ) + (cid:20) x ( τ ) − b a (cid:21) ∂ Ψ τs ( x ) ∂x + o ( ε / ) (cid:27) . The term b/ (2 a ) is the ratio of the first derivative to the second derivativewith respect to x of f . As f is in a Schwartz space, the derivatives of f are rapidlyfalling and they satisfy Assumptions 1 and 2, and therefore it is reasonable to weassume, 0 < | b | ≤ ηε and 0 < | a | ≤ (1 − ξ − ) − . Hence, using x ( s ) − x ( τ ) = ξ we get, x ( τ ) − b a = x ( s ) − b a . and therefore (cid:12)(cid:12)(cid:12)(cid:12) x ( s ) − b a (cid:12)(cid:12)(cid:12)(cid:12) ≤ ηε. Therefore, letting ε →
0, the Wick rotated Schr¨odinger type equation is, ∂ Ψ τs ( x ) ∂s = (cid:20) b a − f [ s, x ( τ ) , u ( s )] (cid:21) Ψ τs ( x ) . (23)If we differentiate Equation (23) with respect to u , then the solution of the newequation will be a Walrasian optimal strategy in the stochastic case. That is, " ∂∂x f ( s, x, u ) ∂ ∂x f ( s, x, u ) ∂ ∂x f ( s, x, u ) ∂∂x∂u f ( s, x, u ) − ∂∂x f ( s, x, u ) ∂ ∂u∂x f ( s, x, u ) (cid:2) ∂ ∂x f ( s, x, u ) (cid:3) ! − ∂∂u f ( s, x, u ) (cid:21) Ψ τs ( x ) = 0 . (24)Therefore, an optimal Walrasian strategy is found by setting Equation (24)equal to zero obtains, ∂∂u f ( s, x, u ) (cid:20) ∂ ∂x f ( s, x, u ) (cid:21) = 2 ∂∂x f ( s, x, u ) ∂ ∂x∂u f ( s, x, u ) . A unique solution to Equation (23) can be found using a Fourier transformation,as Ψ s ( x ) = I ( x ) exp[ sv ( x, u )], which can be verified by direct differentiation. (cid:3) imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Euclidean action function is, A ,t ( K, V ) = Z t E s (cid:26) π (cid:20) s, H [ s, K ( s ) , V ( s )] , u ( s ) (cid:21) ds + λ (cid:2) K ( s + ds ) − K ( s ) − µ [ s, u ( s )] K ( s ) ds − σ [ s, u ( s )] K ( s ) dB ( s ) (cid:3) + λ (cid:2) V ( s + ds ) − V ( s ) − µ [ s, u ( s )] V ( s ) ds − σ [ s, u ( s )] V ( s ) dB ( s ) (cid:3)(cid:27) . Following arguments similar to those used to prove Proposition 1, define ∆ s = ε >
0, and for L ε > s,s + ε ( K, V ) = 1 L ε Z R exp (cid:26) − ε A s,s + ε ( K, V ) (cid:27) Ψ s ( K, V ) dK ( s ) × dV ( s ) , (25)as ε → s ( K, V ) is the wave function at time s and states K ( s ) and V ( s ) respectively with initial condition Ψ ( K, V ) = Ψ .The action function in [ s, τ ] where τ = s + ε with the Lagrangian is, A s,τ ( K, V ) = Z τs E s (cid:26) π (cid:20) ν, H [ ν, K ( ν ) , V ( ν )] , V ( ν ) , u ( ν ) (cid:21) dν + λ (cid:2) K ( ν + dν ) − K ( ν ) − µ [ ν, u ( ν )] K ( ν ) dν − σ [ ν, u ( ν )] K ( ν ) dB ( ν ) (cid:3) + λ (cid:2) V ( ν + dν ) − V ( ν ) − µ [ ν, u ( ν )] V ( ν ) dν − σ [ ν, u ( ν )] V ( ν ) dB ( ν ) (cid:3)(cid:27) , with initial conditions K (0) = K and V (0) = V , where λ and λ are twoLagrangian multipliers corresponding to the two constraints. The conditionalexpectation is valid when the strategy u ( ν ) is determined at time ν , and henceonly depends on the initial time point of this time interval. Let ∆ K ( ν ) = K ( ν + dν ) − K ( ν ) and, ∆ V ( ν ) = V ( ν + dν ) − V ( ν ), then Fubini’s Theorem implies, A s,τ ( K, V ) = Z τs E s (cid:26) π (cid:20) ν, H [ ν, K ( ν ) , V ( ν )] , V ( ν ) , u ( ν ) (cid:21) dν + λ (cid:2) ∆ K ( ν ) − µ [ ν, u ( ν )] K ( ν ) dν − σ [ ν, u ( ν )] K ( ν ) dB ( ν ) (cid:3) + λ (cid:2) ∆ V ( ν ) − µ [ ν, u ( ν )] V ( ν ) dν − σ [ ν, u ( ν )] V ( ν ) dB ( ν ) (cid:3)(cid:27) . (26)Because K ( ν ) and V ( ν ) are Itˆo processes, Theorem 4.1.2 of Øksendal (2003)implies that there exists a function g [ ν, K ( ν ) , V ( ν )] ∈ C ([0 , ∞ ) × R × R ) that satisfies Theorem 1 in the Appendix,Assumptions 1 and 2, such that Y ( ν ) = g [ ν, K ( ν ) , V ( ν )] where Y ( ν ) is an Itˆo imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral process. If we assume g [ ν + ∆ ν, K ( ν ) + ∆ K ( ν ) , V ( ν ) + ∆ V ( ν )]= λ (cid:2) ∆ K ( ν ) − µ [ ν, u ( ν )] K ( ν ) dν − σ [ ν, u ( ν )] K ( ν ) dB ( ν )+ λ (cid:2) ∆ V ( ν ) − µ [ ν, u ( ν )] V ( ν ) dν − σ [ ν, u ( ν )] V ( ν ) dB ( ν ) + o (1) , Equation (26) becomes, A s,τ ( K, V ) = E s (cid:26) Z τs π (cid:20) ν, H [ ν, K ( ν ) , V ( ν )] , V ( ν ) , u ( ν ) (cid:21) dν + g [ ν + ∆ ν, K ( ν ) + ∆ K ( ν ) , V ( ν ) + ∆ V ( ν )] (cid:27) . (27)Itˆo’s Lemma and Equation (27) of Baaquie (1997) imply A s,τ ( K, V ) = π (cid:20) s, H [ s, K ( s ) , V ( s )] , V ( s ) , u ( s ) (cid:21) + g [ s, K ( s ) , V ( s )]+ ∂∂s g [ s, K ( s ) , V ( s )] + ∂∂S g [ s, K ( s ) , V ( s )] µ [ s, u ( s )] K ( s )+ ∂∂V g [ s, K ( s ) , V ( s )] µ [ s, u ( s )] V ( s )+ (cid:20) σ [ s, u ( s )] K ( s ) ∂ ∂K g [ s, K ( s ) , V ( s )]+ 2 ρσ [ s, u ( s )] K ( s ) ∂ ∂K∂V g [ s, K ( s ) , V ( s )]+ σ [ s, u ( s )] V ( s ) ∂ ∂V g [ s, K ( s ) , V ( s )] (cid:21) + o (1) , where we have used the fact that [∆ K ( s )] = ∆ V ( s )] = ε , and E s [∆ B ( s )] = E s [∆ B ( s )], as ε → K and V . Using Equation (25),the transition wave function in [ s, τ ] becomes,Ψ s,τ ( K, V )= 1 L ε Z R exp (cid:26) − ε (cid:20) π (cid:20) s, H [ s, K ( s ) , V ( s )] , V ( s ) , u ( s ) (cid:21) + g [ s, K ( s ) , V ( s )]+ ∂∂s g [ s, K ( s ) , V ( s )] + ∂∂K g [ s, K ( s ) , V ( s )] µ [ s, u ( s )] K ( s )+ ∂∂V g [ s, S ( s ) , V ( s )] µ [ s, u ( s )] V ( s ) + (cid:20) σ [ s, u ( s )] K ( s ) ∂ ∂K g [ s, K ( s ) , V ( s )]+ 2 ρσ [ s, u ( s )] K ( s ) ∂ ∂K∂V g [ s, K ( s ) , V ( s )]+ σ [ s, u ( s )] V ( s ) ∂ ∂V g [ s, K ( s ) , V ( s )] (cid:21)(cid:21)(cid:27) Ψ s ( K, V ) dK ( s ) dV ( s ) + o ( ε / ) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral as ε → τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε )= 1 L ε Z R exp (cid:26) − ε (cid:20) π (cid:20) s, H [ s, K ( s ) , V ( s )] , V ( s ) , u ( s ) (cid:21) + g [ s, K ( s ) , V ( s )]+ ∂∂s g [ s, K ( s ) , V ( s )] + ∂∂K g [ s, K ( s ) , V ( s )] µ [ s, u ( s )] K ( s )+ ∂∂V g [ s, K ( s ) , V ( s )] µ [ s, u ( s )] V ( s ) + (cid:20) σ [ s, u ( s )] K ( s ) × ∂ ∂K g [ s, K ( s ) , V ( s )] + 2 ρσ [ s, u ( s )] K ( s ) ∂ ∂K∂V g [ s, K ( s ) , V ( s )]+ σ [ s, u ( s )] V ( s ) ∂ ∂V g [ s, K ( s ) , V ( s )] (cid:21)(cid:21)(cid:27) Ψ s ( K, V ) dK ( s ) dV ( s ) + o ( ε / ) , as ε → s and τ suppose that K ( s ) = K ( τ ) + ξ , and V ( s ) = V ( τ ) + ξ .For positive numbers η < ∞ and η < ∞ assume that | ξ | ≤ q η εK ( s ) and | ξ | ≤ q η εV ( s ) . Here, security and volatility are K ( s ) ≤ η ε/ξ and V ( s ) ≤ η ε/ξ ,respectively. Furthermore, Theorem 1 and Assumptions 1, 2 in the Appendiximply Ψ τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε )= 1 L ε Z R (cid:20) Ψ τs ( K, V ) + ξ ∂ Ψ τs ( K, V ) ∂K + ξ ∂ Ψ τs ( K, V ) ∂V + o ( ε ) (cid:21) exp (cid:26) − ε (cid:20) π (cid:20) s, H [ s, K ( τ ) + ξ , V ( τ ) + ξ ] , V ( τ ) + ξ , u ( s ) (cid:21) + g [ s, K ( τ ) + ξ , V ( τ ) + ξ ] + ∂∂s g [ s, K ( τ ) + ξ , V ( τ ) + ξ ]+ g K [ s, K ( τ ) + ξ , V ( τ ) + ξ ] µ [ s, u ( s )]( K ( τ ) + ξ )+ ∂∂V g [ s, K ( τ ) + ξ , V ( τ ) + ξ ] µ [ s, u ( s )]( V ( τ ) + ξ )+ (cid:20) σ [ s, u ( s )]( K ( τ ) + ξ ) ∂ ∂K g [ s, K ( τ ) + ξ , V ( τ ) + ξ ]+ 2 ρσ [ s, u ( s )]( K ( τ ) + ξ ) ∂ ∂K∂V g [ s, K ( τ ) + ξ , V ( τ ) + ξ ]+ σ [ s, u ( s )]( V ( τ ) + ξ ) ∂ ∂V g [ s, K ( τ ) + ξ , V ( τ ) + ξ ] (cid:21)(cid:21)(cid:27) Ψ τ [ K ( ξ ) , V ( ξ )] dξ dξ + o ( ε / ) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral as ε → f [ s, ξ , ξ , u ( s )] as in Equation (16), thenΨ τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε )= 1 L ε Ψ τs ( K, V ) Z R exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ + 1 L ε ∂ Ψ τs ( K, V ) ∂K Z R ξ exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ + 1 L ε ∂ Ψ τs ( K, V ) ∂V Z R ξ exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ + o ( ε / ) . Assume that f is a C function, then f [ s, ξ , ξ , u ( s )]= f [ s, K ( τ ) , V ( τ ) , u ( s )] + [ ξ − K ( τ )] ∂∂K f [ s, K ( τ ) , V ( τ ) , u ( s )]+ [ ξ − V ( τ )] ∂∂V f [ s, K ( τ ) , V ( τ ) , u ( s )]+ (cid:20) [ ξ − K ( τ )] ∂ ∂K f [ s, K ( τ ) , V ( τ ) , u ( s )]+ 2[ ξ − K ( τ )][ ξ − V ( τ )] ∂ ∂K∂V g [ s, K ( τ ) , V ( τ ) , u ( s )]+ [ ξ − V ( τ )] ∂ ∂V g [ s, K ( τ ) , V ( τ ) , u ( s )] (cid:21) + o ( ε ) , as ε → u →
0. Define m = ξ − K ( τ ) and m = ξ − V ( τ ) so that dξ = dm and dξ = dm respectively so that Z R exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ = Z R exp (cid:26) − ε (cid:20) f [ s, K ( τ ) , V ( τ ) , u ( s )] + m ∂∂K f [ s, K ( τ ) , V ( τ ) , u ( s )]+ m ∂∂V f [ s, K ( τ ) , V ( τ ) , u ( s )] + m ∂ ∂K g [ s, K ( τ ) , V ( τ ) , u ( s )]+ m m ∂ ∂K∂V f [ s, K ( τ ) , V ( τ ) , u ( s )]+ m ∂ ∂V f [ s, K ( τ ) , V ( τ ) , u ( s )] (cid:21)(cid:27) dm dm . (28)Let Θ = " ∂ ∂K f [ s, K ( τ ) , V ( τ ) , u ( s )] ∂ ∂K∂V g [ s, K ( τ ) , V ( τ ) , u ( s )] ∂ ∂K∂V f [ s, K ( τ ) , V ( τ ) , u ( s )] ∂ ∂V g [ s, K ( τ ) , V ( τ ) , u ( s )] , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral and m = (cid:20) m m (cid:21) , and − v = (cid:20) ∂∂K f [ s, K ( τ ) , V ( τ ) , u ( s )] ∂∂V f [ s, K ( τ ) , V ( τ ) , u ( s )] (cid:21) , where we assume that Θ is positive definite, then the integrand in Equation (28)becomes a shifted Gaussian integral, Z R exp (cid:26) − ε (cid:0) f − v T m + m T Θ m (cid:1) (cid:27) dm = exp ( − εf ) Z R exp (cid:26) ( εv T ) m − m T ( ε Θ) m (cid:27) dm = π p ε | Θ | exp h ε v T Θ − v − εf i , where v T and m T are the transposes of vectors v and m respectively. Therefore,1 L ε Ψ τs ( K, V ) Z R exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ = 1 L ε Ψ τs ( K, V ) π p ε | Θ | exp h ε v T Θ − v − εf i , (29)such that inverse matrix Θ − > L ε ∂ Ψ τs ( K, V ) ∂K Z R ξ exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ dξ = 1 L ε ∂ Ψ τs ( K, V ) ∂K π p ε | Θ | (cid:0) Θ − + K (cid:1) exp h ε v T Θ − v − εf i , (30)and 1 L ε ∂ Ψ τs ( K, V ) ∂V Z R ξ exp (cid:8) − εf [ s, ξ , ξ , u ( s )] (cid:9) dξ × dξ = 1 L ε ∂ Ψ τs ( K, V ) ∂V π p ε | Θ | (cid:0) Θ − + V (cid:1) exp h ε v T Θ − v − εf i . (31)Equations (29), (30) and (31) imply that the Wick rotated Schr¨odinger typeequation is,Ψ τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε )= 1 L ε π p ε | Θ | exp h ε v T Θ − v − εf i (cid:20) Ψ τs ( K, V ) + (cid:0) Θ − + K (cid:1) ∂ Ψ τs ( K, V ) ∂K + (cid:0) Θ − + V (cid:1) ∂ Ψ τs ( K, V ) ∂V (cid:21) + o ( ε / ) , as ε → imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Assuming L ε = π/ p ε | Θ | > τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε )= (cid:20) ε (cid:18) v T Θ − v − f (cid:19)(cid:21) (cid:20) Ψ τs ( K, V ) + (cid:0) Θ − + K (cid:1) ∂ Ψ τs ( K, V ) ∂K + (cid:0) Θ − + V (cid:1) ∂ Ψ τs ( K, V ) ∂V (cid:21) + o ( ε / ) . As K ( s ) ≤ η ε/ξ , assume | Θ − | ≤ η ε (1 − ξ − ) such that | (2Θ) − + K | ≤ η ε . For V ( s ) ≤ η ε/ξ we assume | Θ − | ≤ η ε (1 − ξ − ) such that | (2Θ) − + V | ≤ η ε . Therefore, | Θ − | ≤ ε min (cid:8) η (1 − ξ − ) , η (1 − ξ − ) (cid:9) suchthat, | (2Θ) − + K | → | (2Θ) − + V | →
0. HenceΨ τs ( K, V ) + ε ∂ Ψ τs ( K, V ) ∂s + o ( ε ) = (1 − εf )Ψ τs ( K, V ) + o ( ε / ) . Therefore, the Wick rotated Schr¨odinger type Equation is, ∂ Ψ τs ( K, V ) ∂s = − f [ s, ξ , ξ , u ( s )] Ψ τs ( K, V ) . Therefore, the solution of − ∂f [ s, ξ , ξ , u ( s )] ∂u Ψ τs ( K, V ) = 0 , (32)is a Walrasian optimal strategy, which has the formΨ s ( K, V ) = exp {− sf [ s, ξ , ξ , u ( s )] } I ( K, V ) . As the transition function Ψ τs ( K, V ) is the solution to Equation (32), the resultfollows. (cid:3)
The Euclidean action function for firm ρ under Pareto optimality in real time[0 , t ] is, A ,t ( x ) = Z t E s (cid:26) k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] ds + λ (cid:2) x ( s + ds ) − x ( s ) − µ [ s, x ( s ) , u ( s )] ds − σ [ s, x ( s ) , u ( s )] dB ( s ) (cid:3)(cid:27) . Following the arguments for the proof of Proposition 1, we have A s,τ ( x ) = E s (cid:26) Z τs k X ρ =1 α ρ π ρ [ ν, x ( ν ) , u ( ν )] dν + λ ( ν + dν ) (cid:2) ∆ x ( ν ) − µ [ ν, x ( ν ) , u ( ν )] dν − σ [ ν, x ( ν ) , u ( ν )] dB ( ν ) (cid:3)(cid:27) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral where τ = s + ε .As x ( ν ) is an Itˆo process then from Theorem 4.1.2 of Øksendal (2003) we knowthere exists a p -dimensional vector valued function g [ ν, x ( ν )] ∈ C ([0 , ∞ ) × R n )that satisfies Theorem 1 in the Appendix, Assumptions 1 and 2, and Y ( ν ) = g [ ν, x ( ν )] where Y ( ν ) is an Itˆo process. Assume g [ ν + ∆ ν, x ( ν ) + ∆ x ( ν )] = λ (cid:2) ∆ x ( ν ) − µ [ ν, x ( ν ) , u ( ν )] dν − σ [ ν, x ( ν ) , u ( ν )] dB ( ν ) + o (1) , as ε →
0, then the generalized Itˆo’s Lemma implies, A s,τ ( x ) ε = E s (cid:26) k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] ε + g [ s, x ( s )] ε + ∂∂s g [ s, x ( s )] ε + nk X i =1 ∂∂x i g [ s, x ( s )] µ [ s, x ( s ) , u ( s )] ε + nk X i =1 ∂∂x i g [ s, x ( s )] σ [ s, x ( s ) , u ( s )] ε ∆ B ( s )+ nk X i =1 nk X j =1 σ ij [ s, x ( s ) , u ( s )] ∂ ∂x i x j g [ s, x ( s )] ε + o ( ε ) (cid:27) , where σ ij [ s, x ( s ) , u ( s )] represents { i, j } th component of the variance-covariancematrix, and we used the conditions ∆ B i ∆ B j = δ ij ε , ∆ B i ε = ε ∆ B i = 0, and∆ x i ( s )∆ x j ( s ) = ε , where δ ij is the Kronecker delta function. Hence A s,τ ( x ) = (cid:20) k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] + g [ s, x ( s )] + ∂∂s g [ s, x ( s )]+ nk X i =1 ∂∂x i g [ s, x ( s )] µ [ s, x ( s ) , u ( s )]+ nk X i =1 nk X j =1 σ ij [ s, x ( s ) , u ( s )] ∂ ∂x i x j g [ s, x ( s )] + o (1) (cid:21) , where E s [∆ B ( s )] = 0 and E s [ o ( ε )] /ε → ε → imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral conditions x nk × . Expanding Ψ s,τ ( x ) yields,Ψ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε Z R n × k exp (cid:26) − ε (cid:20) k X ρ =1 α ρ π ρ [ s, x ( s ) , u ( s )] + g [ s, x ( s )]+ ∂∂s g [ s, x ( s )] + nk X i =1 ∂∂x i g [ s, x ( s )] µ [ s, x ( s ) , u ( s )]+ nk X i =1 nk X j =1 σ ij [ s, x ( s ) , u ( s )] ∂ ∂x i x j g [ s, x ( s )] (cid:21)(cid:27) Ψ s ( x ) dx ( s ) + o ( ε / ) . Let x ( s ) nk × = x ( τ ) nk × + ξ nk × and assume || ξ || ≤ ηε [ x T ( s )] − for some η > τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε Z R n × k (cid:20) Ψ τs ( x ) + ξ ∂ Ψ τs ( x ) ∂x + o ( ε ) (cid:21) exp (cid:26) − ε (cid:20) k X ρ =1 α ρ π ρ [ s, x ( τ ) + ξ, u ( s )]+ g [ s, x ( τ ) + ξ ] + ∂∂s g [ s, x ( τ ) + ξ ] + nk X i =1 ∂∂x i g [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ( s )]+ nk X i =1 nk X j =1 σ ij [ s, x ( τ ) + ξ, u ( s )] ∂ ∂x i ∂x j g [ s, x ( τ ) + ξ ] (cid:21)(cid:27) dξ + o ( ε / ) . Let f [ s, ξ, u ( s )] = k X ρ =1 α ρ π ρ [ s, x ( τ ) + ξ, u ( s )] + g [ s, x ( τ ) + ξ ]+ ∂∂s g [ s, x ( τ ) + ξ ] + nk X i =1 ∂∂x i g [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ( s )]+ nk X i =1 nk X j =1 σ ij [ s, x ( τ ) + ξ, u ( s )] ∂ ∂x i ∂x j g [ s, x ( τ ) + ξ ] , then Ψ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε Ψ τs ( x ) Z R n × k exp (cid:26) − εf [ s, ξ, u ( s )] (cid:27) dξ + 1 L ε ∂ Ψ τs ( x ) ∂x Z R n × k ξ exp (cid:26) − εf [ s, ξ, u ( s )] (cid:27) dξ + o ( ε / ) . (33) imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Expanding f [ s, ξ, u ( s )] and defining m nk × = ξ nk × − x ( τ ) nk × so that dξ = dm , first integral on the right hand side of Equation (33) becomes, Z R n × k exp (cid:26) − εf [ s, ξ, u ( s )] (cid:27) dξ = exp (cid:26) − εf [ s, x ( τ ) , u ( s )] (cid:27) Z R n × k exp (cid:26) − ε (cid:20) nk X i =1 ∂∂x i f [ s, x ( τ ) , u ( s )] m i + nk X i =1 nk X j =1 ∂ ∂x i ∂x j f [ s, x ( τ ) , u ( s )] m i m j (cid:21)(cid:27) dm + o ( ε ) . Assume there exists a symmetric, positive definite and non-singular Hessianmatrix θ nk × nk and a vector w nk × such that, Z R n × k exp (cid:26) − εf [ s, ξ, u ( s )] (cid:27) dξ = s (2 π ) nk ε | θ | exp (cid:26) − εf [ s, x ( τ ) , u ( s )] + ε w T θ − w (cid:27) , where, θ = ∂ ∂x ∂x f ∂ ∂x ∂x f . . . ∂ ∂x ∂x nk f ∂ ∂x ∂x f ∂ ∂x ∂x f . . . ∂ ∂x ∂x nk f ... ... . . . ... ∂ ∂x nk ∂x f ∂ ∂x nk ∂x f . . . ∂ ∂x nk ∂x nk f , and w [ s, x ( τ ) , u ( s )] = − ∂∂x f [ s, x ( τ ) , u ( s )] − ∂∂x f [ s, x ( τ ) , u ( s )]... − ∂∂x nk f [ s, x ( τ ) , u ( s )] . The second integral on the right hand side of Equation (33) becomes, Z R n × k ξ exp (cid:26) − εf [ s, ξ, u ( s )] (cid:27) dξ = s (2 π ) nk ε | θ | exp (cid:26) − εf [ s, x ( τ ) , u ( s )] + ε w T θ − w (cid:27)(cid:20) x ( τ ) + ( θ − w ) (cid:21) . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral So that Ψ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= 1 L ε s (2 π ) nk ε | θ | exp (cid:26) − εf [ s, x ( τ ) , u ( s )] + εw T θ − w (cid:27) × (cid:20) Ψ τs ( x ) + (cid:2) x ( τ ) + ( θ − w ) (cid:3) ∂ Ψ τs ( x ) ∂x (cid:21) + o ( ε / ) . Assume L ε = p (2 π ) nk / ( ε | θ | ) >
0, thenΨ τs ( x ) + ε ∂ Ψ τs ( x ) ∂s + o ( ε )= (cid:8) − εf [ s, x ( τ ) , u ( s )] + εw T θ − w (cid:9) × (cid:20) Ψ τs ( x ) + (cid:2) x ( τ ) + ( θ − w ) (cid:3) ∂ Ψ τs ( x ) ∂x (cid:21) + o ( ε / ) . For any finite positive number η we know x ( τ ) ≤ ηε | ξ T | − , and there exists | θ − w | ≤ ηε | − ξ T | − such that for ε → (cid:12)(cid:12) x ( τ ) + ( θ − w ) (cid:12)(cid:12) ≤ ηε andhence ∂ Ψ τs ( x ) ∂s = (cid:8) − f [ s, x ( τ ) , u ( s )] + w T θ − w (cid:9) Ψ τs ( x ) . Taking ε →
0, the Wick rotated Schr¨odinger type equation is ∂ Ψ τs ( x ) ∂s = − f [ s, x ( τ ) , u ( s )] Ψ τs ( x ) , with the Wheeler-Di Witt type equation, − ∂f [ s, x ( τ ) , u ( s )] ∂u ρ Ψ τs ( x ) = 0 , whose solution with respect to u ρ gives ρ th firm’s cooperative Pareto Optimalstrategy φ ρ ∗ p [ s, x ∗ ( s )]. (cid:3) Following Chow (1996) the Euclidean action function of firm ρ in [0 , t ] is, A ρ ,t ( x ) = Z t E s (cid:26) π ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] ds + λ ρ (cid:2) x ( s + ds ) − x ( s ) − µ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] ds − σ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] dB ( s ) (cid:3)(cid:27) . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Let ∆ s = ε >
0, and for L ε > ρ isΨ ρs,s + ε ( x ) = 1 L ε Z R n exp (cid:26) − ε A ρs,s + ε ( x ) (cid:27) Ψ ρs ( x ) dx ( s ) , (34)for a time interval [ s, s + ε ] where ε → ρs ( x ) is the value of firm ρ ’stransition function at time s and states x ( s ) with initial conditions Ψ ρ ( x ) = Ψ ρ .In Equation (34), R n represents n -dimensional strategy space of firm ρ . Let∆ x ( ν ) = x ( ν + dν ) − x ( ν ) then the Euclidean action function of firm ρ is, A ρs,τ ( x ) = E s (cid:26) Z τs π i [ ν, x ( ν ) , u ρ ( ν ) , u ∗− ρ ( ν )] dν + λ i (cid:2) ∆ x ( ν ) − µ [ ν, x ( ν ) , u ( ν )] dν − σ [ ν, x ( ν ) , u ρ ( ν ) , u ∗− ρ ( ν )] dB ( ν ) (cid:3)(cid:27) . (35)By Theorem 4.1.2 of Øksendal (2003) we know there exists a p -dimensionalvector valued function g ρ [ ν, x ( ν )] ∈ C ([0 , ∞ ) × R n ) that satisfies Theorem 1in the Appendix, Assumptions 1 and 2, and Y ρ ( ν ) = g ρ [ ν, x ( ν )] where Y ρ ( ν ) isfirm ρ ’s Itˆo process. If we assume g ρ [ ν + ∆ ν, x ( ν ) + ∆ x ( ν )] = λ ρ (cid:2) ∆ x ( ν ) − µ [ ν, x ( ν ) , u ρ ( ν ) , u ∗− ρ ( ν )] dν − σ [ ν, x ( ν ) , u ρ ( ν ) , u ∗− ρ ( ν )] dB ( ν ) + o (1) , Equation (35) becomes, A ρs,τ ( x ) = E s (cid:26) Z τs π ρ [ ν, x ( ν ) , u ρ ( ν ) , u ∗− ρ ( ν )] dν + g ρ [ ν + ∆ ν, x ( ν ) + ∆ x ( ν )] (cid:27) . Generalized Itˆo’s Lemma implies A ρs,τ ( x ) = (cid:20) π ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] + g ρ [ s, x ( s )]+ ∂∂s g ρ [ s, x ( s )] + n X i =1 ∂∂x i g ρ [ s, x ( s )] µ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )]+ n X i =1 n X j =1 σ ij (cid:2) s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s ) (cid:3) ∂ ∂x i ∂x j g ρ [ s, x ( s )] + o (1) (cid:21) , where E s [∆ B ( s )] = 0 and E s [ o ( ε )] /ε → ε → x ρ , where σ ij [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] represents { i, j } th component of the variance-covariancematrix, and ∆ B i ∆ B j = δ ij ε , ∆ B i ε = ε ∆ B i = 0, and ∆ x i ( s )∆ x j ( s ) = ε . A Tay- imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral lor series expansion of the vector valued transition function Ψ ρs,τ impliesΨ τ,ρs ( x ) + ε ∂ Ψ τ,ρs ( x ) ∂s + o ( ε )= 1 L ε Z R n exp (cid:26) − ε (cid:20) π ρ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )]+ g ρ [ s, x ( s )] µ [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )]+ n X i =1 n X j =1 σ ij [ s, x ( s ) , u ρ ( s ) , u ∗− ρ ( s )] × ∂ ∂x i ∂x j g ρ [ s, x ( s )] (cid:21)(cid:27) Ψ ρs ( x ) dx ( s ) + o ( ε / ) , as ε →
0. Let x ( s ) n × = x ( τ ) n × + ξ n × . There exists a positive number η < ∞ such that, || ξ || ≤ ηε [ x T ( s )] − , and [ x T ( s )] − exists and not equal to zero.Following our previous argumentsΨ τ,ρs ( x ) + ε ∂ Ψ τ,ρs ( x ) ∂s + o ( ε )= 1 L ε Z R n (cid:20) Ψ τ,ρs ( x ) + ξ ∂ Ψ τ,ρs ( x ) ∂x + o ( ε ) (cid:21) × exp (cid:26) − ε (cid:20) π ρ [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )] + g ρ [ s, x ( τ ) + ξ ]+ ∂∂s g ρ [ s, x ( τ ) + ξ ] + n X i =1 ∂∂x i g ρ [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )]+ n X i =1 n X j =1 σ ij [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )] × ∂ ∂x i ∂x j g ρ [ s, x ( τ ) + ξ ] (cid:21)(cid:27) dξ + o ( ε / ) . (36)Let f ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] = π ρ [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )] + g ρ [ s, x ( τ ) + ξ ]+ ∂∂s g ρ [ s, x ( τ ) + ξ ] + n X i =1 ∂∂x i g ρ [ s, x ( τ ) + ξ ] µ [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )]+ n X i =1 n X j =1 σ ij [ s, x ( τ ) + ξ, u ρ ( s ) , u ∗− ρ ( s )] ∂ ∂x i ∂x j g ρ [ s, x ( τ ) + ξ ] . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Equation (36) thenΨ τ,ρs ( x ) + ε ∂ Ψ τ,ρs ( x ) ∂s + o ( ε )= 1 L ε Ψ τ,ρs ( x ) Z R n exp (cid:26) − εf ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] (cid:27) dξ + 1 L ε ∂ Ψ τ,ρs ( x ) ∂x Z R n ξ exp (cid:26) − εf ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] (cid:27) dξ + o ( ε / ) , where f ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] = f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )]+ n X i =1 ∂∂x i f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )][ ξ i − x i ( τ )]+ n X i =1 n X j =1 ∂ ∂x i ∂x j f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )][ ξ i − x i ( τ )][ ξ j − x j ( τ )] + o ( ε ) , as ε →
0. Define m n × = ξ n × − x ( τ ) n × so that dξ = dm , then Z R n exp (cid:26) − εf ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] (cid:27) dξ = s (2 π ) n ε | ˜ θ | exp (cid:26) − εf ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] + ε b w T ˜ θ − b w (cid:27) , where we assume ˜ θ n × n the symmetric, positive definite and non-singular Hessianmatrix ˜ θ = ∂ ∂x ∂x f ρ ∂ ∂x ∂x f ρ . . . ∂ ∂x ∂x n f ρ∂ ∂x ∂x f ρ ∂ ∂x ∂x f ρ . . . ∂ ∂x ∂x n f ρ ... ... . . . ... ∂ ∂x n ∂x f ρ ∂ ∂x n ∂x f ρ . . . ∂ ∂x n ∂x n f ρ , and b w [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] = − ∂∂x f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] − ∂∂x f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )]... − ∂∂x n f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] . Similarly, Z R n ξ exp (cid:26) − εf ρ [ s, ξ, u ρ ( s ) , u ∗− ρ ( s )] (cid:27) dξ = s (2 π ) n ε | ˜ θ | exp (cid:26) − εf ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )]+ ε b w T ˜ θ − b w (cid:27)(cid:20) x ( τ )+ ( θ − b w ) (cid:21) , imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral and henceΨ τ,ρs ( x ) + ε ∂ Ψ τ,ρs ( x ) ∂s + o ( ε )= 1 L ε s (2 π ) n ε | ˜ θ | exp (cid:26) − εf ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] + ε b w T ˜ θ − b w (cid:27) × (cid:20) Ψ τ,ρs ( x ) + h x ( τ ) + (˜ θ − b w ) i ∂ Ψ τ,ρs ( x ) ∂x (cid:21) + o ( ε / ) . Assuming L ε = q (2 π ) n / ( ε | ˜ θ | ) >
0, the Wick rotated Schr¨odinger type equationis,Ψ τ,ρs ( x ) + ε ∂ Ψ τ,ρs ( x ) ∂s + o ( ε ) = n − εf ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] + ε b w T ˜ θ − b w o × (cid:20) Ψ τ,ρs ( x ) + h x ( τ ) + (˜ θ − b w ) i ∂ Ψ τ,ρs ( x ) ∂x (cid:21) + o ( ε / ) . For any finite positive number η we know x ( τ ) ≤ ηε | ξ T | − , and there exists a | θ − w | ≤ ηε | − ξ T | − such that for ε → (cid:12)(cid:12) x ( τ ) + ( θ − w ) (cid:12)(cid:12) ≤ ηε .Hence, ∂ Ψ τ,ρs ( x ) ∂s = − f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] Ψ τ,ρs ( x ) , with the Wheeler-Di Witt type equation is, − ∂f ρ [ s, x ( τ ) , u ρ ( s ) , u ∗− ρ ( s )] ∂u ρ Ψ τ,ρs ( x ) = 0 , whose solution with respect to u ρ gives ρ th firm’s non-cooperative feedback Nashequilibrium strategy φ ρ ∗ N [ s, x ∗ ( s ) , u ∗ ρ ( s ) , u ∗− ρ ( s )]. (cid:3)
4. Discussion
In this paper we use a Feynman type path integral method to find optimalstrategies for dynamic profit functions quadratic in time with a stochastic dif-ferential market dynamics for infinite dimensional vector spaces (i.e. Walrasianequilibrium) and finite dimensional vector spaces (i.e. Pareto optimality andNash equilibrium). In Proposition 2 we show in the generalized non-linear caselike the Merton-Garman-Hamiltonian (Baaquie, 2007; Merton, 1973) Equationwe are able find an optimal strategy where traditional Pontryagin’s maximumprinciple does not work. Furthermore, in Example 1 where both the profit func-tion and market dynamics are linear to strategy we are still able to find optimalstrategy of a firm. Again in this case, we cannot use Pontryagin’s maximum imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral principle because after doing differentiation with respect to control, the strategyterm vanishes and optimal strategy cannot be found. According to the Gener-alized Weierstrass Theorem we know solution exists when both the objectivefunction and market dynamics are linear in terms of control (Intriligator, 1971).Under Proposition 4 we calculate a non-cooperative feedback Nash equilibriumand in the future we plan to extend this result to cooperative Nash equilibria. Appendix A: Appendix section
This appendix outlines the complete assumptions required to develop the results.Throughout this paper we are considering Euclidean quantum field theory whichrequires further assumptions on Equation (1). A quantum field is an operatedvalued distribution F [ s, x ( s )] to the unbounded operators on a Hilbert spacefollowing the Garding-Wightman axioms (Simon, 1979). Consider a measure dξ ≡ dx ( s ) on an Euclidean free field L ( R ) (The dimension is two for space-time under a Walrasian system), whose moments are the candidates of Schwingerfunctions. For a real valued tempered distribution T , let (cid:18)Z F ( y ) f ( y ) d y (cid:19) ( T ) = T ( f ) , be a random variable, where f is a real valued test function of y ∈ R takesSchwinger function S n ( y , ..., y n ) = Z F ( y ) ... F ( y n ) dx which implies Z S n ( y , ..., y n ) f ( y ) ... f n ( y n ) d n y = Z T ( f ) ... T ( f n ) dx ( T ) . Theorem 1. [Fr¨ohlich’s Reconstruction Theorem (Simon, 1979)] Let dξ be acylindrical measure on Euclidean free field L ( R ) obeying the following proper-ties:(i) The measure dξ is invariant with respect to proper Euclidean motions of theform T ( x ) T ( Ex + h ) , where h ∈ R and E ∈ SO (2) , where SO (2) repre-sents Lie Special orthogonal group of dimension .(ii) Osterwalder-Schrader positivity: For a given real valued test function f inGarding-Wightman field L ′ ( R ) or f ∈ L ′ ( R ) with support f ⊂ { ( s, x ) , s > } , let ( θf )( s, x ) = f ( − s, x ) , where θ is a parameter. Then for real valued f , f , ..., f n with the above support and for the set of complex numbers z , z , ..., z n ∈ C wehave n X j =1 n X k =1 z k z j Z exp { i [ F ( f k ) − F ( θf j )] } dξ ≥ . imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral (iii) For any real valued test function in Euclidean free field f ∈ L ( R ) , Z exp[ F ( f )] dξ < ∞ , and the action of the translations [ s, x ( s )] → [ s + ε, x ( s + ε )] is ergodic. Assume Equation (1) is in Euclidean free field and it satisfies three conditionsin above Theorem 1. Hence, the measure dξ is cylindrical and the feasible set ofEquation (1) satisfies dx ( s ) ≥ µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] dB ( s ) . (37)As G [ s, x ( s ) , u ( s )] = dx ( s ) − µ [ s, x ( s ) , u ( s )] ds − σ [ s, x ( s ) , u ( s )] dB ( s ), Equation(37) implies G [ s, x ( s ) , u ( s )] ≥
0. The dynamic Walrasian system then satisfiesmax u ∈ U Π( u, t ) = max u ∈ U E Z t π [ s, x ( s ) , u ( s )] ds, with constraint dx ( s ) = µ [ s, x ( s ) , u ( s )] ds + σ [ s, x ( s ) , u ( s )] dB ( s ), and initialcondition x (0) = x . Following Chow (1996) at time s ′ ∈ [0 , t ′ ], the stochasticLagrangian function is Z t ′ E s ′ (cid:26) π [ s ′ , x ( s ′ ) , u ( s ′ )] − λ ˜ G [ s ′ , x ( s ′ ) , u ( s ′ )] (cid:27) ds ′ , (38)where λ is the non-negative Lagrangian multiplier,˜ G [ s ′ , x ( s ′ ) , u ( s ′ )] ds ′ = G [ s ′ , x ( s ′ ) , u ( s ′ )] , and E s ′ is the conditional expectation on time s ′ , E s ′ ( . ) = E [ . | x ( s ′ )]. As weare interested in a forward looking process, at time s ′ only the informationup to s ′ is available, and based on this we forecast the state of the systemat time s ′ + ds ′ . Furthermore, in the path integral approach, Equation (38)corresponds to the Lagrangian function of the Feynman action functional inMinkowski space-time with imaginary time s ′ . In order to get a Euclideanpath integral we need to perform the Wick rotation on Equation (38). Sup-pose, there exists dynamic non-negative measurable profit function ˆ π such that π [ s ′ , x ( s ′ ) , u ( s ′ )] = d ˆ π [ s ′ , x ( s ′ ) , u ( s ′ )] / ( ds ′ ) . For imaginary time s ′ , the Feyn-man action functional becomes A F ,t ′ ( x ) = Z t ′ E s ′ (cid:26) π [ s ′ , x ( s ′ ) , u ( s ′ )] − λ ˜ G [ s ′ , x ( s ′ ) , u ( s ′ )] (cid:27) ds ′ . (39)Multiplying both sides of Equation (39) by i and substituting s ′ = − i s (so that ds ′ = − i ds ) yields, i A F ,t ′ ( x ) = i Z t E s (cid:26) (cid:18) d − i ds (cid:19) ˆ π [ s, x ( s ) , u ( s )] − λ ˜ G [ s, x ( s ) , u ( s )] (cid:27) ( − i ds ) (40) imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral so that A ,t ( x ) = Z t E s (cid:26) π [ s, x ( s ) , u ( s )] + λ ˜ G [ s, x ( s ) , u ( s )] (cid:27) ds. In Equation (40), A ,t ( x ) is defined as Euclidean action functional after theWick rotation. Theorem 1 and Condition (37) imply that if G [ s, x ( s ) , u ( s )] ≥ G [ s, x ( s ) , u ( s )] ≥ s is always non-negative. Now we assume some further conditions on G [ s, x ( s ) , u ( s )]. Assumption 1.
Suppose G [ s, x ( s ) , u ( s )] is a non-negative real valued contin-uous function of ( s, x, u ) ∈ [0 , t ] × X × R and infinitely differentiable with re-spect to x and u if s ∈ [0 , t ] is fixed and α th order derivatives ∂ αx G [ s, x, u ] and ∂ αu G [ s, x, u ] respectively are continuous functions of ( s, x, u ) for any α . More-over, for any integer m ≥ there exist positive constants v m and v m such that, (cid:12)(cid:12) ∂ αx G [ s, x, u ] (cid:12)(cid:12) ≤ v m , (cid:12)(cid:12) ∂ αu G [ s, x, u ] (cid:12)(cid:12) ≤ v m , if α is an integer with ≤ α ≤ m and ( s, x, u ) ∈ [0 , t ] × X × R . From Assumption 1 there exists positive constants v , v and v such that forall ( s, x, u ) ∈ [0 , t ] × X × R , (cid:12)(cid:12) G ( s, x, u ) (cid:12)(cid:12) ≤ v (1+ | x | ) , (cid:12)(cid:12) ∂ x G ( s, x, u ) (cid:12)(cid:12) ≤ v (1+ | x | )and, (cid:12)(cid:12) ∂ u G ( s, x, u ) (cid:12)(cid:12) ≤ v (1 + | x | ). Definition 4.
For all x ∈ [0 , define a Wick rotated wave integral I (Ψ) withEuclidean action function A ( x ) such that I (Ψ) = Z R exp {−A ( x ) } Ψ( x ) dx, where Ψ( x ) is a real valued wave function of x . The integration defined in Definition 4 may not converge absolutely, and weneed following definition (Fujiwara, 2017).
Definition 5.
For ε > consider a family of C ∞ , ω ε ( x ) which follows theproperties given in Definition 3.1 of Fujiwara (2017). The Wick rotated waveintegral is I (Ψ) = lim ε → Z R ω ε exp {−A ( x ) } Ψ( x ) dx, as long as(i) For any family of ω ε ( x ) the integral I ( ω ε ) converges absolutely and,(ii) The right hand side limit of Equation (2) exists and independent of choiceof { ω ε } . After using Proposition 3 . I (Ψ) in Definition 4 is absolutely convergent. imsart-generic ver. 2014/10/16 file: manuscript.tex date: February 24, 2020 . Pramanik and A. M. Polansky/Euclidean Path Integral Assumption 2.
Suppose, x ∈ X such that;(i) The Euclidean action A ( x ) is a C ∞ function. If | α | ≥ , then there exists apositive constant C α such that, (cid:12)(cid:12) ∂ αx A ( x ) (cid:12)(cid:12) ≤ C α . (ii) The wave function Ψ( x ) which depends on x is infinitely differentiable withrespect to x . There exists a constant ρ ≥ such that for any α sup x ∈ X (1 + | x | ) − ρ (cid:12)(cid:12) ∂ αx Ψ( x ) (cid:12)(cid:12) < ∞ . Lemma 2. [Convergence of Euclidean path integral (Fujiwara, 2017)] Considersmall real time interval [ s, s + ∆ s ] ⊂ [0 , t ] such that for some positive number δ > we have | ∆ s | ≤ δ and let ∆ : s = s < s < ... < s J < s J +1 = s + ∆ s be an arbitrary division of interval [ s, s + ∆ s ] . Suppose τ j = s j − s j − , | ∆ | =max ≤ j ≤ j +1 τ j and for x ∈ X define transition function Ψ ,t ( x ) = Z A exp (cid:2) − A ,t ( x ) (cid:3) D x , (41) where A is the space of all paths that connect x (0) to x ( t ) and D x is a uniformmeasure on the space A . Let us define a local transition function in the interval [ s, s + ∆] such that Ψ s,s +∆ s ( x ) := 1 L ε Z R exp (cid:26) − A s,s +∆ s ( x ) (cid:27) Ψ s ( x ) dx (42) which satisfies Definitions 4 and 5 with I (∆ , x, s, s + ∆ s ) := 1( L ε ) n Z R n exp (cid:26) n X j =1 −A (cid:2) x ( s j − , s j ) (cid:3)(cid:27) Ψ s ( x ) n Y j =1 dx ( s j ) , (43) where A s j − ,s j ( x ) is the Euclidean action function in [ s, s + ∆ s ] and it is theEuclidean action function of τ j . If Equations (41)-(43) satisfy Assumptions 1and 2 then the following limit exists Ψ ,t ( x ) = lim | ∆ |→ I (∆ , x, s, s + ∆ s ) . References
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