The Stochastic Balance Equation for the American Option Value Function and its Gradient
aa r X i v : . [ q -f i n . P R ] F e b The Stochastic Balance Equation for theAmerican Option Value Function and itsGradient
Malkhaz Shashiashvili
School of Mathematics and Computer Science, Kutaisi International University,33a, Ilia Chavchavadze Ave., Floor 4, Apartment 53, Tbilisi 0179, GeorgiaE-mail address: [email protected]
Abstract
In the paper we consider the problem of valuation and hedgingof American options written on dividend-paying assets whose pricedynamics follow the multidimensional diffusion model. We derive astochastic balance equation for the American option value functionand its gradient. We prove that the latter pair is the unique solutionof the stochastic balance equation as a result of the uniqueness in therelated adapted future-supremum problem.
Key words and phrases.
Snell envelope, optimal stopping, parabolicobstacle problem, adapted future-supremum problem.
In this paper we study American options written on dividend-paying assets.We assume that the underlying asset dynamics follow the multidimensionaldiffusion model. It is well known that the arbitrage-free value of Americanoption can be expressed in terms of the optimal stopping problem (Bensous-san [1], Karatzas [2]). We consider American options with convex Lipschitz1ontinuous payoff functions ψ ( x ), x = ( x , . . . , x n ) ∈ [0 , ∞ ) n . The class of op-tions with this kind of payoff functions includes index options, spread options,call on max options, put on min options, multiple strike options and others.We will assume that the time horizon T is finite T < ∞ , and under the risk-neutral probability measure P the underlying stock prices S t = ( S t , . . . , S nt ),0 ≤ t ≤ T evolve according to stochastic differential equation dS it = S it (cid:0) r − d i ( t, S t ) (cid:1) dt + S it (cid:16) n X j =1 σ ij ( t, S t ) dW jt (cid:17) , i = 1 , . . . , n, ≤ t ≤ T, with the initial condition S i = S i > , i = 1 , . . . , n. (1.1)Here W t = ( W t , . . . , W nt ), 0 ≤ t ≤ T , is a standard n -dimensional Wienerprocess on the probability space (Ω , F , F t , P ), where ( F t ) ≤ t ≤ T is a com-pletion of the natural filtration ( F Wt ) ≤ t ≤ T by all P -null sets A ∈ F with P ( A ) = 0.Further, r ≥ d i ( t, S ) is the dividendyield for the stock S i , i = 1 , . . . , n and σ ( t, S ) = ( σ ij ( t, S )), i, j = 1 , . . . , n isthe n -dimensional local volatility matrix.We assume that there exist continuous bounded functions b d i ( t, x ), i =1 , . . . , n , b σ ( t, x ) = ( b σ ij ( t, x )), i, j = 1 , . . . , n , ( t, x ) ∈ [0 , T ] × ( −∞ , ∞ ) n , suchthat b d i ( t, x ) ≥ , i = 1 , . . . , n, (cid:12)(cid:12) b d i ( t, x ) − b d i ( u, y ) (cid:12)(cid:12) ≤ c (cid:0) | t − u | / + | x − y | (cid:1) , i = 1 , . . . , n, (cid:12)(cid:12)b σ ij ( t, x ) − b σ ij ( u, y ) (cid:12)(cid:12) ≤ c (cid:0) | t − u | / + | x − y | (cid:1) , i, j = 1 , . . . , n, (1.2)with c ≥
0, and the matrix b a ( t, x ) = b σ ( t, x ) b σ ⊤ ( t, x ) is uniformly positivedefinite: n X i,j =1 b a ij ( t, x ) y i y j ≥ λ n X i =1 y i , λ > , ∀ x, y ∈ ( −∞ , ∞ ) n , (1.3)and the latter functions are related with the dividend yield and the localvolatility functions by the following change of variable relationship: for S =( S , . . . , S n ) with S i > i = 1 , . . . , n , we have d i ( t, S ) = b d i ( t, ln S ) , i = 1 , . . . , n,σ ij ( t, S ) = b σ ij ( t, ln S ) , i, j = 1 , . . . , n, where ln S = (ln S , . . . , ln S n ) . (1.4)2e will need to introduce several spaces of real valued functions. C = C ([0 , T ] × [0 , ∞ ) n ) the space of continuous real valued functions f = f ( t, S ), 0 ≤ t ≤ T , S = ( S , . . . , S n ) ∈ [0 , ∞ ) n .We say that f ∈ C is Lipschitz continuous in S ∈ [0 , ∞ ) n , uniformly in t , if there exists a constant L ≥
0, such that (cid:12)(cid:12) f ( t, S ) − f ( t, e S ) (cid:12)(cid:12) ≤ L | S − e S | , ≤ t ≤ T, S, e S ∈ [0 , ∞ ) n . (1.5)We denote L ∞ ((0 , T ) × (0 , ∞ ) n ) the space of functions ϕ = ϕ ( t, S ), suchthat | ϕ ( t, S ) | ≤ K a.e. in dt × dS for some K ≥ . (1.6)If f = f ( t, S ) ∈ C ([0 , T ] × [0 , ∞ ) n ) is continuously differentiable withrespect to S for each t , 0 ≤ t < T , then we write ∇ f ( t, S ) = (cid:16) ∂f ( t, S ) ∂S , . . . , ∂f ( t, S ) ∂S n (cid:17) for the gradient of f = f ( t, S ) . (1.7) ∇ f ∈ L ∞ means that ∂f∂S i ∈ L ∞ for every i = 1 , . . . , n .We introduce also the parabolic differential operator Lf ( t, S ) associatedwith the multidimensional stock prices process S t = ( S t , . . . , S nt ), 0 ≤ t ≤ T ,satisfying the stochastic differential equation (1.1) Lf ( t, S ) = 12 n X i,j =1 a ij ( t, S ) S i S j ∂ f∂S i ∂S j + n X i =1 (cid:0) r − d i ( t, S ) (cid:1) S i ∂f∂S i + ∂f∂t − rf, (1.8)where a ij ( t, S ) = b a ij ( t, ln S ) , i, j = 1 , . . . , n and also a ( t, S ) = σ ( t, S ) · σ ⊤ ( t, S ) . Consider now the solution of stochastic differential equation (1.1) S u ( t, S ) =( S u ( t, S ) , . . . , S nu ( t, S )), t ≤ u ≤ T , started at time t from the position S ∈ (0 , ∞ ) n , that is S t ( t, S ) = S and define the value function v ( t, S ), 0 ≤ t ≤ T , S ∈ (0 , ∞ ) n , of the following optimal stopping problem v ( t, S ) = sup t ≤ τ ≤ T E p (cid:0) e − r ( τ − t ) ψ ( S τ ( t, S )) (cid:1) , (1.9)3here the supremum is taken over all ( F u ), t ≤ u ≤ T stopping times τ , t ≤ τ ≤ T .As already mentioned above, it was discovered by Bensoussan [1]and Karatzas [2], that the arbitrage-free value of American options with thepayoff function ψ ( S ), S ∈ (0 , ∞ ) n , written on the underlying stock prices S t = ( S t , . . . , S nt ), 0 ≤ t ≤ T , coincides with the value function v ( t, S ) ofthe optimal stopping problem (1.9). For the rigorous mathematical proofof this and some other facts we will refer to the monograph by Pascucci [3,Chapters 8, 9 and 11].For arbitrary function ϕ ( t, S ) = ( ϕ ( t, S ) , . . . , ϕ n ( t, S )) ∈ L ∞ ((0 , T ) × (0 , ∞ ) n ) let us introduce the notation z ϕ ( θ, S θ ) = (cid:0) S θ · ϕ ( θ, S θ ) , . . . , S nθ · ϕ n ( θ, S θ ) (cid:1) , ≤ θ ≤ T,z ∇ v ( θ, S θ ) = (cid:16) S θ · ∂v∂S ( θ, S θ ) , . . . , S nθ · ∂v∂S n ( θ, S θ ) (cid:17) , ≤ θ ≤ T. (1.10)Throughout the paper we assume that the payoff function ψ ( S ) = ψ ( S , . . . , S n ) , S = ( S , . . . , S n ) ∈ (0 , ∞ ) n is the convex Lipschitz continuous function. Theorem 1.1 (The stochastic balance equation) . Suppose the conditions (1.2) – (1.4) are satisfied. Then the pair ( v ( t, S ) , ∇ v ( t, S )) is the unique solu-tion – pair ( f ( t, S ) , ϕ ( t, S )) , where f ( t, S ) ∈ C ([0 , T ] × [0 , ∞ ) n ) , ϕ ( t, S ) = ( ϕ ( t, S ) , . . . , ϕ n ( t, S )) ,ϕ i ( t, S ) ∈ L ∞ ((0 , T ) × (0 , ∞ ) n ) , i = 1 , . . . , n, of the following stochastic balance equation for the American option valuefunction and its gradient f ( t, S t ) = sup t ≤ u ≤ T (cid:20) e − r ( u − t ) ψ ( S u ) − u Z t e − r ( θ − t ) z ϕ ( θ, S θ ) σ ( θ, S θ ) dW θ (cid:21) , ≤ t ≤ T, (1.11) where the multidimensional stock prices process S t = ( S t , . . . , S nt ) , ≤ t ≤ T ,satisfies the stochastic differential equation (1.1) with the initial condition S = S , ∀ S ∈ (0 , ∞ ) n . emark. We would like to note that an inspiration to derive the stochasticbalance equation came to us after careful reading of the paper by Davisand Karatzas [4], in which, in the proof of Theorem 3 the so called future-supremum process appeared for the first time in the literature on optimalstopping problems.For the American option value function v ( t, S ) and its gradient ∇ v ( t, S )the stochastic balance equation takes the following form v ( t, S t ) = sup t ≤ u ≤ T (cid:20) e − r ( u − t ) ψ ( S u ) − u Z t e − r ( θ − t ) (cid:16) S θ ∂v∂S ( θ, S θ ) , . . . , S nθ ∂v∂S n ( θ, S θ ) (cid:17) σ ( θ, S θ ) dW θ (cid:21) , ≤ t ≤ T. (1.12)Notice that equation (1.12) does not contain partial derivative with respectto time ∂v∂t , nor the second order partial derivatives ∂ v∂S i ∂S j with respect tothe state argument S = ( S , . . . , S n ).The proof of Theorem 1.1 is based essentially on the introduction andanalyses of the new problem in stochastic analyses which we call the adaptedfuture-supremum problem. Let (Ω , F , P ) be a complete probability space with a filtration ( F t ) ≤ t ≤ T sat-isfying the usual conditions (that is, F contains all P -null sets and F t = F t + ,0 ≤ t ≤ T ). Consider a real-valued stochastic process X = ( X t ), 0 ≤ t ≤ T ,adapted to the given filtration ( F t ) ≤ t ≤ T with trajectories X t ( ω ), 0 ≤ t ≤ T ,which are right continuous and possess left limits. We shall assume that thefollowing basic condition is satisfied: there exists a nonnegative uniformly in-tegrable martingale U = ( U t ), 0 ≤ t ≤ T , adapted to the filtration ( F t ) ≤ t ≤ T ,such that | X t | ≤ U t , ≤ t ≤ T, (2.1)it is well known in this case (see Thompson [6])that there exists a smallestsupermartingale Y = ( Y t ), 0 ≤ t ≤ T , greater or equal then X , which is called5he Snell envelope of X and which possesses the following representation asthe value process of the optimal stopping problem Y t = ess sup τ t E ( X τ t | F t ) for fixed t, ≤ t ≤ T, P -a.s. , (2.2)where the essential supremum is taken over all ( F t ) ≤ t ≤ T -stopping times τ t with t ≤ τ t ≤ T . It is easy to see, that under condition (2.1)the supermartin-gale ( Y t , F t ), 0 ≤ t ≤ T , is of class ( D ) and therefore it has a Doob–Meyerdecomposition Y t = M t + B t , ≤ t ≤ T, (2.3)where M = ( M t , F t ), 0 ≤ t ≤ T , is a uniformly integrable martingale withthe initial value M = 0 and B = ( B t , F t ), 0 ≤ t ≤ T , is an integrablepredictable nonincreasing process and such a decomposition is unique.For the stochastic process X = ( X t , F t ), 0 ≤ t ≤ T , satisfying condition(2.1) let us introduce the so-called future-supremum process C t = sup t ≤ s ≤ T X s , ≤ t ≤ T. (2.4)It is evident that in general C t is not an ( F t ) ≤ t ≤ T -adapted process. In-deed, consider the case when the process X t , 0 ≤ t ≤ T is increasing. Thenwe have C t = X T , 0 ≤ t ≤ T , but X T is not ( F t ) ≤ t ≤ T -adapted. Next, let usassume that the process X t , 0 ≤ t ≤ T , is nonincreasing, then we obtain C t = X t , ≤ t ≤ T, and hence the process C t is ( F t ) ≤ t ≤ T -adapted.Now the question arises: can we adjust the process X t , 0 ≤ t ≤ T , by azero mean martingale M = ( M t , F t ), 0 ≤ t ≤ T , M = 0, in such a way, thatthe corresponding future-supremum process C t = sup t ≤ s ≤ T ( X s − M s ) , ≤ t ≤ T, (2.5)becomes ( F t ) ≤ t ≤ T -adapted?Consider the martingale M t = E ( X T | F t ) − E ( X T | F ), 0 ≤ t ≤ T , M = 0 and let us consider the future-supremum process C t = sup t ≤ s ≤ T ( X s − M s ) = E ( X T | F ) + sup t ≤ s ≤ T (cid:0) X s − E ( X T | F s ) (cid:1) .
6f the initial process ( X t , F t ), 0 ≤ t ≤ T , is a submartingale (in particular,increasing process), then it is evident that X s − E ( X T | F s ) ≤ , ≤ s ≤ T, while at s = T , X T − E ( X T | F T ) = 0, hence C t = E ( X T | F ) is ( F t ) ≤ t ≤ T -adapted.It is a remarkable and intuitively unexpected fact that the followingproposition holds true in a quite general framework. Theorem 2.1.
Let X = ( X t , F t ) ≤ t ≤ T be a real valued stochastic processhaving right-continuous trajectories X t ( ω ) , ≤ t ≤ T , with left-hand limits,satisfying condition (2.1) . Then there exists one and only one martingale ( M t , F t ) ≤ t ≤ T with M = 0 , such that the future-supremum process, C t = sup t ≤ s ≤ T ( X s − M s ) , ≤ t ≤ T, is ( F t ) ≤ t ≤ T -predictable . (2.6)We will show at first that the right-continuity of the process ( X − M )implies the right-continuity of the nonincreasing process C . Lemma 2.1.
The nonincreasing future–supremum process C = ( C t ) ≤ t ≤ T isright continuous, that is C t + = C t , ≤ t < T. (2.7) Proof.
From the definition of the process C we have C t = max (cid:16) sup t ≤ s ≤ t + δ ( X s − M s ) , C t + δ (cid:17) , ≤ t < t + δ ≤ T. (2.8)We pass to limit δ ↓ C t = max( X t − M t , C t + ) , ≤ t < T. (2.9)If C t > X t − M t , then C t = C t + , hence we should consider the case C t = X t − M t . In this case we have X t − M t = C t ≥ C t + ≥ C t + δ ≥ X t + δ − M t + δ . After passing to limit δ ↓ X t − M t = C t ≥ C t + ≥ X t − M t , thus C t = C t + as required. 7 roof of Theorem 2.1.(a) Existence of the desired martingale. Consider the Snell envelope Y = ( Y t , F t ), 0 ≤ t ≤ T , of the stochasticprocess X = ( X t , F t ), 0 ≤ t ≤ T . It has the Doob–Meyer decomposition(2.3) Y t = M t + B t , ≤ t ≤ T. We have X T − M T ≤ sup t ≤ s ≤ T ( X s − M s ) ≤ sup t ≤ s ≤ T ( Y s − M s ) = sup t ≤ s ≤ T B s = B t , ≤ t ≤ T, (2.10)hence the random variable C t = sup t ≤ s ≤ T ( X s − M s )is integrable for each t , 0 ≤ t ≤ T .From the equality (2.2) we get B t = Y t − M t = ess sup τ t E ( X τ t − M τ t | F t ) ≤ E (cid:16) sup t ≤ s ≤ T ( X s − M s ) | F t (cid:17) = E ( C t | F t ) , ≤ t ≤ T, and, therefore, EB t ≤ EC t , ≤ t ≤ T. (2.11)We have from the inequalities (2.10), (2.11) for each t , 0 ≤ t ≤ T , thecoincidence C t = B t , ( P -a.s.)but both nonincreasing processes C = ( C t ) ≤ t ≤ T and B = ( B t ) ≤ t ≤ T are rightcontinuous, thus they are indistinguishable C t = B t for all t, ≤ t ≤ T ( P -a.s.) (2.12)and as the filtration ( F t ) ≤ t ≤ T satisfies the usual conditions, the future-supremum process C = ( C t ) ≤ t ≤ T turns out to be ( F t ) ≤ t ≤ T -predictable.8t is interesting to note that the Snell envelope Y of the process X canbe written as Y t = M t + sup t ≤ s ≤ T ( X s − M s )= sup t ≤ s ≤ T ( X s − ( M s − M t )) for all t, ≤ t ≤ T (2.13)(see Shashiashvili [5]). (b) The uniqueness of the required martingale. Suppose ( c M t , F t ) ≤ t ≤ T with c M = 0, is a martingale such that the future-supremum process sup t ≤ s ≤ T ( X s − c M s ) is ( F t )-predictable . (2.14)We will show that c M t = M t , 0 ≤ t ≤ T , where ( M t , F t ) ≤ t ≤ T is themartingale part in the Doob–Meyer decomposition (2.3) of the Snell envelope( Y t , F t ) ≤ t ≤ T of the stochastic process ( X t , F t ) ≤ t ≤ T .Denote b C t = sup t ≤ s ≤ T ( X s − c M s ) , b Y t = c M t + b C t , ≤ t ≤ T. (2.15)Then it is evident that as ( b C t , F t ) ≤ t ≤ T is the right continuous nonin-creasing predictable process, the stochastic process ( b Y t , F t ) ≤ t ≤ T is the rightcontinuous (with left-hand limits) supermartingale and (2.15) is its Doob–Meyer decomposition.We have b C t ≥ X t − c M t , b Y t ≥ X t , ≤ t ≤ T, ( P -a.s.) (2.16)thus ( b Y t , F t ) ≤ t ≤ T is a supermartingale which majorizes ( X t , F t ) ≤ t ≤ T .We have from (2.15) b C t − δ = max (cid:16) sup t − δ ≤ s 0, we will obtain b C t − = max( X t − − c M t − , b C t ) , < t ≤ T. (2.18)9et us show now the following key property T Z I ( b Y s − >X s − ) d b C s = 0 . (2.19)Take s such that b Y s − > X s − . Then b C s − = b Y s − − c M s − > X s − − c M s − and fromthe equality (2.18) we get b C s − = b C s , that is, ∆ b C s = 0, otherwise I ( b Y s − >X s − ) · ∆ b C s = 0 . (2.20)Consider the set ( s : b Y s − − X s − > s : b Y s − − X s − > 0) = (cid:16) s : b Y s − − X s − > , ∆( X − c M ) s = 0 (cid:17) ∪ (cid:16) s : b Y s − − X s − > , ∆( X − c M ) s = 0 (cid:17) . (2.21)The first set in (2.21) is countable, hence T Z I ( b Y s − >X s − , ∆( X − c M ) s =0) d b C s = 0 (2.22)by the equality (2.20).Consider now the second set of (2.21) and its arbitrary point s : b Y s − − X s − > , ∆( X − c M ) s = 0 . (2.23)We have b C s − = b Y s − − c M s − > X s − − c M s − , hence from (2.18) and (2.23) we get b C s = b C s − > X s − c M s (2.24)thus there exists ε > 0, such that b C s > X s − c M s + ε. (2.25)As a point s is the continuity point of ( X s − c M s ), then ∃ δ > 0, such that (cid:12)(cid:12)(cid:12) ( X s − c M s ) − sup s − δ ≤ u ≤ s + δ ( X u − c M u ) (cid:12)(cid:12)(cid:12) < ε . (2.26)10herefore, we can write the inequality b C s > sup s − δ ≤ u ≤ s + δ ( X u − c M u ) + ε . At the same time from the definition (2.15) we have b C s − δ = max (cid:16) sup s − δ ≤ u 1. Moreover,(1) for all ( t, S ) ∈ [0 , T ] × (0 , ∞ ) n we have f ( t, S ) = v ( t, S ) , (3.3)where v ( t, S ) (see (1.9)) is the value function of the American optionwith the payoff ψ ( S ),(2) the function f ( t, S ) admits spacial gradient ∇ f ( t, S ) = (cid:16) ∂f ( t, S ) ∂S , . . . , ∂f ( t, S ) ∂S n (cid:17) in the classical sense and ∇ f ( t, S ) ∈ C ([0 , T ) × (0 , ∞ ) n ) ∩ L ∞ ([0 , T ) × (0 , ∞ ) n ) . (3.4)Let us write the Itˆo formula for the discounted function e − rt f ( t, S ) andthe n -dimensional stock prices diffusion process S t = ( S t , . . . , S nt ), 0 ≤ t ≤ Te − rt f ( t, S t ) = f (0 , S ) + t Z e − ru · Lf ( u, S u ) du + t Z e z − ru z ∇ f ( u, S u ) σ ( u, S u ) dW u , ≤ t ≤ T, (3.5) e − ru f ( u, S u ) − e − rt f ( t, S t ) = u Z t e − rθ · Lf ( θ, S θ ) dθ + u Z t e − rθ z ∇ f ( θ, S θ ) σ ( θ, S θ ) dW θ , t ≤ u ≤ T, (3.6)13here z ∇ f ( θ, S θ ) = (cid:16) S θ ∂f∂S ( θ, S θ ) , . . . , S nθ ∂f∂S n ( θ, S θ ) (cid:17) , ≤ θ ≤ T. Denote M f = ( M ft , F t ) ≤ t ≤ T the martingale part in the It¨o formula (3.6) M ft = t Z e − rθ z ∇ f ( θ, S θ ) σ ( θ, S θ ) dW θ , ≤ t ≤ T. (3.7)Consider the American option discounted payoff process X = ( X t , F t ) ≤ t ≤ T , where X t = e − rt ψ ( S t ) , (3.8)and its Snell envelope Y = ( Y t , F t ) ≤ t ≤ T . (3.9) Proposition 3.1. The Snell envelope Y of the American option discountedpayoff process X is given in the following form Y t = e − rt f ( t, S t ) = e − rt v ( t, S t ) , ≤ t ≤ T. (3.10) Proof. We have from (3.1), that f ( t, S ) ≥ ψ ( S ) and Lf ( t, S ) ≤ , Lf ( t, S ) I ( f ( t,S ) >ψ ( S )) = 0 a.e. dt × dS. (3.11)As ∇ f ( t, S ) is a bounded function (see (3.4)) together with the componentsof the volatility matrix σ ( t, S ), it is easy to see, that the local martin-gale (3.7) is actually square integrable martingale and hence the stochasticprocess ( e − rt f ( t, S t ) , F t ) ≤ t ≤ T is a supermartingale dominating the process( e − rt ψ ( S t ) , F t ) ≤ t ≤ T . Consider the stopping times τ t ( ω ) = inf n u ≥ t : f ( u, S u ( ω )) = ψ ( S u ( ω )) o ∧ T, ≤ t ≤ T, (3.12)and write the equality (3.6) at stopping time τ t ( ω ) e − rτ t f ( τ t , S τ t ) − e − rt f ( t, S t )= τ t Z t e − rθ · Lf ( θ, S θ ) dθ + ( M fτ t − M ft ) , ≤ t ≤ T. (3.13)14sing the definition of the stopping time τ t and taking the conditionalexpectation in (3.13) with respect to F t , we get E (cid:16) e − rτ t ψ ( S τ t ) | F t (cid:17) − e − rt f ( t, S t )= E (cid:18) τ t Z t e − rθ · Lf ( θ, S θ ) dθ | F t (cid:19) , ≤ t ≤ T. (3.14)We have (cid:12)(cid:12)(cid:12)(cid:12) τ t Z t e − rθ · Lf ( θ, S θ ) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ T Z t I ( θ<τ t ) | Lf ( θ, S θ ) | dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ T Z I ( f ( θ,S θ ) >ψ ( S )) | Lf ( θ, S θ ) | dθ. (3.15)Let us calculate the expectation of the latter integral E T Z I ( f ( θ,S θ ) >ψ ( S )) | Lf ( θ, S θ ) | dθ = T Z Z (0 , ∞ ) n I ( f ( θ,S ) >ψ ( S )) | Lf ( θ, S ) | p (0 , S ; θ, S ) dS dθ = 0 (3.16)according to the property (3.11), where p (0 , S ; θ, S ) is the probability densityof the random variable S θ = ( S θ , . . . , S nθ ), 0 < θ < T .Thus we conclude that T Z I ( f ( θ,S θ ) >ψ ( S θ )) | Lf ( θ, S θ ) | dθ = 0 ( P -a.s.) (3.17)and hence from the relations (3.14), (3.15) we get e − rt f ( t, S t ) = E (cid:16) e − rτ t ψ ( S τ t ) | F t (cid:17) ≤ Y t , ≤ t ≤ T, (3.18)15here ( Y t , F t ) ≤ t ≤ T is the Snell envelope, that is, the smallest supermartingaledominating the process ( e − rt ψ ( S t ) , F t ) ≤ t ≤ T .Ultimately we have Y t = e − rt f ( t, S t ), 0 ≤ t ≤ T , hence we have shownthe assertion (3.10) of the Proposition 3.1.Thus we get the following representation of the Snell envelope Y t = e − rt v ( t, S t ), 0 ≤ t ≤ T , of the discounted payoff process X t = e − rt ψ ( S t ),0 ≤ t ≤ T , Y t = e − rt v ( t, S t ) = v (0 , S ) + t Z e − ru · Lv ( u, S u ) du + M vt , ≤ t ≤ T, (3.19)where M vt = t Z e − rθ · z ∇ v ( θ, S θ ) σ ( θ, S θ ) dW θ , ≤ t ≤ T. (3.20) Now, we are ready to prove Theorem 1.1. Consider the Doob–Meyer decomposition (2.3) of the Snell envelope( Y t , F t ) ≤ t ≤ T Y t = M vt + B vt , B vt = v (0 , S ) + t Z e − ru · Lv ( u, S u ) du, ≤ t ≤ T. (3.21)We know from (2.12) that B vt = sup t ≤ u ≤ T ( X u − M vu ) = sup t ≤ u ≤ T (cid:0) e − ru ψ ( S u ) − M vu (cid:1) (3.22)and hence Y t = sup t ≤ u ≤ T (cid:16) e − ru ψ ( S u ) − ( M vu − M vt ) (cid:17) , ≤ t ≤ T (3.23)after multiplying the latter equality by e rt we obtain v ( t, S t ) = sup t ≤ u ≤ T (cid:18) e − r ( u − t ) ψ ( S u ) − u Z t e − r ( θ − t ) z ∇ v ( θ, S θ ) σ ( θ, S θ ) dW θ (cid:19) , (3.24)0 ≤ t ≤ T, f, t, S ) , ϕ ( t, S )) satisfies the stochastic balanceequation (1.11). Multiplying this equation by e − rt we get e − rt f ( t, S t ) = sup t ≤ u ≤ T (cid:2) e − ru ψ ( S u ) − ( M u − M t ) (cid:3) = M t + B t , (3.25)where M t = t Z e − rθ z ϕ ( θ, S θ ) σ ( θ, S θ ) dW θ , ≤ t ≤ T,B t = sup t ≤ u ≤ T (cid:0) e − ru ψ ( S u ) − M u (cid:1) , ≤ t ≤ T. (3.26)We see from the equality (3.25) that the future-supremum process B t , 0 ≤ t ≤ T , is ( F t ) ≤ t ≤ T -predictable and hence we can apply the uniqueness state-ment of Theorem 2.1, which asserts that the martingale ( M t , F t ) ≤ t ≤ T (3.26)coincides with the martingale part in the Doob–Meyer decomposition of theSnell envelope e − rt v ( t, S t ), 0 ≤ t ≤ T , that is M t = M vt , ≤ t ≤ T, otherwise t Z e − rθ z ϕ ( θ, S θ ) σ ( θ, S θ ) dW θ = t Z e − rθ z ∇ v ( θ, S θ ) σ ( θ, S θ ) dW θ , ≤ t ≤ T. (3.27)Comparing (3.23) and (3.25) we conclude f ( t, S t ) = v ( t, S t ) , ≤ t ≤ T, (3.28)but the functions f ( t, S ) and v ( t, S ) are continuous and the random variable S t , t > , ∞ ) n (see Bogach¨ev, R¨ekner,Shaposhnikov [7, Theorem 3.2]), hence we get the coincidence f ( t, S ) = v ( t, S ) , ( t, S ) ∈ [0 , T ] × [0 , ∞ ) n . (3.29)We have from (3.27) E (cid:18) T Z e − ru z ( ϕ −∇ v ) ( u, S u ) σ ( u, S u ) dW u (cid:19) = 0 , (3.30)17hat is E T Z e − ru (cid:18) n X i,j =1 a ij ( u, S u ) z ( ϕ −∇ v ) i z ( ϕ −∇ v ) j (cid:19) du = 0 , (3.31)where a ( u, S ) = σ ( u, S ) σ ⊤ ( u, S ) . We recall now that the symmetric matrix a ( u, S ) is uniformly positive definite(see condition (1.3)) and hence we get from the latter equality T Z E (cid:12)(cid:12) z ( ϕ −∇ u ) ( u, S u ) (cid:12)(cid:12) du = 0 , (3.32)which can be written in the following form T Z Z (0 , ∞ ) n (cid:12)(cid:12) z ( ϕ −∇ u ) ( u, S ) (cid:12)(cid:12) p (0 , S ; u, S ) dS du = 0 , (3.33)where the transition probability density p (0 , S ; u, S ) is strictly positive (seeBogach¨ev, R¨ekner, Shaposhnikov [7, Theorem 3.2]) in (0 , T ) × (0 , ∞ ) n . There-fore we conclude z ( ϕ −∇ v ) ( u, S ) = 0 a.e. du × dS, (3.34)which gives us the uniqueness assertion of Theorem 1.1: f ( t, S ) = v ( t, S ) , ( t, S ) ∈ [0 , T ] × [0 , ∞ ) n ,ϕ ( t, S ) = ∇ v ( t, S ) , a.e. dt × dS in (0 , T ) × (0 , ∞ ) n . (3.35) References [1] A. Bensoussan, On the theory of option pricing. Acta Appl. Math . (1984), no. 2, 139–158.[2] I. Karatzas, On the pricing of American options. Appl. Math. Optim. (1988), no. 1, 37–60.[3] A. Pascucci, PDE and Martingale Methods in Option Pricing . Bocconi& Springer Series, 2. Springer, Milan; Bocconi University Press, Milan,2011. 184] M. H. A. Davis and I. Karatzas, A deterministic approach to optimalstopping. Probability, statistics and optimisation , 455–466, Wiley Ser.Probab. Math. Statist. Probab. Math. Statist., Wiley, Chichester, 1994.[5] M. Shashiashvili, Representation of the Snell envelope as the futuresupremum process. Seminar of I. Vekua Inst. Reports (2020), 37–40.[6] M. E. Thompson, Continuous parameter optimal stopping problems. Z.Wahrscheinlichkeitstheorie und Verw. Gebiete (1971), 302–318.[7] V. I. Bogach¨ev, M. R¨ekner and S. V. Shaposhnikov, Positive densities oftransition probabilities of diffusion processes. (Russian) Teor. Veroyatn.Primen. (2008), no. 2, 213–239; translation in Theory Probab. Appl.53 , ∆( X − c M ) s = 0 (cid:17) ⊆ [ r d ( − b C s ) | F t (cid:19) = 0 , hence b Y t = E ( b Y τ εt | F t ) ≤ E ( X τ εt | F t ) + ε ≤ Y t + ε. (2.31)Thus we get b Y t ≤ Y t , but ( Y t , F t ) ≤ t ≤ T is the smallest supermartingale thatmajorizes ( X t , F t ) ≤ t ≤ T , therefore we have the equality b Y t = Y t , ≤ t ≤ T, and from the uniqueness of the Doob–Meyer decomposition we obtain c M t = M t , ≤ t ≤ T, that is, we have shown the uniqueness of the required martingale( c M t , F t ) ≤ t ≤ T . We shall be based on the obstacle problem for parabolic operators consid-ered in Pascucci [3] in Sections 8.2, 9.4 and 11.3. Parabolic Sobolev spaces S ploc ((0 , T ) × (0 , ∞ ) n ) are introduced for any p ≥ f ( t, S ) belonging to thespace S ploc ((0 , T ) × (0 , ∞ ) n ) for the exponents p > n +22 .The obstacle problem is formulated in the following manner:find a function f ( t, S ), which belongs to the space S loc ((0 , T ) × (0 , ∞ ) n ) ∩ C ([0 , T ] × [0 , ∞ ) n ) and satisfies the equationmax n Lf ( t, S ) , ψ ( S ) − f ( t, S ) o = 0 a.e. dt × dS in (0 , T ) × (0 , ∞ ) n (3.1)with the terminal condition f ( T, s ) = ψ ( S ) . (3.2)12ere Lf ( t, S ) is the second order parabolic differential operator (1.8) andsuch a function f ( t, S ) is called a strong solution of the obstacle problem.The basic result Theorem 11.13 about the obstacle problem in Pascucci [3]asserts the existence and the uniqueness of the strong solution of the obstacleproblem (3.1), (3.2) belonging to the parabolic Sobolev space S ploc ((0 , T ) × (0 , ∞ ) n ) for any p ≥