A Stochastic Model of Optimal Debt Management and Bankruptcy
|
|
- Douglas McLaughlin
- 6 years ago
- Views:
Transcription
1 A Stochastic Model of Otimal Debt Management and Bankrutcy Alberto Bressan (, Antonio Marigonda (, Khai T. Nguyen (, and Michele Palladino ( (* Deartment of Mathematics, Penn State University University Park, PA 1680, USA. (** Diartimento di Informatica, Università di Verona, Italy (*** Deartment of Mathematics, North Carolina State University, USA Raleigh, NC 7695, USA. s: bressan@math.su.edu, antonio.marigonda@univr.it, khai@math.ncsu.edu, mu6@su.edu May, 017 Abstract A roblem of otimal debt management is modeled as a noncooerative interaction between a borrower and a ool of lenders, in infinite time horizon with exonential discount. The yearly income of the borrower is governed by a stochastic rocess. When the debt-to-income ratio x(t reaches a given size x, bankrutcy instantly occurs. The interest rate charged by the risk-neutral lenders is recisely determined in order to comensate for this ossible loss of their investment. For a given bankrutcy threshold x, existence and roerties of otimal feedback strategies for the borrower are studied, in a stochastic framework as well as in a limit deterministic setting. The aer also analyzes how the exected total cost to the borrower changes, deending on different values of x. 1 Introduction We consider a roblem of otimal debt management in infinite time horizon, modeled as a noncooerative interaction between a borrower and a ool of risk-neutral lenders. Since the debtor may go bankrut, lenders charge a higher interest rate to offset the ossible loss of art of their investment. 1
2 In the models studied in [8, 9], the borrower has a fixed income, but large values of the debt determine a bankrutcy risk. Namely, if at a given time t the debt-to-income ratio x(t is too big, there is a ositive robability that anic sreads among investors and bankrutcy occurs within a short time interval [t, t + ε]. This event is similar to a bank run. Calling T b the random bankrutcy time, this means { } Prob T b [t, t + ε] T b > t = ρ(x(t ε + o(ε. Here the instantaneous bankrutcy risk ρ( is a given, nondecreasing function. At all times t, the borrower must allocate a ortion u(t [0, 1] of his income to service the debt, i.e., aying back the rincial together with the running interest. Our analysis will be mainly focused on the existence and roerties of an otimal reayment strategy u = u (x in feedback form. In the alternative model roosed by Nuño and Thomas in [16], the yearly income Y (t is modeled as a stochastic rocess: dy (t = µy (t dt + σy (t dw. (1.1 Here µ 0 is an exonential growth rate, while W denotes Brownian motion on a filtered robability sace. Differently from [8, 9], in [16] it is the borrower himself that chooses when to declare bankrutcy. This decision will be taken when the debt-to-income ratio reaches a certain threshold x, beyond which the burden of servicing the debt becomes worse than the cost of bankrutcy. At the time T b when bankrutcy occurs, we assume that the borrower ays a fixed rice B, while lenders recover a fraction θ(x(t b [0, 1] of their outstanding caital. Here x θ(x is a nonincreasing function of the debt size. For examle, the borrower may hold an amount R 0 of collateral (gold reserves, real estate... which will be roortionally divided among creditors if bankrutcy occurs. In this case, when bankrutcy occurs each investor will receive a fraction { } R0 θ(x(t b = max x(t b, 1 (1. of his outstanding caital. Aim of the resent aer is to rovide a detailed mathematical analysis of some models closely related to [16]. We stress that these roblems are very different from a standard roblem of otimal control. Indeed, the interest rate charged by lenders is not given a riori. Rather, it is determined by the exected evolution of the debt at all future times. Hence it deends globally on the entire feedback control u(. An otimal solution for the borrower must be understood as the best trade-off between the sustainability of his debt, related to the interest rate charged by the lenders, and the need to kee the reayment rate as small as ossible. A recise descrition of our model is given in Section. Here the strategy of the borrower comrises a feedback control u = u(x, determining the fraction of income allocated to servicing the debt, and a stoing set S IR +, where bankrutcy is declared. In a way, this
3 resembles the classical roblem of stochastic control with stoing time, as in [7]. We remark that, in a naive formulation, the otimization roblem always admits the trivial solution u(x 0, S =. (1.3 This corresonds to a Ponzi scheme: no ayment is ever made, bankrutcy is never declared, and the interest on old loans is ayed by initiating more ad more new loans. This strategy guarantees zero cost, and is clearly otimal. To rule out the trivial solution and achieve a more realistic model, we assume that some uer bound x for the debt is given, beyond which bankrutcy must instantly occur. For examle, one can think of x as the maximum amount of cash that all financial markets can rovide. It can be very large, but certainly finite. In this modified setting, the otimization roblem is formulated for x [0, x ], and the stoing set S [0, x ] must contain the oint x. The main results of the aer can be summarized as follows. Given an uer bound x for the debt, we show that the otimal choice of the stoing set is S = {x }. In other words, it is never convenient for the borrower to declare bankrutcy, unless he is forced to do it. We then seek an otimal feedback control u = u (x, x [0, x ] which minimizes the exected cost to the borrower. For any value σ 0 of the diffusion coefficient in (1.1, we rove that the roblem admits at least one solution, in feedback form. In the deterministic case where σ = 0, the solution can be constructed by concatenating solutions of a system of two ODEs, with terminal data given at x = x. We then study how the exected total cost of servicing the debt together with the bankrutcy cost (exonentially discounted in time deends on the uer bound x. Let θ(x [0, 1] be the salvage rate, i.e., the fraction of outstanding caital that will be ayed back to lenders if bankrutcy occurs when the debt-to-income ratio is x. If lim s + θ(s s = +, (1.4 then, letting x +, the total exected cost to the borrower goes to zero. On the other hand, if θ(s s < +, (1.5 lim s + then the total exected cost to the borrower remains uniformly ositive as x +. Our analysis shows that, if the debtor can access a large amount x of credit, when (1.4 holds he can ostone the bankrutcy time far into the future. Due to the exonential discount, as x + his exected cost will thus aroach zero. On the other hand, when (1.5 holds, after the debt has reached a certain threshold, bankrutcy must occur within a fixed time regardless of the size of x. We remark that the assumtion (1.5 is more realistic. For examle, if (1. holds, then θ(x x = R 0 for all x large enough. The remainder of the aer is organized as follows. In Section we describe more carefully the model, deriving the equations satisfied by the value function V and the discounted bond 3
4 rice. In Sections 3 and 4 we construct equilibrium solutions in feedback form, in the stochastic case (σ > 0 and in the deterministic case (σ = 0, resectively. Finally, Sections 5 and 6 contain an analysis of how the exected cost to the borrower changes, deending on the bankrutcy threshold x. In the economics literature, some related models of debt and bankrutcy can be found in [1, 3, 8, 1, 13]. For the basic theory of otimal control and viscosity solutions of Hamilton- Jacobi equations we refer to [4, 10]. A model with stochastic growth We consider a slight variant of the model in [16]. We denote by X(t the total debt of a borrower (a government, or a rivate comany at time t. The annual income Y (t of the borrower is assumed to be a random rocess, governed by the stochastic evolution equation (1.1. The debt is financed by issuing bonds. When an investor buys a bond of unit nominal value, he receives a continuous stream of ayments with intensity (r + λe λt. Here r is the interest rate ayed on bonds, which we assume coincides with the discount rate, λ is the rate at which the borrower ays back the rincial. If no bankrutcy occurs, the ayoff for an investor will thus be 0 e rt (r + λe λt dt = 1. In case of bankrutcy, a lender recovers only a fraction θ [0, 1] of his outstanding caital. Here θ can deend on the total amount of debt at the time on bankrutcy. To offset this ossible loss, the investor buys a bond with unit nominal value at a discounted rice [0, 1]. As in [9, 16], at any time t the value (t is uniquely determined by the cometition of a ool of risk-neutral lenders. We call U(t the rate of ayments that the borrower chooses to make to his creditors, at time t. If this amount is not enough to cover the running interest and ay back art of the rincial, new bonds are issued, at the discounted rice (t. As in [16], the nominal value of the outstanding debt thus evolves according to Ẋ(t = λx(t + (λ + rx(t U(t. (.1 (t For a more detailed derivation of equation (.1 from the economic rimitives, we refer the reader to [16]. 4
5 The debt-to-income ratio is defined as x = X/Y. In view of (1.1 and (.1, Ito s formula [17, 18] yields the stochastic evolution equation [( λ + r dx(t = (t λ + σ µ x(t u(t ] dt σ x(t dw. (. (t Here u = U/Y is the fraction of the total income allocated to ay for the debt. Throughout the following we assume r > µ. In this model, the borrower has two controls. At each time t he can decide the ortion u(t of the total income which he allocates to reay the debt. Moreover, he can decide at what time T b bankrutcy is declared. Throughout the following, we assume that an uer bound x for the debt is a riori given (as an external constraint, imosed by the size of the markets, and consider strategies in feedback form. These comrise: (i a closed set S [0, x ], with x S, where bankrutcy is declared, and (ii a feedback control determining the reayment rate u = u (x [0, 1] for x [0, x ] \ S. (.3 For a given choice of the stoing set S, the bankrutcy time is thus the random variable T b. = inf { t 0 ; x(t S }. (.4 Given an initial size x 0 of the debt, the goal of the borrower is to minimize the total exected cost, exonentially discounted in time. Namely, minimize: J ( x 0, u, S [ Tb. = E e rt L ( u (x(t ] dt + e rt b B. (.5 0 x(0=x 0 Here B is a large constant, accounting for the bankrutcy cost, while L(u is the instantaneous cost to the borrower for imlementing the control u. To comlete the model, we need an equation determining the discounted bond rice in the evolution equation (.. For every x > 0, let θ(x be the salvage rate, i.e. the fraction of the outstanding caital that can be recovered by lenders, if bankrutcy occurs when the debt has size x. Given an initial debt size is x 0, the exected ayoff to a lender urchasing a couon with unit nominal value is comuted by the right hand side of [ T b ] (x 0 = E (r + λe (r+λt dt + e (r+λt b θ(x(t b. (.6 x(0=x 0 0 Assuming that the bond rice is determined by the cometition of a large ool of risk-neutral lenders, this exected ayoff should coincide with the discounted bond rice (x 0. This motivates (.6 5
6 Notice that the stoing time T b in (.4, and hence (x 0, deends on the initial state x 0, on the stoing set S, and on all values of the feedback control u (. Since the salvage rate θ( is nonincreasing, we have (x = θ(x if x S, (x [θ(x, 1] for all x [0, x ]. (.7 Having described the model, we can introduce the definition of otimal solution, in feedback form. Definition 1 (stochastic otimal feedback solution. In connection with the above model, we say that a set S [0, x ] and a air of functions u = u (x, = (x rovides an otimal solution to the roblem of otimal debt management (. (.6 if (i Given the function (, for every initial value x 0 [0, x ] the feedback control u ( with stoing time T b as in (.4 rovides an otimal solution to the stochastic control roblem (.5, with dynamics (.. (ii Given the feedback control u ( and the set S, for every initial value x 0 the discounted rice (x 0 satisfies (.6, where T b is the stoing time (.4 determined by the dynamics (.. We emhasize that, in our model, if x(t = x then bankrutcy must instantly occur. The following simle observation shows that, for the borrower, it is never convenient to voluntarily declare bankrutcy at any earlier time. Lemma.1. Let S, u (, ( be an otimal solution to the debt management roblem (. (.6. Then S = {x }. Proof. Assume that, on the contrary, there is a value x 0 < x such that x 0 S. We show that condition (i in the above definition cannot hold. Indeed, consider the otimization roblem with initial datum x(0 = x 0. If x 0 S, then bankrutcy instantly occurs at time T b = 0, and the exected cost in (.5 is J[x 0, u, S] = B. However, the alternative strategy u(t 0, with bankrutcy occurring at the first time where x(t = x, rovides the strictly smaller exected cost J ( x 0, 0, {x } [ ] = E e rt b B < B. Motivated by Lemma.1, from now on we shall always take S = {x } as the stoing set. The random stoing time is thus T b. = inf { t 0 ; x(t = x }. (.8 Concerning the cost function L in (.5, we shall assume 6
7 (A The function L is twice continuously differentiable for u [0, 1[ and satisfies L(0 = 0, L > 0, L > 0, lim L(u = +. (.9 u 1 For examle, for some c, α > 0, one may take L(u = c ln 1 1 u, or L(u = cu (1 u. α For a given function = (x, we denote by V ( the value function for the stochastic otimal control roblem (.5 with dynamics (.. Namely, V (x 0. = inf u( J ( x 0, u, {x }. (.10 Denote by H(x, ξ,. = min ω [0,1] {L(ω ξ } ( λ + r ω + λ + σ µ x ξ (.11 the Hamiltonian associated to the dynamics (. and the cost function L in (.5. Notice that, as long as > 0, the function H is differentiable with Lischitz continuous derivatives w.r.t. all arguments. By standard arguments, the value function V rovides a solution to the second order ODE with boundary conditions rv (x = H ( x, V (x, (x + (σx V (x, (.1 V (0 = 0, V (x = B. (.13 As soon as the function V is determined, the otimal feedback control is recovered by { u (x = argmin L(ω V } (x ω [0,1] (x ω. By (A this yields u (x = 0 if ( V (L 1 (x (x if V (x (x V (x (x L (0, > L (0. (.14 On the other hand, if the feedback control u = u (x is known, then by the Feynman-Kac formula ( satisfies the equation [( ] λ + r (r + λ((x 1 = (x λ + σ µ x u (x (x + (σx (x, (.15 (x 7
8 with boundary values (0 = 1, (x = θ(x. (.16 Combining (.1 and (.15, we are thus led to the system of second order ODEs rv (x = H ( x, V (x, (x + (σx V (x, (r + λ((x 1 = H ξ ( x, V (x, (x (x + (σx with the boundary conditions { V (0 = 0, V (x = B, { (0 = 1, (x = θ(x. (x, (.17 (.18 In the next section, an otimal feedback solution to the roblem (. (.6 will be obtained by solving the above system of ODEs for the value function V ( and for the discounted bond rice (. We close this section by collecting some useful roerties of the Hamiltonian function. Lemma.. Let the assumtions (A hold. Then, for all ξ 0 and 0 < 1, the function H in (.11 satisfies ( ( (λ + rx 1 λ + r + (σ λ µx ξ H(x, ξ, λ + σ µ xξ, (.19 (λ + rx 1 + (σ λ µx H ξ (x, ξ, ( λ + r λ + σ µ x. (.0 Moreover, for every x, > 0 the ma ξ H(x, ξ, is concave down and satisfies Proof. H(x, 0, = 0, (.1 ( λ + r H ξ (x, 0, = λ + σ µ x, (. ( 1 λ + r, if > λ + σ µ x, ( (.3 λ + r lim H(x, ξ, = ξ + +, if 1 λ + σ µ x. 1. Since H(x,, is defined as the infimum of a family of affine functions, it is concave down. We observe that (.11 imlies ( λ + r H(x, ξ, = λ + σ µ xξ if 0 ξ L (0. (.4 This yields the identities (.1-(.. 8
9 . Taking ω = 0 in (.11 we obtain the uer bound in (.19. By the concavity roerty, the ma ξ H ξ (x, ξ, is non-increasing. Hence (. yields the uer bound in ( Since L(w 0 for all w [0, 1], we have H(x, ξ, min { ξ } ( λ + r w + w [0,1] λ + σ µ x ξ and obtain the lower bound in (.19. On the other hand, using the otimality condition, one comutes from (.11 that where H ξ (x, ξ, = (λ + rx u (ξ, u (ξ, = argmin ω [0,1] {L(ω ξ ω } + (σ λ µx (.5 = (L 1 ( ξ < 1. Observe that, as ξ +, one has u (ξ, 1 in (.5. The non-increasing roerty of the ma ξ H ξ (x, ξ, yields the lower bound in ( To rove (.3 we observe that, in the first case, there exists ω 0 < 1 such that ( ω 0 λ + r > λ + σ µ x. Hence, letting ξ + we obtain lim H(x, ξ, lim ξ + ξ + [ L(ω 0 ω 0 ξ + ( λ + r ] λ + σ µ x ξ =. To handle the second case, we observe that, for ξ > 0 large, the minimum in (.11 is attained at the unique oint ω(ξ where L (ω(ξ = ξ/. Hence ω(ξ = 1 and lim H(x, ξ, = lim ξ + lim ξ + [ L(ω(ξ ω(ξ ( ] λ + r ξ + ξ + λ + σ µ x ξ lim L(ω(ξ = +. ξ + 9
10 Remark.3. We summarize here the main differences between the roosed model and the model resented in [16]. In [16], the borrower is a government which can control the rimary surlus ratio, the inflation rate and the time of declaring bankrutcy. The control on the inflation rate can be used by the government as a monetary olicy to temorarily deflate the actual debt value, by aying a rice in terms of welfare cost. While controlling the rimary surlus ratio is actually equivalent in our model to the choice of u(, and in both models the borrower can choose the bankrutcy time, in our model the borrower cannot choose the inflation rate r. This simlification can be motivated assuming either that the borrower is not a government, or that the monetary olicy of the government has been delegated to an indeendent central banker which acts in order to kee it constant (e.g. % in Eurozone, no matter of the consequences on the borrower s debt sustainability. In [16], the instantaneous references of the borrower are exressed by a (discounted utility function of logarithmic tye, while our analysis deals with more general cost functions L(. 3 Existence of solutions Let x > 0 be given. If a solution (V, to the boundary value roblem (.17-(.18 is found, then the feedback control u = u (x defined at (.14 and the function = (x will rovide an equilibrium solution to the debt management roblem, as in Definition 1. To construct a solution to the system (.17-(.18, we consider the auxiliary arabolic system V t (t, x = rv (t, x + H ( x, V x (t, x, (t, x + (σx V xx (t, x, t (t, x = (r + λ(1 (t, x + H ξ ( x, Vx (t, x, (t, x x (t, x + (σx with boundary conditions V (t, 0 = 0, V (t, x = B, (t, 0 = 1, (t, x = θ(x. for all t 0. xx (t, x, (3.1 Following [], the main idea is to construct a comact, convex set of functions (V, : [0, x ] [0, B] [θ(x, 1] which is ositively invariant for the arabolic evolution roblem. A toological technique will then yield the existence of a steady state, i.e. a solution to (.17-(.18. Theorem 3.1. In addition to (A, assume that σ > 0 and θ(x > 0. Then the system of second order ODEs (.17 with boundary conditions (.18 admits a C solution (V,, such that V : [0, x ] [0, B] is increasing and : [0, x ] [θ(x, 1] is decreasing. Proof. 10
11 1. For any ε > 0, consider the arabolic system V t = rv + H(x, V x, + (ε + (σx V xx, t = (r+λ(1 +H ξ (x, V x, x + (ε + (σx xx, V (0 = 0, V (x = B, (0 = 1, (x = θ(x. (3. (3.3 obtained from (3.1 by adding the terms εv xx, ε xx on the right hand sides. For any ε > 0, this renders the system uniformly arabolic, also in a neighborhood of x = 0.. Recalling [, Theorem 1], for every initial data V 0, 0 C ([0, x ], the system (3.-(3.3 with initial data V (0, x = V 0 (x, (0, x = 0 (x. (3.4 admits a unique solution V (t, x, (t, x in C ([0, T ] [0, x ] for all T > 0. Adoting a semigrou notation, let t (V (t,, (t, = S t (V 0, 0 be the solution of the system (3.-(3.3 with initial data (3.4. Consider the closed, convex set of functions in C ([0, x ] { } D = (V, : [0, x ] [0, B] [θ(x, 1] ; V, C, V x 0, x 0, and (.18 holds. (3.5 We claim that the above domain is ositively invariant under the semigrou S, namely S t (D D for all t 0. (3.6 Indeed, consider the constant functions V + (t, x = B, V (t, x = 0, + (t, x = 1, (t, x = θ(x. Recalling (.1, one easily checks that V + is a suersolution and V is a subsolution of the scalar arabolic roblem (3.. Indeed rv + + H(x, V x +, + (ε + (σx V xx + 0, V + (t, 0 0, V + (t, x B. rv + H(x, V x, + (ε + (σx V xx 0, V (t, 0 0, V (t, x B. A standard comarison rincile (see for examle Theorem 9.1 in [15] yields 0 = V (t, x V (t, x V + (t, x = B for all (t, x [0, T ] [0, x ]. 11
12 Similarly, since + is a suersolution and is a subsolution of the scalar arabolic roblem (3.3, one has that θ(x = (t, x (t, x + (t, x = 1 for all (t, x [0, T ] [0, x ]. This roves that, if the initial data V 0, 0 in (3.4 take values in the box [0, B] [θ(x, 1], then for every t 0 the solution of the system (3.-(3.3 will satisfy 0 V (t, x B, θ(x (t, x 1, (3.7 for all x [0, x ]. In turn, this imlies V x (t, 0 0, V x (t, x 0, x (t, 0 0, x (t, x 0. ( Next, we rove that the monotonicity roerties of V (t, and (t, are reserved in time. Differentiating w.r.t. x one obtains V xt = rv x + H x + H ξ V xx + H x + σ xv xx + (ε + (σx V xxx, (3.9 ( d xt = (r + λ x + dx H ξ(x, V x, x + H ξ xx + σ x xx + (ε + (σx xxx. (3.10 By (.1, for every x, one has H x (x, 0, = H (x, 0, = 0. Hence V x 0 is a subsolution of (3.9 and x 0 is a suersolution of (3.10. In view of (3.8, we obtain x (t, x 0 V x (t, x for all t 0, x [0, x ]. This concludes the roof that the set D in (3.5 is ositively invariant for the system (3.-( Thanks to the bounds (.19-(.0, we can now aly Theorem 3 in [] and obtain the existence of a steady state (V ε, ε D for the system (3.-(3.3. We recall the main argument in []. For every T > 0 the ma (V 0, 0 S T (V 0, 0 is a comact transformation of the closed convex domain D into itself in C (R. By Schauder s theorem it has a fixed oint. This yields a eriodic solution of the arabolic system (3.-(3.3, with eriod T. Letting T 0, one obtains a steady state. 1
13 5. It now remains to derive a riori estimates on this stationary solution, which will allow to take the limit as ε 0. Consider any solution to rv + H(x, V, + (ε + (σx V = 0, (3.11 (r + λ(1 + H ξ (x, V, + (ε + (σx = 0, with V increasing, decreasing, and satisfying the boundary conditions (.18. By the roerties of H derived in Lemma., we can find δ > 0 small enough and ξ 0 > 0 such that the following imlication holds: x [0, δ], [θ(x, 1], ξ ξ 0 = H(x, ξ, 0. As a consequence, if V (x > ξ 0 for some x [0, δ], then the first equation in (3.11 imlies V (x 0. We conclude that either V (x ξ 0 for all x [0, δ], or else V attains its maximum on the subinterval [δ, x ]. By the intermediate value theorem, there exists a oint ˆx [δ, x ] where V (ˆx = V (x V (δ x δ B x δ. (3.1 By (.19, the derivative V satisfies a differential inequality of the form V c 1 V + c, x [δ, x ]. (3.13 for suitable constants c 1, c. By Gronwall s lemma, from the differential inequality (3.13 and the estimate (3.1 one obtains a uniform bound on V (x, for all x [δ, ˆx] [ˆx, x ]. Relying on the first equation of (3.11, we also obtain an uniform bound on V (x, for all x [δ, x ]. 6. Similar arguments aly to. By (.0, the term H ξ (x, V, in (3.11 is uniformly bounded. For every δ > 0, by (3.11 shows that satisfies a linear ODE whose coefficients remain bounded on [δ, x ], uniformly w.r.t. ε. This yields the bound (x C δ for all x [δ, x ] (3.14 for some constant C δ, uniformly valid as ε 0. Relying on the second equation of (3.11, we also obtain an uniform bound on (x, for all x [δ, x ] To make sure that, as ε 0, the limit satisfies the boundary value (0 = 1. one needs to rovide a lower bound on also in a neighborhood of x = 0, indeendent of ε. Introduce the constant { γ =. ( } 1 λ + r min 1, (r + λ θ(x λ + σ µ. Then define (x. = 1 cx γ, 13
14 choosing c > 0 so that (x = θ(x. We claim that the convex function is a lower solution of the second equation in (3.11. Indeed, by (3.11, one has [ ( ] λ + r (r + λcx γ H ξ (x, V, cγx γ 1 (r + γ θ(x λ + σ µ γ cx γ Letting ε 0, we now consider a sequence (V ε, ε of solutions to (3.11 with boundary conditions (.18. Thanks to the revious estimates, the functions (V ε and ( ε are uniformly bounded by some constant C 1,δ > 0 on [δ, x ], and ε satisty (x ε (x 1 for all x [0, x ], ε > 0. On the other hand, since H and H ξ are uniformly bounded and uniformly Lischitz on [δ, x ] [ C 1,δ, C 1,δ ] [θ(x, 1], the functions (V ε = ε + σ x [rv ε H(x, (V ε, ε ] and ( ε = ε + σ x [(r + λ (ε 1 H ξ (x, (V ε, ε ( ε ] are also uniformly bounded and uniformly Lischitz on [δ, x ]. By choosing a suitable subsequence, we achieve the uniform convergence (V ε, ε (V,, where V, are twice continuously differentiable on the oen interval ]0, x [, and satisfy the boundary conditions (.18. Having constructed a solution (V, to the boundary value roblem (.17-(.18, a standard result in the theory of stochastic otimization imlies that the feedback control u ( in (.14 is otimal for the roblem (.5 with dynamics (.. For a roof of this verification theorem, see Theorem 4.1,.149 in [14] or Theorem 11..,.41 in [17]. As a consequence of Theorem 3.1 we thus obtain Corollary 3.. Under the same assumtions as in Theorem 3.1, the debt management roblem (. (.6 admits an otimal solution. 4 The deterministic case If σ = 0, then the stochastic equation (. reduces to the deterministic control system ( λ + r ẋ = λ µ x u. (4.1 Throughout the aer, we always assume r > µ. The deterministic Debt Management Problem can be formulated as follows. 14
15 (DMP Given an initial value x(0 = x 0 [0, x ] of the debt, minimize Tb 0 e rt L(u(t dt + e rt b B, (4. subject to the dynamics (4.1, where the bankrutcy time T b is defined as in (.4, while (t = Tb t (r+λe (r+λs ds+e (r+λ(t b t θ(x = 1 (1 θ(x e (r+λ(t b t. (4.3 Since in this case the otimal feedback control u and the corresonding functions V, may not be smooth, a concet of equilibrium solution should be more carefully defined. Definition (deterministic otimal feedback solution. A coule of iecewise Lischitz continuous functions u = u (x and = (x rovide an equilibrium solution to the debt management roblem (DMP, with continuous value function V, if (i For every x 0 [0, x ], V is the minimum cost for the otimal control roblem subject to ẋ(t = minimize: Tb 0 e rt L(u(x(t dt + e rt b B, (4.4 ( λ + r (x(t λ µ x(t u(t (x(t, x(0 = x 0. (4.5 Moreover, every Carathéodory solution of (4.5 with u(t = u (x(t is otimal. (ii For every x 0 [0, x ], there exists at least one solution t x(t of the Cauchy roblem ( λ + r ẋ = (x λ µ x u (x (x, x(0 = x 0, (4.6 such that (x 0 = Tb with T b as in (.4. 0 (r + λe (r+λt dt + e ( r+λt b θ(x = 1 (1 θ(x e (r+λt b, (4.7 In the deterministic case, (.17 takes the form rv (x = H ( x, V (x, (x, (r + λ((x 1 ( = H ξ x, V (x, (x (x, (4.8 with Hamiltonian function (see Figure 1 H(x, ξ, = min {L(ω ξ } ( λ + r ω + ω [0,1] (λ + µ x ξ. (4.9 15
16 We consider solutions to (4.8 with the boundary condition V (0 = 0, (0 = 1, V (x = B, (x = θ(x. (4.10 Let s introduce two functions H max (x,. = su ξ 0 H(x, ξ, and ξ (x,. = argmax H(x, ξ,. ξ 0 Recalling (.5, we have H ξ (x, ξ, = (λ + rx u (x, (λ + µx (4.11 where Two cases may occur: u (ξ, = argmin {L(w ξ } w w [0,1] = (L 1 ( ξ < 1. (4.1 If (λ + rx (λ + µx 1 then the function ξ H(x, ξ, is monotone increasing and H max (x, = lim H(x, ξ, = +. (4.13 ξ In this case, we will define ξ (x,. = +. If (λ + rx (λ + µx < 1, we define u (x, = (λ + rx (λ + µx. (4.14 From (4.11 and (4.1, we have ξ (x, = L (u (x, = L ((λ + rx (λ + µx (4.15 and it yields H max (x, = H ( x, ξ (x,, ( = L (λ + rx (λ + µx. Notice that u is the control that kees the debt x constant in time. achieves the minimum in (4.9 when L ((λ + rx (λ + µx = ξ. This value u 16
17 H max (x, rv O F ξ F + ξ Figure 1: In the case where (λ + rx (λ + µx < 1, the Hamiltonian function ξ H(x, ξ, has a global maximum H max (x,. For rv H max, the values F (x, V, ξ (x, F + (x, V, are well defined. Observe that H ξξ (x, ξ, 0, H ξ (x, ξ, > 0 for all 0 ξ < ξ (x,, H ξ (x, ξ, < 0 for all ξ > ξ (x,. (4.16 We regard the first equation in (4.8 as an imlicit ODE for the function V. For every x 0 and [0, 1], if rv (x > H max (x,, then this equation has no solution. On the other hand, when 0 rv (x H max (x,, the imlicit ODE (4.8 can equivalently be written as a differential inclusion : { } V (x F (x, V,, F + (x, V,. (4.17 where F ±(x,v, are denoted by F (x, V, ξ (x, F + (x, V, and H(x, F ± (x, V,, = rv. Remark 4.1. Recalling (4.1, we observe that The value V = F + (x, V, ξ (x, corresonds to the choice of an otimal control such that ẋ < 0. The value V = F (x, V, ξ (x, corresonds to the choice of an otimal control such that ẋ > 0. When rv = H max (x,, then the value V = F + (x, V, = F (x, V, = ξ (x, corresonds to the unique control such that ẋ = 0. Since ξ H(x, ξ, is concave down, the functions F ± satisfy the following monotonicity roerties (Fig. 1 17
18 (MP For any fixed x,, the ma V F + (x, V, is decreasing, while V F (x, V, is increasing. For V = F, the second ODE in (4.8 can be written as (x = G ( x, V (x, (x, where G (x, V,. = (r + λ( 1 H ξ ( x, F (x, V,, 0. ( Construction of a solution. Consider the function W (x. = 1 r L( (r µx, (4.19 with the understanding that W (x = + if (r µx 1. Notice that W (x is the total cost of keeing the debt constantly equal to x (in which case there would be no bankrutcy and hence 1. Moreover, denote by (V B (, B ( the solution to the system of ODEs V (x = F (x, V (x, (x, (x = G (x, V (x, (x, (4.0 with terminal conditions V (x = B, (x = θ(x. (4.1 Notice that the ODE (4.0 admits a unique local solution around every oint (x 0, 0 with V (x 0, 0 = η 0 rovided that H ξ (x 0, F (x 0, η 0, 0, 0 0, i.e., F (x 0, η 0, 0 < ξ (x 0, 0 or, equivalently, rη 0 < H max (x 0, 0. On the other hand, if V B (x < W (x then H ξ (x, V B (x, B(x > 0. Assume by contradiction that Then we have H ξ (x, V B(x, B (x = 0. V B (x = 1 r Hmax (x, B (x 1 r Hmax (x, 1 = W (x and it yields a contradiction. Thus, (V B, B is uniquely defined on [x 1, x ] where the oint }. x 1 = inf {x [0, x ] ; V B (x < W (x. (4. Call V 1 ( the solution to the backward Cauchy roblem V (x = F (x, V (x, 1, x [0, x 1 ], V (x 1 = W (x 1, (4.3 18
19 B W V B V 1 0 x 1 1 x * r µ Figure : Constructing the equilibrium solution in feedback form. For an initial value of the debt x(0 x 1, the debt increases until it reaches x 1, then it is held at the constant value x 1. If the initial debt is x(0 > x 1, the debt kees increasing until it reaches bankrutcy in finite time. we will show that a feedback equilibrium solution to the debt management roblem is obtained as follows (see Figure. V 1 (x if x [0, x 1 ], V (x = (4.4 V B (x if x [x 1, x ]. 1, if x [0, x 1 ], (x = (4.5 B (x, if x ]x 1, x ]. { argmin L(ω (V } (x u ω [0,1] (x ω, if x x 1, (x = (4.6 (r µx 1, if x = x 1. Theorem 4.. Assume that the cost function L satisfies the assumtions (A, and moreover L((r µx > rb. Then the functions V,, u in (4.4 (4.6 are well defined, and rovide an equilibrium solution to the debt management roblem, in feedback form. Proof. 1. The solution of (4.0-(4.1 satisfies the obvious bounds V 0, 0, V (x B, (x [θ(x, 1]. We begin by roving that the function V B ]x 1, x ]. is well defined and strictly ositive for x 19
20 To rove that V B (x > 0 for all x ]x 1, x ], assume, on the contrary, that V B (y = 0 for some y > x 1 0. From (4.15, it holds ξ (x, C 1 > 0 for all x [y, x ], [θ(x, 1] for some ositive constant C 1. Recalling (4.16, we obtain that H ξ (x, ξ, C for all x [y, x ], [θ(x, 1], ξ [0, C 1 ] for some ositive constant C. Since H(x, 0, = 0, the mean value theorem yields H(x, F (x, V,, C F (x, V, for all x [y, x ], [θ(x, 1] rovided by F (x, V, C 1. The definition of F imlies that there exists a constant δ 1 > 0 small such that F (x, V, r C V, (4.7 for all x [y, x ], [θ(x, 1] and V [0, δ 1 ]. Hence, for any solution of (4.0, V (y = 0 imlies V (x = 0 for all x y, roviding a contradiction. Next, observe that the functions F, G are locally Lischitz continuous as long as 0 V < H max (x,. Moreover, V (x < W (x imlies V (x < W (x = H max (x, 1 H max (x, (x. Therefore, the functions V B, B are well defined on the interval [x 1, x ].. If x 1 = 0 the construction of the functions V,, u is already comleted in ste 1. In the case where x 1 > 0, we claim that the function V 1 is well defined and satisfies 0 < V 1 (x < W (x for 0 < x < x 1. (4.8 Indeed, if V 1 (y = 0 for some y > 0, the Lischitz roerty (4.7 again imlies that V 1 (x = 0 for all x y. This contradicts the terminal condition in (4.3. To comlete the roof of our claim (4.8, it suffices to show that This is true because W (x < F (x, W (x, 1 for all x ]0, x 1 ]. (4.9 W (x = r µ r L ( r µx = r µ r ξ (x, 1 < ξ (x, 1 = F ( x, H max (x, 1, 1 = F (x, W (x, 1. 0
21 3. In the remaining stes, we show that V,, u rovide an equilibrium solution. Namely, they satisfy the roerties (i-(ii in Definition. To rove (i, call V ( the value function for the otimal control roblem (4.4-(4.5. For any initial value, x(0 = x 0, in both cases x 0 [0, x 1 ] and x 0 ]x 1, x ], the feedback control u in (4.6 yields the cost V (x 0. This imlies V (x 0 V (x 0. To rove the converse inequality we need to show that, for any measurable control u : [0, + [ [0, 1], calling t x(t the solution to ( λ + r ẋ = x1 (x λ µ x u(t x1 (x, x(0 = x 0, (4.30 one has where Tb 0 e rt L(u(tdt + e rt b B V (x 0, (4.31 T b = inf { t 0 ; x(t = x } is the bankrutcy time (ossibly with T b = +. For t [0, T b ], consider the absolutely continuous function φ u (t. = t 0 e rs L(u(sds + e rt V (x(t. At any Lebesgue oint t of u(, recalling that (V, solves the system (4.8, we comute d [ ] dt φu (t = e rt L(u(t rv (x(t + (V (x(t ẋ(t (( λ + r = e [L(u(t rt rv (x(t + (V (x(t (x(t λ µ x(t u(t ] (x(t [ { e rt min L(ω (V } (x(t ω + ω [0,1] (x(t ] = e [H rt (x(t, (V (x(t, (x(t rv (x(t Therefore, ( λ + r (x(t λ µ = 0. ] x(t(v (x(t rv (x(t roving (4.31. V (x 0 = φ u (0 lim t T b φu (t = Tb 0 e rt L(u(tdt + e rt b B, 1
22 4. It remains to check (ii. The case x 0 = 0 is trivial. Two main cases will be considered. CASE 1: x 0 ]0, x 1 ]. Then there exists a solution t x(t of (4.6 such that (t = 1 and x(t ]0, x 1 ] for all t > 0. Moreover, In this case, T b = + and (4.7 holds. lim x(t = x 1. t + CASE : x 0 ]x 1, x ]. Then x(t > x 1 for all t [0, T b ]. This imlies From the second equation in (4.8 it follows ẋ(t = H ξ (x(t, V B (x(t, B (x(t. d dt (t = (x(tẋ(t = (r + λ((t 1, d ln(1 (x(t = (r + λ. dt Therefore, for every t [0, T b ] one has Letting t T b we obtain roving (4.7. (x(0 = 1 (1 (x(t e (r+λt. (x 0 = 1 (1 θ(x e (r+λt b, Remark 4.3. In general, however, we cannot rule out the ossibility that a second solution exists. Indeed, if the solution V B, B of (4.0-(4.1 can be rolonged backwards to the entire interval [0, x ], then we could relace (4.4-(4.5 simly by V (x = V B (x, (x = B (x for all x [0, x ]. This would yield a second solution. We claim that no other solutions can exist. This is based on the fact that the grahs of W and V B cannot have any other intersection, in addition to 0 and x 1. Indeed, assume on the contrary that W (x = V B (x for some 0 < x < x 1 (see Figure 3. Since B (x < 1 and W (x V B (x, the inequalities rw (x = H(x, W (x, 1 < H(x, W (x, B (x H(x, V B(x, B (x = rv B (x yield a contradiction. Next, let V, be any equilibrium solution and define Then x. = su { x [0, x ] ; (x = 1 }.
23 B W V B 0 x 1 r µ x 1 x * Figure 3: By the monotonicity roerties of the Hamiltonian function H in (4.9, the grahs of V B and W cannot have a second intersection at a oint x > 0. On ]x, x ] the functions V, rovide a solution to the backward Cauchy roblem (4.0-(4.1. On ]0, x ] the function V rovides the value function for the otimal control roblem minimize: subject to the dynamics (with 1 0 e rt L(u(t dt ẋ = (r µx u, and the state constraint x(t [0, x ] for all t 0. The above imlies V (x = V B (x, if x [x, x ], V (x W (x, if x [0, x ]. Since V must be continuous at the oint x, by the revious analysis this is ossible only if x = 0 or x = x 1. 5 Deendence on the bankrutcy threshold x. In this section we study the behavior of the value function V B when the maximum size x of the debt, at which bankrutcy is declared, becomes very large. From a modeling oint of view, this amount to discuss the ossibility of the otimality of a Ponzi scheme, in which the debt is serviced by initiating more and more new loans. We 3
24 will show that under some natural assumtions on the function θ( exressing the fraction recover by lenders as a function of the debt-to-income ratio at the moment of bankrutcy. For a given x > 0, we denote by V B (, x, B (, x the solution to the system (4.0 with terminal data (4.1. Letting x, we wish to understand whether the value function V B remains ositive, or aroaches zero uniformly on bounded sets. Toward this goal, we introduce the constant M 1. = r µ max Recalling Lemma. for σ = 0, we have H(x, ξ, (r µx 1 Thus, the first equation of (4.8 imlies that rb rv B (x, x In turn, if x > M 1, this yields (r µx 1 B (x, x { 1, rb L (0 }. (5.1 ξ for all x [0, x ], ξ 0. V B(x, x x [0, x ]. V B (x, x B (x, x L (0, for all x [M 1, x ]. Calling u = u (x the otimal feedback control, by (.14 we have u (x = 0, for all x [ M 1, x ]. (5. In this case, the Hamiltonian function takes a simler form, namely H(x, V, = [ (λ + r (λ + µ ] V x, H ξ (x, V, = [ (λ + r (λ + µ ] x. Therefore, the system of ODEs (4.0 can be written as V r = [(λ + r (λ + µ]x V, ( 1 = (λ + r [(λ + r (λ + µ] x. (5.3 The second ODE of in (5.3 is equivalent to d ( (1 (x r µ dx ln (x r+λ = r + λ x. Solving backward the above ODE with the terminal data (x = θ(x, we obtain B (x, x = θ(x x x ( 1 B (x, x r µ r+λ 1 θ(x 4 for all x [ M 1, x ]. (5.4
25 Therefore, B (x, x ( θ(x x x ( θ(x x 1 + x r+λ r µ r+λ r µ for all x [ M 1, x ]. (5.5 Different cases will be considered, deending on the roerties of the function θ(. By obvious modeling considerations, we shall always assume θ(x [0, 1], θ (x x for all x 0. We first study the case where θ has comact suort. Recall that M 1 is the constant in (5.1. Lemma 5.1. Assume that θ(x = 0 for all x M, (5.6 for some constant M M 1. Then, for any x > M, the solution V B (, x, B (, x of (4.0-(4.1 satisfies V B (x, x = B and B (x, x = 0 for all x [ M, x ]. Proof. By (5.4 and (5.6, for every x > M one has B (x, x = 0 for all x [ M, x ]. Inserting this into the first ODE in (5.3, we obtain V B(x, x = 0. In turn, this yields V B (x, x = B for all x [ M, x ]. This means that bankrutcy instantly occurs if the debt reaches M. Next, we now consider that case where θ(x > 0 for all x. Lemma 5.. If x > M 1 and θ(x > 0, then θ(x > 0 for all x [0, [. (5.7 ( V B (x, x B (x, x x = B θ(x x r r µ for all x [ M 1, x ]. (5.8 In articular, for x [ M 1, x ] one has ( ( θ(x x r+λ r r+λ r µ B 1 + x ( V B (x, x x r B θ(x x r µ. (5.9 5
26 Proof. Since B (x, x solves the second equation of (5.3 and B (x, x = θ(x (0, 1, we have that x B (x, x is a strictly decreasing function of x. For a fixed value of x, let χ( : [θ(x, 1[ [0, x ] be the inverse function of B (, x. From (5.3, a direct comutation yields d d V B(χ(, x r = [(λ + r (λ + µ] χ( V B(χ(, x χ (, From (5.10 it follows d d B(χ(, x ( 1 = (λ + r [(λ + r (λ + µ] χ( χ ( = 1. d d ln V B(χ(, x = r λ + r 1 1. (5.10 Solving the above ODE with the terminal data V B (x, x = B, B (x, x = θ(x, we obtain V B (χ(, x = ( r 1 r+λ B, ( θ(x hence V B (x, x = ( 1 B (x, x r r+λ B. 1 θ(x Recalling (5.4, a direct comutation yields (5.8. The uer and lower bounds for V B (x, x in (5.9 now follow from (5.5 and the inequality B (x, x 1. Corollary 5.3. Assume that lim su x + θ(x x = +. (5.1 Then the value function V = V (x, x satisfies lim V (x, x + x = 0 for all x 0. (5.13 Indeed, for x M 1 we have V (x, x = V B (x, x, and (5.13 follows from the second inequality in (5.9. When x < M 1, since the ma x V (x, x is nondecreasing, we have 0 lim x V (x, x lim V (M 1, x = 0. x Corollary 5.4. Assume that R =. lim su x + θ(x x < +. (5.14 Then V B (x, x B Moreover, the followings holds. ( 1 + ( R x r+λ r r+λ r µ for all x > x > M 1. (5.15 6
27 (i If then θ (x θ(x + 1 x 0 and θ (x 0 for all x > 0 (5.16 inf V B(x, x = x >0 lim V B(x, x > 0 for all x M 1. (5.17 x (ii Assume that there exist 0 < δ < 1 such that δ θ (x θ(x + 1 x < 0 (5.18 for all x sufficiently large. Then, for each x > M 1, there exists an otimal value x = x (x such that V B (x, x (x = inf V B(x, x. (5.19 x 0 Proof. It is clear that (5.15 is a consequence of (5.9 and (5.14. We only need to rove (i and (ii. For a fixed x M 1, we consider the functions of the variable x alone: Y (x. = V B (x, x, q(x. = B (x, x. Using (5.8 and (5.4, we obtain Y (x Y (x = ( r q r µ (x [ θ q(x (x θ(x + 1 ], (5.0 x and This imlies q (x q(x [ q (x θ q(x (x θ(x + 1 ] x = θ (x x + θ(x + r µ ( q θ(x x r + λ (x 1 q(x + θ (x. (5.1 1 θ(x = [ r µ r+λ q(x 1 q(x + 1 ] [ θ (x θ(x + 1 ] x r µ ( (r + λ 1 + r µ r+λ q(x 1 q(x θ (x 1 θ(x. (5. If (5.16 holds, then (5.0 and (5. imly Y (x Y (x = q (x q(x [ θ (x θ(x + 1 x ] 0 for all x > x M 1. Hence the function Y is non-increasing. This roves (5.17. To rove (ii, we observe that ( 1 lim su x 1 + r µ r+λ q(x 1 q(x 1 < 0, lim x θ(x = 0. 7
28 Hence (5.18 and (5. imly [ q (x θ q(x (x θ(x 1 ] x > 0, for all x sufficiently large. By (5.0 this yields Y (x Y (x > 0 for all x large enough. Hence there exists some articular value x (x x where the function x Y (x = V B (x, x attains its global minimum. This yields ( Deendence on x in the stochastic case In this section we study how the value function deends on the bankrutcy threshold x, in the stochastic case where σ > 0. Extensions of Corollaries 5.3 and 5.4, will be roved, constructing uer and lower bounds for a solution V (, x, (, x of the system (.17-(.18, in the form where V (x V (x, x V 1 (x, 1 (x (x, x (x, (6.1 (i for any V (, with V x 0, the functions 1 ( and ( are a subsolution and a suersolution of the second equation in (3.1, resectively. (ii for any (, with [0, 1] and x 0, the function V 1 ( and V ( are a suersolution and a subsolution of the first equation in (3.1, resectively. 1. We begin by constructing a suitable air of functions V 1, 1. Let ( 1, Ṽ1 be the solution to the backward Cauchy roblem ( λ + r rṽ1(x = + σ xṽ 1, 1 Ṽ 1 (x = B, (r + λ( 1 1 = This solution satisfies ( λ + r 1 + σ x 1, 1 (x = θ(x. (6. 1 (x = θ(x x x ( σ +λ+r 1 1 (x λ+r, lim 1 θ(x 1(x = 1, (6.3 x 0+ 8
29 Ṽ 1 (x = B ( r 1 1 (x r+λ, 1 θ(x Using (6. and (6.3 one obtains ( x 1 = 1(x 1 (x + σ + r + λ x r + λ 1 1 (x = 1(x x 1 (x + σ + r + λ r + λ 1 θ(x (θ(x x r+λ r+λ+σ lim Ṽ 1 (x = 0. (6.4 x 0+ x σ r+λ+σ 1 (x λ+r λ+r+σ Since 1 is monotone decreasing, it follows that 1(x > 0 for all x ]0, x [. In turn, this yields ( λ + r (r + λ( σ x 1 + σ x 1 1 > 0. Recalling (.0, we have (r + λ(1 1 + H ξ (x, ξ, σ x. 1 > 0 for all ξ 0. (6.5 Next, differentiating both sides of the first ODE in (6., we obtain ( r σ λ + r + (λ + ( r 1 λ + r x Ṽ 1 = + σ xṽ 1 for all x ]0, x [. 1 1 This imlies 1 Recalling (.19 and (6., we obtain Ṽ 1 (x < 0 for all x ]0, x [. rṽ1 + H(x, Ṽ 1, 1 + σ x Ṽ 1 < 0. (6.6 When x 1, the ma H(x, ξ, is monotone decreasing. Defining λ+r [ ] Ṽ ( 1. for x 1 0,, r+λ r+λ V 1 (x = [ ] 1 Ṽ (x for x, r+λ x, we thus have rv 1 (x + H(x, V 1(x, q + σ x V 1 (x 0 for all q 1 (x. (6.7 9
30 . We now construct the functions V,. Defining (x a straightforward comutation yields Set (x = (x x and consider the continuous function. = 1 x For x [0, x [ one has (x = 1 and hence ( θ(x x + r µ, < 0, (x = (x x. x. = θ(x x + r µ, (6.8 (x = min { 1, (x }. (6.9 (r + λ(1 + H ξ (x, ξ, + σ x = 0. On the other hand, for x ]x, x [ and ξ 0, one has (x = (x < 1, and H ξ (x, ξ, (λ + rx 1 (x + (σ λ µx (r µx 1 (x [ ] 1 (r µ θ(x x + r µ = x r µ x > 0. (6.10 Recalling (.0, we get (r + λ(1 + H ξ (x, ξ, + σ x [ (λ + rx 1 (r + λ(1 + [ (λ + rx 1 = (r + λ(1 + (σ λ µx ] (x + σ x ] + (σ λ µx (x + σ x = (r + λ(1 [(λ + r 1 ] x + (σ λ µ + σ In articular, = 1 x (r µ = (r µθ(x x x 1 x < 0. (r + λ(1 + H ξ (x, ξ, + σ x 0 for all x ]0, x [, ξ 0. (
31 Next, define V (x For all x [0, x ], we thus have V (x = 0, and hence. = (1 (x B for all x [0, x ]. (6.1 rv + H(x, V, q + σ x V = H(x, 0, q = 0 for all q ]0, 1]. (6.13 On the other hand, for x ]x, x ] we have V (x = B (x x Recalling (.19, (6.9, (6.10 and (6.8, we estimate > 0 and V (x = B (x x. rv + H(x, V, + σ x ( (λ + rx 1 V rv + + (σ λ µx V + σ x V = B [ ( r r + λ + r 1 ] x + (σ λ µ (x σ = B for all x ]x, x [. Recalling (6.8, one has ( λ 1 x (λ + µ r = B [ λ(1 (x + (r µ (x 1 ] x > 0 (λ + rx > 1 for all x ]x, x ]. Therefore the ma H(x, V (x, is monotone decreasing on [0, 1], for all x ]x, x ]. This imlies rv + H(x, V, q + σ x V 0 for all x ]x, x ], q ]0, (x]. Together with (6.13, we finally obtain rv (x + H(x, V (x, q + σ x V (x 0 for all x ]0, x [, q ]0, (x]. (6.14 Relying on (6.5, (6.6, (6.11 and (6.14, and using the same comarison argument as in the roof of Theorem 3.1 we now rove Theorem 6.1. In addition to (A1, assume that σ > 0 and θ(x > 0. Then the system (.17 with boundary conditions (.18 admits a solution (V (, x, (, x satisfying the bounds (6.1 for all x [0, x ]. Proof. 31
32 1. Recalling D in (3.5, we claim that the domain { } D 0 = (V, D (V (x, (x [V (x, V 1 (x] [ 1 (x, (x], for all x [0, x ] (6.15 is ositively invariant for the semigrou {S t } t 0, generated by the arabolic system (3.- (3.3. Namely: S t (D 0 D 0 for all t 0. Indeed, from the roof of Theorem 3.1, we have We now observe that x (t, x 0 V x (t, x for all t > 0, x ]0, x [. (6.16 (i For any V (, with V x 0, by (6.5 the function (t, x = 1 (x is a subsolution of the second equation in (3.1. Similarly, by (6.11, the function (t, x = (x is a suersolution. (ii For any (, with [0, 1] and x 0, by (6.7 the function V (t, x = V 1 (x is a suersolution of the first equation in (3.1. Similarly, by (6.14, the function V (t, x = V (x is a subsolution. Together, (i-(ii imly the ositive invariance of the domain D 0.. Using the same argument as in ste 4 of the roof of Theorem 3.1, we conclude that the system (.17-(.18 admits a solution (V, P D 0. Corollary 6.. Let the assumtions in Theorem 6.1 hold. If lim su s + θ(s s = +, then, for all x 0, the value function V (, x satisfies lim V (x, x x = 0. (6.17 Proof. Using (4.3, (6.4 and Theorem 6.1, we have the estimate ( r 1 V (x, x 1 (x r+λ ( V 1 (x = B B 1 θ(x for all x 1 r+λ increasing, we then have x θ(x x r r+λ+σ. This imlies that (6.17 holds for all x 1 r+λ. Since x V (x, x is monotone 0 lim x V (x, x lim V x This comletes the roof of (6.17. ( 1 r + λ, x 3 = 0 for all x [ 0, 1 ]. r + λ
33 Corollary 6.3. Let the assumtions in Theorem 6.1 hold. If C 1. = lim su s + θ(s s < +, then ( lim inf V (x, x x B 1 C x where the constants C, M are defined as for all x > M, (6.18 C. = C1 + r µ. λ + µ r and M = C 1 + λ + µ r + 1. λ λ(r µ Proof. This follows from (6.9, (6.1 and Theorem Concluding remarks If the uer bound for the debt size (beyond which bankrutcy instantly occurs is allowed to be x = +, then the equations (4.8 admit the trivial solution V (x = 0, (x = 1, for all x 0. This corresonds to a Ponzi scheme, roducing a debt whose size grows exonentially, without bounds. In ractice, this is not realistic because there is a maximum amount of liquidity that the market can suly. It is interesting to understand what haens when this bankrutcy threshold x is very large. Our analysis has shown that three cases can arise, deending on the fraction θ of outstanding caital that lenders can recover. (i If lim θ(s s = +, then for the borrower it is convenient to have s + x as large as ossible. Indeed, the exected total cost for servicing the debt aroaches zero as x +. (iii If lim θ(s s < + and (5.16 holds then for the borrower it is still convenient to have s + x as large as ossible. However, as x +, the exected total cost for servicing the debt remains uniformly ositive. (iii If lim θ(s s < + and (5.18 holds, then for every initial value x 0 of the debt there s + is a value x (x 0 of the bankrutcy threshold which is otimal for the borrower. Examles corresonding to three cases (i (iii are obtained by taking { θ(s = min 1, R } 0 s α (7.1 with 0 < α < 1, α = 1, or α > 1, resectively. 33
34 In case (iii we observe that, even if the bankrutcy threshold x were not imosed by the external market but could be selected by the borrower in an otimal way, this choice of x could never time consistent. Indeed, assume that at the initial time t = 0 the borrower announces that he will declare bankrutcy when the debt reaches size x. Based on this information, the lenders determine the discounted rice of bonds. However, when the time T b comes when x(t b = x, it is never convenient for the borrower to declare bankrutcy. It is the creditors, or an external authority, that must actually enforce bankrutcy. To see this, assume that at time T b when x(t b = x the borrower announces that he has changed his mind, and will declare bankrutcy only at the later time T b when the debt reaches x(t b = x. If he chooses a control u(t = 0 for t > T b, his discounted cost will be e (T b T br B < B. This new strategy is thus always convenient for the borrower. On the other hand, it can be much worse for the lenders. Indeed, consider an investor having a unit amount of outstanding caital at time T b. If bankrutcy is declared at time T b, he will recover the amount θ(x. However, if bankrutcy is declared at the later time T b, his discounted ayoff will be T b (r + λe (r+λ(t Tb dt + e (r+λ(t b Tb θ(x. T b To areciate the difference, consider the deterministic case, assuming that θ( is the function in (7.1, with α 1, and that x M 1. By the analysis at the beginning of Section 5, we have u (x = 0 for all x [x, x ]. Relacing x with x in (5.4 we obtain that the solution to (5.3 with terminal data satisfies ( 1 B (x, x = θ(x B (x, x 1 θ(x (x = θ(x = R 0 (x α r µ r+λ < θ(x = 1 α θ(x θ(x. If the investors had known in advance that bankrutcy is declared at x = x (rather than at x = x, the bonds would have fetched a smaller rice. In conclusion, if the bankrutcy threshold x is chosen by the debtor, the only equilibrium can be x = +. In this case, the model still allows bankrutcy to occur, when total debt aroaches infinity in finite time. Acknowledgments. The authors would also like to thank the anonymous referees, whose suggestions and comments heled to imrove many sections of the aer. References [1] M. Aguiar and G. Goinath, Defaultable debt, interest rates and the current account. J. International Economics 69 (006,
Supplemental Material: Buyer-Optimal Learning and Monopoly Pricing
Sulemental Material: Buyer-Otimal Learning and Monooly Pricing Anne-Katrin Roesler and Balázs Szentes February 3, 207 The goal of this note is to characterize buyer-otimal outcomes with minimal learning
More informationBrownian Motion, the Gaussian Lévy Process
Brownian Motion, the Gaussian Lévy Process Deconstructing Brownian Motion: My construction of Brownian motion is based on an idea of Lévy s; and in order to exlain Lévy s idea, I will begin with the following
More informationInformation and uncertainty in a queueing system
Information and uncertainty in a queueing system Refael Hassin December 7, 7 Abstract This aer deals with the effect of information and uncertainty on rofits in an unobservable single server queueing system.
More information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationDynamic Stability of the Nash Equilibrium for a Bidding Game
Dynamic Stability of the Nash Equilibrium for a Bidding Game Alberto Bressan and Hongxu Wei Deartment of Mathematics, Penn State University, University Park, Pa 16802, USA e-mails: bressan@mathsuedu, xiaoyitangwei@gmailcom
More informationForward Vertical Integration: The Fixed-Proportion Case Revisited. Abstract
Forward Vertical Integration: The Fixed-roortion Case Revisited Olivier Bonroy GAEL, INRA-ierre Mendès France University Bruno Larue CRÉA, Laval University Abstract Assuming a fixed-roortion downstream
More informationarxiv: v1 [q-fin.pm] 13 Mar 2014
MERTON PORTFOLIO PROBLEM WITH ONE INDIVISIBLE ASSET JAKUB TRYBU LA arxiv:143.3223v1 [q-fin.pm] 13 Mar 214 Abstract. In this paper we consider a modification of the classical Merton portfolio optimization
More informationWorst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
Worst-case evaluation comlexity for unconstrained nonlinear otimization using high-order regularized models E. G. Birgin, J. L. Gardenghi, J. M. Martínez, S. A. Santos and Ph. L. Toint 2 Aril 26 Abstract
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationRisk Neutral Modelling Exercises
Risk Neutral Modelling Exercises Geneviève Gauthier Exercise.. Assume that the rice evolution of a given asset satis es dx t = t X t dt + X t dw t where t = ( + sin (t)) and W = fw t : t g is a (; F; P)
More informationVolumetric Hedging in Electricity Procurement
Volumetric Hedging in Electricity Procurement Yumi Oum Deartment of Industrial Engineering and Oerations Research, University of California, Berkeley, CA, 9472-777 Email: yumioum@berkeley.edu Shmuel Oren
More informationBuyer-Optimal Learning and Monopoly Pricing
Buyer-Otimal Learning and Monooly Pricing Anne-Katrin Roesler and Balázs Szentes January 2, 217 Abstract This aer analyzes a bilateral trade model where the buyer s valuation for the object is uncertain
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationThe Stigler-Luckock model with market makers
Prague, January 7th, 2017. Order book Nowadays, demand and supply is often realized by electronic trading systems storing the information in databases. Traders with access to these databases quote their
More informationSUBORDINATION BY ORTHOGONAL MARTINGALES IN L p, 1 < p Introduction: Orthogonal martingales and the Beurling-Ahlfors transform
SUBORDINATION BY ORTHOGONAL MARTINGALES IN L, 1 < PRABHU JANAKIRAMAN AND ALEXANDER VOLBERG 1. Introduction: Orthogonal martingales and the Beurling-Ahlfors transform We are given two martingales on the
More informationA Multi-Objective Approach to Portfolio Optimization
RoseHulman Undergraduate Mathematics Journal Volume 8 Issue Article 2 A MultiObjective Aroach to Portfolio Otimization Yaoyao Clare Duan Boston College, sweetclare@gmail.com Follow this and additional
More informationA Controlled Optimal Stochastic Production Planning Model
Theoretical Mathematics & Applications, vol.3, no.3, 2013, 107-120 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2013 A Controlled Optimal Stochastic Production Planning Model Godswill U.
More informationAsymmetric Information
Asymmetric Information Econ 235, Sring 2013 1 Wilson [1980] What haens when you have adverse selection? What is an equilibrium? What are we assuming when we define equilibrium in one of the ossible ways?
More informationStatistics and Probability Letters. Variance stabilizing transformations of Poisson, binomial and negative binomial distributions
Statistics and Probability Letters 79 (9) 6 69 Contents lists available at ScienceDirect Statistics and Probability Letters journal homeage: www.elsevier.com/locate/staro Variance stabilizing transformations
More informationLecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.
STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationRegression estimation in continuous time with a view towards pricing Bermudan options
with a view towards pricing Bermudan options Tagung des SFB 649 Ökonomisches Risiko in Motzen 04.-06.06.2009 Financial engineering in times of financial crisis Derivate... süßes Gift für die Spekulanten
More informationIntroduction to Probability Theory and Stochastic Processes for Finance Lecture Notes
Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes Fabio Trojani Department of Economics, University of St. Gallen, Switzerland Correspondence address: Fabio Trojani,
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationAsian Economic and Financial Review A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION. Ben David Nissim.
Asian Economic and Financial Review journal homeage: htt://www.aessweb.com/journals/5 A MODEL FOR ESTIMATING THE DISTRIBUTION OF FUTURE POPULATION Ben David Nissim Deartment of Economics and Management,
More informationPricing Dynamic Solvency Insurance and Investment Fund Protection
Pricing Dynamic Solvency Insurance and Investment Fund Protection Hans U. Gerber and Gérard Pafumi Switzerland Abstract In the first part of the paper the surplus of a company is modelled by a Wiener process.
More informationOn Existence of Equilibria. Bayesian Allocation-Mechanisms
On Existence of Equilibria in Bayesian Allocation Mechanisms Northwestern University April 23, 2014 Bayesian Allocation Mechanisms In allocation mechanisms, agents choose messages. The messages determine
More informationLECTURE NOTES ON MICROECONOMICS
LECTURE NOTES ON MCROECONOMCS ANALYZNG MARKETS WTH BASC CALCULUS William M. Boal Part : Consumers and demand Chater 5: Demand Section 5.: ndividual demand functions Determinants of choice. As noted in
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationMatching Markets and Social Networks
Matching Markets and Social Networks Tilman Klum Emory University Mary Schroeder University of Iowa Setember 0 Abstract We consider a satial two-sided matching market with a network friction, where exchange
More informationA discretionary stopping problem with applications to the optimal timing of investment decisions.
A discretionary stopping problem with applications to the optimal timing of investment decisions. Timothy Johnson Department of Mathematics King s College London The Strand London WC2R 2LS, UK Tuesday,
More informationCapital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows
Caital Budgeting: The Valuation of Unusual, Irregular, or Extraordinary Cash Flows ichael C. Ehrhardt Philli R. Daves Finance Deartment, SC 424 University of Tennessee Knoxville, TN 37996-0540 423-974-1717
More informationSINGLE SAMPLING PLAN FOR VARIABLES UNDER MEASUREMENT ERROR FOR NON-NORMAL DISTRIBUTION
ISSN -58 (Paer) ISSN 5-5 (Online) Vol., No.9, SINGLE SAMPLING PLAN FOR VARIABLES UNDER MEASUREMENT ERROR FOR NON-NORMAL DISTRIBUTION Dr. ketki kulkarni Jayee University of Engineering and Technology Guna
More informationAnalysis of pricing American options on the maximum (minimum) of two risk assets
Interfaces Free Boundaries 4, (00) 7 46 Analysis of pricing American options on the maximum (minimum) of two risk assets LISHANG JIANG Institute of Mathematics, Tongji University, People s Republic of
More informationForecast Horizons for Production Planning with Stochastic Demand
Forecast Horizons for Production Planning with Stochastic Demand Alfredo Garcia and Robert L. Smith Department of Industrial and Operations Engineering Universityof Michigan, Ann Arbor MI 48109 December
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
More informationTHE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION
THE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION SILAS A. IHEDIOHA 1, BRIGHT O. OSU 2 1 Department of Mathematics, Plateau State University, Bokkos, P. M. B. 2012, Jos,
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationNon-Exclusive Competition and the Debt Structure of Small Firms
Non-Exclusive Cometition and the Debt Structure of Small Firms Aril 16, 2012 Claire Célérier 1 Abstract This aer analyzes the equilibrium debt structure of small firms when cometition between lenders is
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationLiquidation of a Large Block of Stock
Liquidation of a Large Block of Stock M. Pemy Q. Zhang G. Yin September 21, 2006 Abstract In the financial engineering literature, stock-selling rules are mainly concerned with liquidation of the security
More informationOn the pricing equations in local / stochastic volatility models
On the pricing equations in local / stochastic volatility models Hao Xing Fields Institute/Boston University joint work with Erhan Bayraktar, University of Michigan Kostas Kardaras, Boston University Probability
More informationPrudence, risk measures and the Optimized Certainty Equivalent: a note
Working Paper Series Department of Economics University of Verona Prudence, risk measures and the Optimized Certainty Equivalent: a note Louis Raymond Eeckhoudt, Elisa Pagani, Emanuela Rosazza Gianin WP
More informationQuantitative Aggregate Effects of Asymmetric Information
Quantitative Aggregate Effects of Asymmetric Information Pablo Kurlat February 2012 In this note I roose a calibration of the model in Kurlat (forthcoming) to try to assess the otential magnitude of the
More informationWorst-case evaluation complexity of regularization methods for smooth unconstrained optimization using Hölder continuous gradients
Worst-case evaluation comlexity of regularization methods for smooth unconstrained otimization using Hölder continuous gradients C Cartis N I M Gould and Ph L Toint 26 June 205 Abstract The worst-case
More informationA TRAJECTORIAL INTERPRETATION OF DOOB S MARTINGALE INEQUALITIES
A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME Abstract. We resent a unified aroach to Doob s L maximal inequalities for 1
More informationA NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION
A NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION JYH-JIUAN LIN 1, CHING-HUI CHANG * AND ROSEMARY JOU 1 Deartment of Statistics Tamkang University 151 Ying-Chuan Road,
More informationSensitivity Analysis on Long-term Cash flows
Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
More information4: SINGLE-PERIOD MARKET MODELS
4: SINGLE-PERIOD MARKET MODELS Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 4: Single-Period Market Models 1 / 87 General Single-Period
More informationLogarithmic derivatives of densities for jump processes
Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July
More informationFunctional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA,
More informationAsymptotic results discrete time martingales and stochastic algorithms
Asymptotic results discrete time martingales and stochastic algorithms Bernard Bercu Bordeaux University, France IFCAM Summer School Bangalore, India, July 2015 Bernard Bercu Asymptotic results for discrete
More informationStability in geometric & functional inequalities
Stability in geometric & functional inequalities A. Figalli The University of Texas at Austin www.ma.utexas.edu/users/figalli/ Alessio Figalli (UT Austin) Stability in geom. & funct. ineq. Krakow, July
More informationMidterm Exam: Tuesday 28 March in class Sample exam problems ( Homework 5 ) available tomorrow at the latest
Plan Martingales 1. Basic Definitions 2. Examles 3. Overview of Results Reading: G&S Section 12.1-12.4 Next Time: More Martingales Midterm Exam: Tuesday 28 March in class Samle exam roblems ( Homework
More informationOPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF FINITE
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 005 Seville, Spain, December 1-15, 005 WeA11.6 OPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF
More informationSharpe Ratios and Alphas in Continuous Time
JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 39, NO. 1, MARCH 2004 COPYRIGHT 2004, SCHOOL OF BUSINESS ADMINISTRATION, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195 Share Ratios and Alhas in Continuous
More informationOptimal Dividend Policy of A Large Insurance Company with Solvency Constraints. Zongxia Liang
Optimal Dividend Policy of A Large Insurance Company with Solvency Constraints Zongxia Liang Department of Mathematical Sciences Tsinghua University, Beijing 100084, China zliang@math.tsinghua.edu.cn Joint
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationCS522 - Exotic and Path-Dependent Options
CS522 - Exotic and Path-Deendent Otions Tibor Jánosi May 5, 2005 0. Other Otion Tyes We have studied extensively Euroean and American uts and calls. The class of otions is much larger, however. A digital
More informationTwin Deficits and Inflation Dynamics in a Mundell-Fleming-Tobin Framework
Twin Deficits and Inflation Dynamics in a Mundell-Fleming-Tobin Framework Peter Flaschel, Bielefeld University, Bielefeld, Germany Gang Gong, Tsinghua University, Beijing, China Christian R. Proaño, IMK
More informationEfficiency in Decentralized Markets with Aggregate Uncertainty
Efficiency in Decentralized Markets with Aggregate Uncertainty Braz Camargo Dino Gerardi Lucas Maestri December 2015 Abstract We study efficiency in decentralized markets with aggregate uncertainty and
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationEconomic Performance, Wealth Distribution and Credit Restrictions under variable investment: The open economy
Economic Performance, Wealth Distribution and Credit Restrictions under variable investment: The oen economy Ronald Fischer U. de Chile Diego Huerta Banco Central de Chile August 21, 2015 Abstract Potential
More informationStochastic Volatility Effects on Defaultable Bonds
Stochastic Volatility Effects on Defaultable Bonds Jean-Pierre Fouque Ronnie Sircar Knut Sølna December 24; revised October 24, 25 Abstract We study the effect of introducing stochastic volatility in the
More informationCash-in-the-market pricing or cash hoarding: how banks choose liquidity
Cash-in-the-market ricing or cash hoarding: how banks choose liquidity Jung-Hyun Ahn Vincent Bignon Régis Breton Antoine Martin February 207 Abstract We develo a model in which financial intermediaries
More informationNon-semimartingales in finance
Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology
More informationAn Introduction to Point Processes. from a. Martingale Point of View
An Introduction to Point Processes from a Martingale Point of View Tomas Björk KTH, 211 Preliminary, incomplete, and probably with lots of typos 2 Contents I The Mathematics of Counting Processes 5 1 Counting
More informationChapter 1: Stochastic Processes
Chater 1: Stochastic Processes 4 What are Stochastic Processes, and how do they fit in? STATS 210 Foundations of Statistics and Probability Tools for understanding randomness (random variables, distributions)
More information1 < = α σ +σ < 0. Using the parameters and h = 1/365 this is N ( ) = If we use h = 1/252, the value would be N ( ) =
Chater 6 Value at Risk Question 6.1 Since the rice of stock A in h years (S h ) is lognormal, 1 < = α σ +σ < 0 ( ) P Sh S0 P h hz σ α σ α = P Z < h = N h. σ σ (1) () Using the arameters and h = 1/365 this
More informationOn the smallest abundant number not divisible by the first k primes
On the smallest abundant number not divisible by the first k rimes Douglas E. Iannucci Abstract We say a ositive integer n is abundant if σ(n) > 2n, where σ(n) denotes the sum of the ositive divisors of
More informationForward Dynamic Utility
Forward Dynamic Utility El Karoui Nicole & M RAD Mohamed UnivParis VI / École Polytechnique,CMAP elkaroui@cmapx.polytechnique.fr with the financial support of the "Fondation du Risque" and the Fédération
More informationHedging under arbitrage
Hedging under arbitrage Johannes Ruf Columbia University, Department of Statistics AnStAp10 August 12, 2010 Motivation Usually, there are several trading strategies at one s disposal to obtain a given
More informationContinuous-time Stochastic Control and Optimization with Financial Applications
Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer Some elements of stochastic analysis 1 1.1 Stochastic processes 1 1.1.1 Filtration and processes 1
More informationQuality Regulation without Regulating Quality
1 Quality Regulation without Regulating Quality Claudia Kriehn, ifo Institute for Economic Research, Germany March 2004 Abstract Against the background that a combination of rice-ca and minimum uality
More informationSTOCHASTIC REPUTATION DYNAMICS UNDER DUOPOLY COMPETITION
STOCHASTIC REPUTATION DYNAMICS UNDER DUOPOLY COMPETITION BINGCHAO HUANGFU Abstract This paper studies a dynamic duopoly model of reputation-building in which reputations are treated as capital stocks that
More informationShort-time-to-expiry expansion for a digital European put option under the CEV model. November 1, 2017
Short-time-to-expiry expansion for a digital European put option under the CEV model November 1, 2017 Abstract In this paper I present a short-time-to-expiry asymptotic series expansion for a digital European
More information4 Martingales in Discrete-Time
4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Preliminary Examination: Macroeconomics Fall, 2009
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Preliminary Examination: Macroeconomics Fall, 2009 Instructions: Read the questions carefully and make sure to show your work. You
More informationPart 1: q Theory and Irreversible Investment
Part 1: q Theory and Irreversible Investment Goal: Endogenize firm characteristics and risk. Value/growth Size Leverage New issues,... This lecture: q theory of investment Irreversible investment and real
More informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationWe connect the mix-flexibility and dual-sourcing literatures by studying unreliable supply chains that produce
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 7, No. 1, Winter 25,. 37 57 issn 1523-4614 eissn 1526-5498 5 71 37 informs doi 1.1287/msom.14.63 25 INFORMS On the Value of Mix Flexibility and Dual Sourcing
More informationMultiname and Multiscale Default Modeling
Multiname and Multiscale Default Modeling Jean-Pierre Fouque University of California Santa Barbara Joint work with R. Sircar (Princeton) and K. Sølna (UC Irvine) Special Semester on Stochastics with Emphasis
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationMartingales. by D. Cox December 2, 2009
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (X t : t T) defined on a
More informationProfessor Huihua NIE, PhD School of Economics, Renmin University of China HOLD-UP, PROPERTY RIGHTS AND REPUTATION
Professor uihua NIE, PhD School of Economics, Renmin University of China E-mail: niehuihua@gmail.com OD-UP, PROPERTY RIGTS AND REPUTATION Abstract: By introducing asymmetric information of investors abilities
More informationC (1,1) (1,2) (2,1) (2,2)
TWO COIN MORRA This game is layed by two layers, R and C. Each layer hides either one or two silver dollars in his/her hand. Simultaneously, each layer guesses how many coins the other layer is holding.
More informationBasic Arbitrage Theory KTH Tomas Björk
Basic Arbitrage Theory KTH 2010 Tomas Björk Tomas Björk, 2010 Contents 1. Mathematics recap. (Ch 10-12) 2. Recap of the martingale approach. (Ch 10-12) 3. Change of numeraire. (Ch 26) Björk,T. Arbitrage
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationNon replication of options
Non replication of options Christos Kountzakis, Ioannis A Polyrakis and Foivos Xanthos June 30, 2008 Abstract In this paper we study the scarcity of replication of options in the two period model of financial
More informationRisk minimizing strategies for tracking a stochastic target
Risk minimizing strategies for tracking a stochastic target Andrzej Palczewski Abstract We consider a stochastic control problem of beating a stochastic benchmark. The problem is considered in an incomplete
More informationOptimizing Portfolios
Optimizing Portfolios An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Investors may wish to adjust the allocation of financial resources including a mixture
More informationU. Carlos III de Madrid CEMFI. Meeting of the BIS Network on Banking and Asset Management Basel, 9 September 2014
Search hfor Yield David Martinez-MieraMiera Rafael Reullo U. Carlos III de Madrid CEMFI Meeting of the BIS Network on Banking and Asset Management Basel, 9 Setember 2014 Motivation (i) Over the ast decade
More informationConvexity Theory for the Term Structure Equation
Convexity Theory for the Term Structure Equation Erik Ekström Joint work with Johan Tysk Department of Mathematics, Uppsala University October 15, 2007, Paris Convexity Theory for the Black-Scholes Equation
More informationAnalysis on Mergers and Acquisitions (M&A) Game Theory of Petroleum Group Corporation
DOI: 10.14355/ijams.2014.0301.03 Analysis on Mergers and Acquisitions (M&A) Game Theory of Petroleum Grou Cororation Minchang Xin 1, Yanbin Sun 2 1,2 Economic and Management Institute, Northeast Petroleum
More information